FSR Forum jaargang 12, editie 5

12e Jaargang September 2010 editie #5

Options DN/SB; het vertrouwen voorbij

Interview Nout Wellink

K(r)anttekening

‘Maar het blijft ingewikkeld’

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p38

FSR International Research Project 2010 India

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Wij begrijpen dat je voor je eigen ďŹ nanciĂŤn ook een goede bestemming wilt.

Startende accountants en belastingadviseurs Als startend accountant of belastingadviseur begin je na je studentenleven aan een verantwoordelijke baan die veel van je vraagt. Bij Berk houden we rekening met die overgang van studeren naar werken en proberen we jouw baan zo goed mogelijk te laten aansluiten op jouw leven.

Wij begrijpen als geen ander dat je naast je werk nog andere ambities hebt en dat je na een inspannende werkdag soms behoefte hebt om je op je eigen manier te ontspannen. Bij Berk gaan hard werken, verantwoordelijkheid en vrijheid

www.werkenbijberk.nl

hand in hand. Net als een breed werkterrein en de mogelijkheid van inhoudelijke verdieping via opleidingen. Spreekt de persoonlijke aanpak van Berk jou aan?

Kijk dan eens op www.werkenbijberk.nl voor meer informatie en een overzicht van onze vacatures.

Maakt werk persoonlijk.

Binnen vier jaar al sparringpartner van de CFO? YEP!

Het Young Executive Programme van TNT. Al snel meedoen op hoog niveau. Gevraagd en ongevraagd advies geven. Dat kan bij TNT. Onze organisatie is continu in beweging, er is dus altijd behoefte aan nieuwe ideeĂŤn en een onverwachte visie. Vaak is dit zelfs cruciaal. Daarom is TNT altijd op zoek naar jong talent. FinanciĂŤle starters die binnen een paar jaar met de CFO aan tafel willen, kiezen nu voor het YEP van TNT. Kijk voor meer informatie op www.werkenbijtnt.nl

fsrforum • jaargang 12 • editie #5

Options

Voorwoord

Waarde lezer, Terwijl ik de twijfelachtige eer heb om voor de laatste keer het voorwoord van het FSR Forum te schrijven is het nieuwe bestuur van de Financiële Studievereniging Rotterdam druk bezig zich in te werken om zo komend jaar goed van start te kunnen gaan.

Want na wederom een succesvolle editie van onder andere de Corporate Finance Competition en het International Research Project zijn wij begonnen met de werving van een nieuw bestuur dat de kans zal krijgen om een jaar lang aan het roer te staan van de mooiste studievereniging van Rotterdam. Bent u ook zo benieuwd wie deze gelukkigen zijn? Lees dan zeker verder, zij stellen zich namelijk in deze editie van het FSR Forum vast aan u voor!

Een twijfelachtige eer omdat al weer de laatste editie van deze jaargang van het FSR Forum voor u ligt. Deze editie heeft het thema ‘opties’ meegekregen en uiteraard kunt u zich ook deze editie weer verdiepen in dit onderwerp aan de hand van verschillende wetenschappelijke artikelen.

Daarnaast gaat de heer Groeneveld in zijn k(r)anttekening van deze editie in op de actualiteiten rondom de bank van Dirk Scheringa en het rapport Scheltema dat hier onlangs over verscheen. De Nederlandsche Bank krijgt het hierin naar zeggen zwaar te verduren. Of dat aan het Frederiksplein in Amsterdam ook zo wordt ervaren kunt u lezen in het interview met Nout Wellink verderop in dit magazine. Tevens geeft hij hierin een visie op zijn toekomstplannen.

Zoals gezegd hebben de laatste activiteiten van dit jaar ook plaatsgevonden. Als eerste vertrok een groep van 20 enthousiaste accounting en finance studenten naar Mumbai. Zij hebben daar veel onderzoek verricht en velen van hen hebben dit kunnen koppelen aan bachelor dan wel master scriptie.

2 • Voowoord

En terwijl deze studenten in India verbleven was een andere groep ambititeuze studenten te vinden bij een andere activiteit van de FSR, namelijk de Corporate Finance Competition. Traditiegetrouw vindt dit evenement plaats in een vijfsterrenhotel en dit jaar was dan ook gekozen voor het landgoed Duin en Kruidberg in Santpoort. De twintig deelnemers hebben kennis kunnen maken met vijf verschillende bedrijven die actief zijn in de corporate finance wereld. Zij hebben door middel van gevarieerde cases kunnen zien wat dit werk precies inhoudt.

Van deze activiteiten wordt uitgebreid verslag gedaan in deze editie van het FSR Forum, dus wanneer u benieuwd bent waar de deelnemers van het IRP precies geweest zijn dan vindt u dat in de activiteitenverslagen in dit magazine. Was u net te laat met inschrijven dit jaar dan vindt u in deze editie een overzicht van de activiteiten die komend jaar georganiseerd zullen worden en hun data. Zorg dat u er dit jaar op tijd bij bent!

Op 2 september a.s. zal het voor het XIe bestuur van de Financiele Studievereniging Rotterdam dan echt over zijn. Dan vindt namelijk de ALV plaats waarin wij – als er geen bezwaren zijn – het spreekwoordelijke stokje overdragen aan het XIIIe bestuur. Graag wil ik hen – en in het bijzonder degene die komend jaar het FSR Forum mag maken; Kim de Vries – alvast veel succes en plezier wensen voor het komende jaar!

Met vriendelijke groet, Karin Knegt.

Commissaris Interne Betrekkingen FSR Bestuur 2009-2010

× Voowoord • 3

fsrforum • jaargang 12 • editie #5

Options

Inhoudsopgave

Rules of Thumb in Real Options Analysis Giuseppe Alesii In this normative paper, we derive payback period (PBP) and internal rate of return (IRR) in the presence of real options. In a Kulatilaka - Trigeorgis General Real Option Pricing Model, we derive the expected value of these two decision rules that corresponds to the expected NPV Bellman dynamic programming maximizing strategy in the presence of the options to wait, to mothball and to abandon. A number of original results are derived for an all equity financed firm. Expected PBP and IRR at time 0 are derived together with their distribution. These new methods are applied to a case study in shipping finance.

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Discrete Space-Time Options Pricing Ilya Gikhman In this tutorial we present a framework of the options pricing. This method is alternative to the benchmark binomial scheme and could not be reduce to it. The method does not use expected present reduction which is a unique low that provides finance experts to price derivatives. Price of the derivative contracts are defined based on payoff of an option contract. In other words, if for a particular scenario payoff is a strictly positive number then option premium should offer the same rate of return as underlying security. On the other hand if for a particular scenario payoff is equal to 0 then option price is equal to 0 for this scenario.

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DN/SB; het vertrouwen voorbij K(r)anttekening drs. Joost G. Groeneveld RA RV Dat kun je toch niet serieus menen/nemen? Volgens mij houdt verantwoordelijkheid in dat bij de verantwoordelijke gehoudenheid bestaat tot het beantwoorden van bepaalde vragen van bepaalde personen of instanties op bepaalde momenten. Serieuze verantwoording berust op systematisch onderzoek. De Commissie Scheltema heeft zulk onderzoek verricht. Daarbij is gebleken dat de verantwoordelijke ernstige fouten heeft gemaakt. De antwoorden maakten dat duidelijk. En dan ontstaat een ragfijn spel. Zadel hem op met een onmogelijke opdracht en dan zal hij wel voor de eer bedanken. Maar dat doet hij niet! Hij belooft beterschap. En hij mag doorgaan. Daarmee nemen we als het ware het vertrouwensverlies voor lief. Want lief is het wèl.

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Colofon FSR FORUM is een vijfmaal per jaar verschijnende uitgave van de Financiële Studievereniging Rotterdam KvK Rotterdam nr.: V 40346422 BTW nr.: NL 805159125 B01 ISSN-nr.: 1389-0913 12e jaargang, nummer 4, Oplage 1200

4 • Inhoudsopgave

Redactie Karin Knegt Redactie Advies Commissie Dr. M. B. J. Schauten Dr. O. W. Steenbeek Prof. Dr. M. A. Van Hoepen RA Drs. R. Van der Wal RA Dr. W. F. C. Verschoor Prof. Dr. F. Hartmann Prof. Dr. G. Mertens

Met medewerking van Drs. J. G. Groeneveld RA RV

Redactieadres Redactie FSR FORUM, Kamer H14-06 Erasmus Universiteit Rotterdam Postbus 1738, 3000 DR Rotterdam Tel. 010-408 1830 Fax. 010-408 9061 E-mail forum@fsr.nu

Verenigingsnieuws Woord van de voorzitter

Adverteerders 57

Berk (www.werkenbijberk.nl) Ernst & Young (www.ey.nl/carriere) Grant Thornton (www.carrierebijgt.nl) KPMG (www.gaaan.nu) Kempen & Co (www.kempen.nl) Mazars (www.mazars.nl) Ministerie van Financien (www.werkenbijhetrijk.nl/financieel) PWC (www.werkenbijpwc.nl) TNT (www.werkenbijtnt.nl)

Activiteitenverslag 58

International Research Project 2010

Activiteitenverslag Corporate Finance Competition (CFC)

62

XIIIe FSR Bestuur

64

FSR Oud Bestuurder Geert van Roon

67

FSR Activiteitenoverzicht

71

FSR Alumnivereniging 72

Options

Bedrijfspresentaties Bedrijfspresentatie Flow Traders

17

Bedrijfspresentatie All Options

55

Bedrijfspresentatie Optiver

69

Abonnement Studenten EUR via lidmaatschap FSR en EFR; kosten € 5,-. Overigen via abonnement op FSR FORUM, inlichtingen te verkrijgen op redactieadres; kosten € 27,50 (inclusief BTW en verzendkosten).

Bank: ABN-AMRO 50.15.61.3311 Bank ABN-AMRO 50.15.61.331

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Fotografie en illustraties Coverillustratie: Roger Peters, Haveka www.istockphoto.com, pagina’s: cover, 4, 5, 7, 10/11, 14, 18, 21, 22/23, 25, 26/27, 29, 30/31, 33, 35, 42, 51, 65, 71

Overnemen of nadrukken van artikelen uit het FSR FORUM is uitsluitend toegestaan na schriftelijke toestemming van de redactie.

Hoewel bij deze uitgave de uiterste zorg is nagestreefd kan voor de aanwezigheid van eventuele druk(fouten) en andere onvolledigheden niet worden ingestaan en aanvaard(en) auteur(s), redacteur(en) en uitgever in dezen geen aansprakelijkheid.

Inhoudsopgave • 5

Mazars is ontstaan uit een fusie tussen Mazars en Paardekooper&Hoffman

ª « :( .U1 %,- 0D< $56 x

Ga verder met Mazars.

fsrforum • jaargang 12 • editie #5

Rules of Thumb in Real Options Analysis

Giuseppe Alesii

Introduction Although real options were proposed more than twenty years ago,1 they are still a side product or an application of financial options models. As the latters, real options literature focuses on project evaluation, i.e. expanded NPV computation, neglecting all the other parameters widely used in capital budgeting practice, namely IRR and PBP.2 In this normative paper, we compute these decision rules in the presence of real options providing practitioners with figures which, most of the times, they are more familiar with. After twenty years real options have been proposed for the first time, one of the factors that prevents their becoming widely used in practice is that they are perceived as a valuation criterion on its own. Instead, in this paper we show how even other capital budgeting parameters, such as those previously mentioned, can be computed in a real options framework. In other words, it is possible to translate the active dynamic management of an industrial investment project even in thumb rules. The model we have devised is simple and it can be easily adapted by practitioners to the various contingencies of the business life. We have derived NPVe, the usual expanded NPV, in a version of the Kulatilaka - Trigeorgis General Real Option Pricing Model (GROPM). Together with this expanded NPV we have derived optimal exercise thresholds of the real options to wait, to mothball, to restart and to abandon for the whole life of the project. Having established a Bellman’s Dynamic Programming (DP) optimal policy, we were able to compute forward the same expanded NPV that was previously derived in the usual backward induction process running an Euler Scheme Monte Carlo simulation. The average of these Monte Carlo experiments corresponds to the expected expanded NPV computed in the original backward induction. This fact assures that both backward and forward computations are modeling the same DP optimal policy.

Using the DP optimally managed time series of cash flows and the corresponding NPVes, a number of original results can be derived. Value at risk of the project at time t = 0, V aRNPVe , and the Cash Flow at Risk for each epoch of the investment project horizon, CFaRCFDP can be easily derived, see (Alesii, 2003). Moreover, investment criteria which are intrinsically path dependent like IRR and PBP at time t = 0 can be computed taking into account the optimal exercise of real options.

This paper is organized as follows. In section 1 the use of thumb rules is briefly examined both from an empirical and a theoretical point of view. As a matter of fact a short review in positive capital budgeting literature is provided in order to show that, after all, NPV and a fortiori real options, are not

very used in practice while IRR and PBP are. Moreover, theoretical justifications for the use of these thumb rules is provided. In section 2 it is described the method adopted to compute forward the same E(NPVe) usually computed backward in a GROPM framework. This same method is used to compute the two thumb rules object of this paper. In section 3 the method previously devised is applied to a shipping finance stylized case study. In section 4, conclusions are drawn and several extensions are proposed. The appendix reports an extensive numerical proof of convergence of expected forward computed NPV to the values obtained in the usual backward induction process.

1 Practice and Theory of Thumb Rules in Capital Budgeting The main justification of this paper is that thumb rules, such as PBP and IRR, are widely used in practice and their use can be upheld from a theoretical point of view. The widespread use of thumb rules in the practice of capital budgeting is described taking some evidence from the positive capital budgeting literature. Instead, theoretical justifications for the use of these non-NPV decision rules have been devised in both a behavioral and a rational framework, respectively in organizational behavior literature and in agency theory and real options analysis. To begin with, the most recent surveys in positive capital budgeting have shown how rules of thumb are still widely used in practice, see for instance (Graham and Harvey, 2001) and (Ryan and Ryan, 2002). Although there is an increasing number of firms that use DCF methods and expecially NPV,3 the use of thumb rules is still important: according to (Graham and Harvey, 2001) page 197, more than 75% of US firms still use IRR as an individual criterion, inasmuch as NPV, and more than 50% use PBP. Those figures are more or less the same as those reported by (Ryan and Ryan, 2002). The use of less sophisticated capital budgeting rules is more widespread among small and medium sized firms with an older CEO (Graham and Harvey, 2001). Although that is true, even for large firms, such as the Fortune 1000, thumb rules are important for small sized capital budgets, see Exhibit 2 in (Ryan and Ryan, 2002). The US evidence is confirmed by a stack of studies in other countries. 4 Another pattern in the practice of thumb rules in capital budgeting that emerges from these studies is that they are used in combination with NPV to explore the many faceted aspects of investment performance (Pike, 1996), or because some Non-NPV investment parameters, such as IRR, are a

»

Rules of Thumb in Real Options Analysis • 7

fsrforum • jaargang 12 • editie #5

more cognitively efficient measure of comparison, (Binder and Chaput, 1996), or simply because they are provided in any spreadsheet package (Pike, 1996). PBP and IRR are often found as one of the most popular combinations, see (Mills and Herbert, 1987) and (Cullinane and Panayides, 2000) page 323. The use of thumb rules in combination with NPV can be explained from two points of view. From an agency theory, organizational behavior point of view, in small firms or in large ones but for small capital budgets, quite often investments are mostly defensive and sometimes they are not even evaluated (Runyon, 1983). In divisionalized companies, managers mostly implement headquarters decisions although they exert a certain influence and interact informally with them. In this bargaining process, there is limited role for the rigorous textbook like NPV only evaluation technique, (Scapens and Sale, 1981). In conclusion, formal capital budgeting is only one of the performance control tools and, sometimes, it is not the most important, (Segelod, 2000). As a consequence, informal bargaining takes place better on a whole string of investment project parameters instead of only one, i.e. NPV. This has been given a theoretical rational justification by (Berkovitch and Israel, 1998) who show how in a bargaining process between headquarters and divisions managers, IRR and PI are useful in curbing empire building because, when selecting between mutually exclusive projects, they tend to bias against large scale projects, expecially when allocation according to these criteria is in conflict with the NPV criterion. Hence, thumb rules can be rational on their own when taking into account agency considerations. For instance, a manager may maximize her utility increasing early cash flows at the expense of ultimate overall profitability of the firm, visibility bias, see (Hirshleifer, 1993). In pursuing this kind of optimization, she may find more convenient to use PBP instead of NPV favoring the former investment projects that produce early cash flows (Narayanan, 1985).

Our model differs from those just mentioned in a number of ways.

On the other hand, leaving aside agency theory and organizational behavior considerations, the use of thumb rules alone or in combination with passive NPV has been shown to produce the same allocation as real options models, see (McDonald, 1998) in (Brennan and Trigeorgis, 1998) for a general view about these hybrid rules. This has been proved for models with time homogeneous cash flows with individual irreversible real options for IRR, see (Dixit, 1992), PBP, see (Boyle and Guthrie, 1997) and PI, see (McDonald, 1998).

Basically, an hybrid rule is stated deriving endogenously to the real option model the level of the threshold for the thumb rule which corresponds to the real option exercise, using it, in the case of PBP, in combination with a passive NPV. Our model differs from those just mentioned in a

8 • Rules of Thumb in Real Options Analysis

number of ways. To begin with, our model is numerical while the previous models are based on elegant but difficult to adapt to reality stochastic algebra. Moreover, since the method chosen is numerical, we were able to apply it to a general model with reversible, switching, and irreversible options. Finally, our aim in this paper has been to provide practitioners with the same string of investment parameters they crave for in the bargaining process to get funds to invest. The crucial difference is that in this model IRR and PBP are the translation in a different metrics of NPV maximizing Bellman Dynamic optimal strategy with the exercise of real options to wait, mothball, restart and abandon. This paper and (Alesii, 2003) strive toward establishing a solid link between real options analysis and capital budgeting methods that are commonly considered as an alternative to the former. As a matter of fact, being perceived as a criterion on its own, real options analysis is still used by very few companies and in very few cases, 25% in (Graham and Harvey, 2001) but much less, 1.6% in (Ryan and Ryan, 2002), 0.5% in (Leliveld and Jeffery, 2003), nil in (Sandahl and Sj¨ogren, 2003), and with a decreasing trend, see (Rigby, 2002). Proposing to the practitioners’ audience the Kulatilaka-Trigeorgis GROPM extended for its risk dimensions and translated in other capital budgeting thumb rules mostly used in practice would probably help real options to take root in corporate culture.

2 The Computation of Thumb Rules in the Presence of Real Options The procedure we have followed is numerical and it can be easily adapted by practitioners to any kind of investment project. The model of (Kulatilaka and Trigeorgis, 1994), (hence after KT), has been used to derive both the expanded NPVs of the investment project at time zero and the real options optimal exercise thresholds throughout the whole life of the project. Then, an Euler Scheme Monte Carlo simulation of the same state variable is performed with the same discretization. On each simulated path observation the exposure mapping equation of the firm is computed taking into account real options exercise thresholds.

Hence, a simulated CF history, which has been managed exercising optimally real options, corresponds to each path of the state variable. On these CFs histories it is possible not only to compute expected expanded net present value and its distribution but also to assess CF variability for each epoch, see (Alesii, 2003).5 Moreover, on the same CF time series it is possible to compute any path dependent investment parameter such as PBP and IRR at time t = 0.

In the remaining part of this section, a minor extension of the KT real option pricing model is proposed motivating its choice among the vast variety of real options models, see section 2.1. Moreover, the scenario construction method is described giving a graphic portrayal of the CF computation according to optimal exercise real options thresholds, see section 2.2. Methods adopted to compute PBP and IRR are reported in section 2.3.

2.1 An Extension of the Kulatilaka Trigeorgis Model KT general real options pricing model (GROPM) has been chosen for a variety of reasons.6 To begin with, it is a general model for pricing simultaneously, or individually, a variety of real options while most of the other models are ad hoc individual options pricing models. Moreover, it is one of the few not based on a pseudo asset approach but on a running present value computation of expanded NPV. For these reasons GROPM accommodates general specifications of the

On each simulated path observation the exposure mapping equation of the ďŹ rm is computed taking into account real options exercise thresholds.

exposure mapping and it enables to assess the variability of CF in each epoch of the investment project.

The version of the KT model we have used here is univariate, with a stochastic state variable speciďŹ ed as an arithmetic Ornstein-Uhlenbeck process, see equation (1), discretized in a grid (Kulatilaka, 1993). The choice of this speciďŹ cation is instrumental to the numerical example we have developed in section 3 where the state variable we have chosen evolves like a mean reverting process.

(1)

d θ t = Ρ ¡ θ â&#x2C6;&#x2019; θt

d t + Ď&#x192;θ d Z

where, Ρ the speed of reversion, e.g. for Ρ = 0 the process becomes a geometric Brownian motion while for 0 < Ρ < 1 the process tends to be mean reverting, negative levels are excluded to avoid mean aversion, one is excluded to avoid overshooting ; θ the normal level of θ, i.e. the level at which θ tends to revert; Ď&#x192;2 Ď&#x192;θ2 instantaneous variance rate; d t time differential; d Z standard Wiener process, normally distributed with

E (d Z) = 0 and V ar (d Z) = E (d Z)2 = d t.

We claim that the derivation of the latter for the whole life of the project is an original contribution of this paper. These thresholds have been constructed simply recording the argmax function during each backward induction recursion, hence in a numerical procedure that can be easily generalized to any number of operating modes.8

F θ t , , t

=

â&#x17D;Ť

θj,t =â&#x2021;&#x2019; â&#x17D;Ş â&#x17D;Ź

j,t â&#x17D;Ş â&#x17D;­ θj,t â&#x2021;?â&#x2021;&#x2019;

=

max (2)

θâ&#x2C6;&#x2014;

Πθt , , t â&#x2C6;&#x2019; c , + Ď Âˇ Et t+1 F θt+1 , , , t + 1

argmax F θ t , , t

(3)

where, :=

F (θt , , t)

θâ&#x2C6;&#x2014;

:=

Et t+1 [ ]

Ď = 1/ 1 + i1/m

:=

value of the plant for the level of the state variable θt, for an optimizing operating mode â&#x201E;&#x201C; at time t; expectation operator on equivalent martingale measure, hence starred, of the process θt; present value factor, in which

The solution of a dynamic optimization problem has two faces: the max argument derived from recursions of the Bell-man equation and the argmax argument or optimal policy.

In essence, GROPM is based on a Bellman Dynamic Programming (hence after DP) method on a ďŹ nite horizon solving an impulse control problem. Controls available to the ďŹ rm are its various operating modes, namely, in this case, being idle before investing, operating, being mothballed, abandoned.

Real options, then, are the capabilites to pass from one operating mode to the other, respectively option to wait, option to mothball, to restart and option to abandon. Because of this, GROPM accommodates several degrees of irreversibility of investment decisions being possible to specify a transition cost for each passage between operating modes.7 The solution of a dynamic optimization problem has two faces: the max argument derived from recursions of the Bellman equation, see expression (2); and the argmax argument or optimal policy , see expression (3).

c ,

:=

Π(θt , , t)

:=

i1/m = (1 + rf )1/m â&#x2C6;&#x2019; 1. operating mode transition cost, being l the beginning mode and lâ&#x20AC;˛ the ending mode; individual period operating cash flow.

(Kulatilaka, 1993) shows that under the restrictive condition of the investment project having a β = 0, lemma 4 of (Cox et al., 1985) is applicable and the drift of the arithmetic Ornstein Uhlenbeck can be considered a certainty equivalent drift rate. Hence, EMM and natural probability measure coincide under this restrictive hypothesis. Without loss of generality we consider an individual risk free rate. However, the model could easily accommodate a whole term structure of interest rates.9

Âť

Rules of Thumb in Real Options Analysis â&#x20AC;˘ 9

fsrforum â&#x20AC;˘ jaargang 12 â&#x20AC;˘ editie #5

2.2 Scenario Construction The optimal exercise thresholds partition the discretized space of the state variable in regions in which different operating modes are optimal according to a Bellman DP procedure, see upper graph in ďŹ gure 1. We run a Monte Carlo simulation of an Euler Scheme approximation of the solution of the SDE in expression (1), see equation (4). We obtain the path of the levels of θt â&#x2C6;&#x20AC;t = 0, . . . , T . 10 This path meanders on the grid going through the thresholds, passing from an hysteresis, θj,t =â&#x2021;&#x2019; j,t in (3), to a one mode region,θj,t =â&#x2021;&#x2019; j,t in (3), and the other way around, see upper graph in ďŹ gure 1.

θt = θtâ&#x2C6;&#x2019;1 ¡ eâ&#x2C6;&#x2019;ΡÎ&#x201D; t + θ ¡ 1 â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;ΡÎ&#x201D; t + t

(4)

where, in addition to the previous notation:

t â&#x2C6;ź N 0,

2 Ď&#x192;θ 2¡Ρ

¡ 1 â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;ΡÎ&#x201D; t

: noise term distributed normally with mean zero and variance as a fraction of Ď&#x192;2 θ .

Figure 1: Path of O and CF computation 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1

For each θt in each path we were able to compute _(θt, l, t) netting these cash ďŹ&#x201A;ows of the transition costs when and if they are due. Therefore, a θt path corresponds to a time series of CFs optimally managed according to Bellman Optimum Principle, see lower graph in ďŹ gure 1 in which also a mode indicator function is reported between the two graphs. In the example represented in ďŹ gure 1, the investment project, e.g. a ship, is implemented at time t = 0 since the θt series starts at a level higher than the investment threshold, (operating mode=2).

The very low CF at the beginning is the result of the lump sum initially invested and the ďŹ rst operating CF. After three epochs the time charter rate goes below the mothballing threshold.

Hence the ship is laid up for four periods (operating mode=3) until the time charter rate reaches the restarting threshold.

Even if in the following epoch it goes below it, the project is kept in operating mode, hysteresis situation. The same kind of situation takes place just before epoch 20 and 30. At period 47 the time charter rate goes below the abandonment threshold and the ship is scrapped for its salvage value (operating mode=4). On this CFt time series it is possible to compute NPV expanded for real options using the same risk free rate used in the backward induction process. Averaging these results across different Montecarlo experiments, expected values converge to those found using expression (2). This has been thoroughly shown in the Appendix. The extension for a passively managed project is trivial implying simply the computation of CFt without taking into account real options exercise thresholds.

0 0

10

20

30

40

20 2 2 2 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

50 4 0 0 0

10

0 41

31

21

11

1

-10

-20

2.3 Forward Computation of IRR and PBP On these cash ďŹ&#x201A;ows it is possible to compute several thumb rules widely used in the practice of capital budgeting. Payback period (PBP) and Internal Rate of Return (IRR) can be computed in a path wise way only going forward on each optimally managed cash ďŹ&#x201A;ow time series. PBP computation is straightforward, see for instance (LeďŹ&#x201A;ey, 1996), (Kruschwitz and LofďŹ&#x201A;er, 1999), (Yard, 2000). As a matter of fact, it implies simply the cumulation of CFt until the initial investment is completely recovered, see expression (5).

(5) int(pb) P B = pb â&#x2021;&#x201D; I =

-30

where: PB

-40 Legend: The upper graph in the ďŹ gure represents the real options optimal exercise thresholds in a time - state variable Cartesian space, time unit is the dynamic system resetting period. To be speciďŹ c, the highest represents the investment trigger threshold, the lowest, in bold, represents the abandonment threshold. In the middle, the higher is the restarting threshold while the lower is the mothballing threshold. An indicative path of Ot is represented on the same graph. In the lower graph in the ďŹ gure the corresponding time series of CFt is represented in bold, together with an indicator function, in bars, representing the operating mode in which CFs at that epoch were generated. 2:= operating mode; 3:= mothballed mode; 4:= abandoned mode.

10 â&#x20AC;˘ Rules of Thumb in Real Options Analysis

int (pb) CFint(pb+1)

t=1

CFt + [pb â&#x2C6;&#x2019; int (pb)] ¡ CFint(pb+1)

:= payback period, expressed in the same frequency units as CFt; := integer number of periods before the last; := cash flow of the last period. Under the hypothesis of equally distributed cash flows within each period, a fraction of the last cash flow covers the initial capital which is still to be recovered at time int (pb).

In CFt histories in which investment does not pay back, i.e. inflows do not recover completely initial investment, we have computed a fractionary recovery ratio to show how much of the initial capital is actually recovered.

IRR computation, instead, can be computationally burdensome.11 A textbook like version of this investment parameter is simply not applicable to the generality of the CFt histories both with and without real options. As a matter of fact, CFt series change sign more than once. We have discarded the truncation theorem solution and we have adopted a generalized version of the internal rate of return according to Teichroew, (Teichroew et al., 1965b), (Teichroew et al., 1965a), (Teichroew, 1964). In essence, IRR according to Teichroew (IRRT) is the internal rate of return which equals the running compound value (RCV) to the last cash flow produced by the investment project, taken with the negative sign, compounding positive RCV with the opportunity cost of capital, in our case rf , and negative ones with IRRT , see expressions (6)-(8).

In other words, IRR according to Teichroew sets to zero the compound value of cash flows produced by an investment at the end of its life. This is equivalent to setting up to zero its net present value. Because of this, the simple textbook like IRR is a particular case of IRRT. Results have been found using a simple grid search combined with a secants algorithm. IRRT does not exist in the cases in which cash flows have all negative signs, or are not “well behaved”, see last column in table 1. The probability of the latter occurrence is very low.

at t = 0: ⎧ ⎪ ⎨ CF0 · (1 + r)

RCV0, 1 =

⎪ ⎩ CF · (1 + IRR ) 0 T

⎧ ⎪ ⎨ >0

for CF0

∀ t = 1, . . . N − 2: ⎧ ⎪ ⎨

RCVt,t+1 =

for [CFt + RCVt−1,t ]

⎪ ⎩ <0

at t = N : RCVN −1,N = −CFN

3 Numerical example in shipping finance In this section we apply to a stylized case study in shipping finance the methods previously devised for computing PBP and IRR in the presence of real options. The choice of the shipping industry has been motivated both on positive and normative capital budgeting grounds. A short comparison with other case studies in recent literature is drawn to show that our approach is definitely a new contribution to the field. Setup and motivation of the numerical example and results obtained conclude the section. The main conclusion is that real options are effective in reducing downside risk in IRR and in increasing its expected value. Moreover, real options accelerate recovery of capitals invested, reducing both payback period and the expected life of the project and increasing the expected recovery rate.

IRR according to Teichroew sets to zero the compound value of cash flows produced by an investment at the end of its life.

(7) ⎧ ⎪ ⎨ >0

[CFt + RCVt−1,t ] · (1 + r)

⎪ ⎩ [CF + RCV t t−1,t ] (1 + IRRT )

(6)

⎪ ⎩ <0

carlo experiments were averaged to get expected payback period and internal rate of return of the project. Moreover, we could provide the whole distribution of these investment parameters.

(8)

In order to get the annualized interest rate for IRR, linear compounding was used, i.e. IRR = IRR1/h ·h for a subperiod Δ t = 1/h . As a matter of fact, subannual compounding does not allow to compute equivalent annual rates for IRR < −100%. Both PBP and IRR computed on each of the Monte-

Investments in the shipping industry has been often studied using real options, see for instance (Dixit and Pindyck, 1994) on page 237, (Goncalves de Oliveira, 1999) page 185. This is due to the fact that in tramp shipping services, strategic interactions among competitors are really meaningless. This allows us to apply a reduced model in which a representative agent is faced by a whimsical nature generating the most important profitability driver of the industry, time charter rates. The data generating process of this series has been specified as a driftless GBM by (Dixit and Pindyck, 1994) and a GBM with drift by (Goncalves de Oliveira, 1999). Instead, it is possible to show, see appendix A in (Alesii, 2003), that tramp shipping time charter rates are well described by an Arithmetic Ornstein Uhlenbeck.

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Although any textbook in financial management shows that the use of these thumb rules does not lead to shareholders’ value maximization, practitioners in the shipping industry use them thoroughly.

Moreover, both these models are derived in a infinite time horizon, being based on stationary dynamic programming.

Being a ship a finite lived asset, we have preferred to study the opportunity to invest in a ship over a finite time horizon, T=10 years. Because of these specific features, we were able to derive exercise thresholds for the whole life of the project while this was not the case for the two papers previously mentioned in which only individual levels of the state variable are given as thresholds. Finally, while (Dixit and Pindyck, 1994) on page 237 evaluates the vessel with all the real options we have considered here, namely option to wait, to mothball and to restart, and option to abandon, (Goncalves de Oliveira, 1999) page 185 studies only the switching options to mothball and to restart. Is it worth emphasizing the fact that both the previous models are based on expanded NPV only and that, to our knowledge, there is not in the current literature any extension of the elegant symbolic stochastic algebra that derives the implied values of PBP and IRR. Although any textbook in financial management shows that the use of these thumb rules does not lead to shareholders’ value maximization, practitioners in the shipping industry use them thoroughly. For instance, leading shipping management consultants propose together with second hand ship evaluations also PBP and IRR, see (Drewry and Jupe, 2001) and (Drewry and Kellock & Co, 1999). A recent survey by (Cullinane and Panayides, 2000) reports IRR as the mostly used decision rule, followed by NPV and PBP. These analyses are usually performed on an expected scenario discounted using a risk adjusted rate. Risk is taken into account through sensitivity analysis. Optimal dynamic decisions are simply ignored.

The widespread use of IRR in the evaluation of shipping investment is mirrored in some normative literature that applies risk analysis `a la Hertz to IRR distribution, see (Haralambides, 1993) in (Gwilliam, 1993). There a distribution of returns is derived under a probability assigned by the representative agent, more or less subjectively. In conclusion, the derivation of the whole distribution of PBP and IRR in the presence of real options for a shipping industry investment can be of some interest to ship owners and financiers being these thumb rules more easily understood by both the industrial and the banking practice. The numerical example has been set up as follows.

The initial investment is c1,2 = 40, or cost to move the dynamic system from mode 1, wait, to mode 2, operate. Costs to mothball the project are c2,3 = 2, cost to move from mode 2, to mode 3, laid up ship, e.g. Fujairah anchorage off Oman coast. Costs to restart the project are c3,2 = 4. If the project is abandoned it yields c3,4 = −5, cost to move from mode 3 to mode 4, abandoned, e.g. Bangladesh wrecking yard.12

12 • Rules of Thumb in Real Options Analysis

The project has an expected technical life of 10 years and its operating mode can be revised every six months exercising the options to start the project, to mothball, to restart or to abandon it. In operating mode the profit is πO = 20 · θs − 7 while in mothballing mode it is πM = −1.5. In both waiting mode and abandonment mode cash flows are nil.13 The state variable has been specified as an arithmetic Ornstein Uhlenbeck with the following parameters: η = .125, θ = .5, σθ = .125 , in a grid with θmin = 0 and θmax = 1 with Δ θt = 1% . This process has been chosen after estimating the process parameters on 53 years of monthly time series reported in appendix A of (Alesii, 2003).

The proportions between the normal value and volatility are equivalent to those of dry bulk time charter computed in appendix B of (Alesii, 2003). Reversion speed resembles that of the same time series.

We have derived the value of the investment project at time t = 0 in a backward induction procedure applied to equation (2). Results are represented by the smoothed lines without markers in figure 2. The same procedure has been run both for dynamic active management and passive management.

From the same procedure we have derived the real options exercise thresholds, see equation (3), for the whole life of the project as represented in figure 1. We have performed 80,000 Euler Scheme Monte Carlo simulations of the state variable t approximating equation (4) with the same discretization used in the backward induction. Then, on each of these time series we have computed CFt net of transition costs taking into due account the optimal operating modes indicated by real options exercise thresholds. Results are reported in figure 1 as smoothed lines with markers. As a matter of fact, while the RPVs have been derived for 100 different initial values simultaneously, Monte Carlo simulations have been performed for 10 initial values θ0 = 0.0, 0.1, 0.2, . . . , 0.8, 0.9, 1.0.

At a first glance on figure 2 forward and backward computed expected values seem to be the same both with and without real options. This is confirmed by convergence tests performed in Appendix A in which expected values for actively managed projects differ on the average at most 1.5% for θ0 = 0 and absolute differences are really meaningless with respect to the sum initially invested. This assures that both backward induction and forward computation are modeling the same optimal dynamic behavior.

Convergence for passively managed projects expected values is, instead, complete. Results on NPV distributions are thoroughly commented in (Alesii, 2003) where it is shown that real options are effective not only in enhancing value but also in taming risk, reducing the so called project at risk.

Figure 2: RPV and Monte Carlo Markov Chain Expected Expanded NPVs 120 100

Value_{0,m=1}

80

Value_{0,m=2, without all options} M_1_pass

60

M_1_act 40 20 0 0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

-20 -40 -60 -80 -100

Legend: The graph reports RPVs for both the active and passive management of the project, with and without real options, smoothed line without markers. Together with these values, the corresponding averages from the Monte Carlo Markov Chain for selected levels of θt=0 are reported, namely θt=0 = 0, .1, .2, . . . , .9, 1. These are represented with markers, circles and squares respectively.

Table 1: IRR Variability Passive management Ð0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Pr(IRR < 0) 92.3% 86.8% 77.4% 65.2% 51.4% 36.4% 22.9% 12.7% 6.3% 2.6% 1.0%

Pr(IRR < rf ) 96.7% 92.9% 85.4% 73.8% 59.6% 42.9% 27.0% 14.8% 7.2% 2.9% 1.1%

q.99 -493% -486% -481% -482% -481% -453% -392% -318% -223% -82% 0%

q.95 -430% -417% -401% -386% -361% -304% -217% -84% -11% 22% 46%

q.50 -30% -25% -18% -10% -1% 10% 21% 33% 47% 63% 79%

M(IRR) -100.6% -92.4% -81.2% -69.5% -52.4% -28.8% -4.7% 17.6% 38.0% 57.1% 74.1%

Pr(IRR < rf ) 30.7% 30.3% 29.9% 28.8% 28.4% 26.6% 25.0% 12.3% 4.9% 1.4% 0.3%

q.99 -79% -75% -74% -71% -71% -72% -70% -56% -34% -2% 27%

q.95 -43% -42% -45% -46% -47% -47% -49% -21% 6% 27% 48%

q.50 18% 18% 19% 20% 20% 21% 21% 34% 47% 63% 79%

M(IRR) 13.4% 13.3% 13.8% 14.1% 14.1% 14.9% 14.8% 29.1% 44.1% 60.1% 75.5%

std(IRR) 143.5% 139.2% 134.7% 131.3% 123.5% 106.6% 85.5% 65.5% 47.8% 34.2% 22.9%

max(IRR) 28.8% 35.4% 40.5% 47.3% 52.3% 62.2% 68.5% 76.0% 83.1% 89.4% 96.1%

Skco -1.53 -1.59 -1.69 -1.82 -2.12 -2.65 -3.37 -4.31 -5.38 -6.77 -7.02

Kuco 3.88 4.12 4.54 5.10 6.50 9.61 15.03 24.54 40.41 71.90 89.46

Pr(CF < 0 8t) 18.6% 14.9% 10.5% 5.1% 1.5% 0.2% 0.0% 0.0% 0.0% 0.0% 0.0%

Pr(IRR = n.a.)

0.026% 0.025% 0.013% 0.015% 0.015% 0.015% 0.000% 0.003% 0.001% 0.001% 0.003%

Pr(Not taken) 88.9% 84.5% 77.1% 65.3% 49.7% 29.4% 0.0% 0.0% 0.0% 0.0% 0.0%

Pr(IRR = n.a.)

0.000% 0.000% 0.000% 0.000% 0.000% 0.000% 0.000% 0.000% 0.000% 0.000% 0.003%

Active management Ð0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Pr(IRR < 0) 25.1% 24.8% 24.5% 23.7% 23.7% 22.0% 20.5% 10.0% 3.9% 1.1% 0.2%

std(IRR) 28.7% 28.2% 28.5% 28.4% 28.4% 28.4% 27.5% 24.2% 21.3% 18.3% 14.8%

max(IRR) 75.9% 84.6% 76.5% 79.9% 83.1% 80.8% 68.5% 76.0% 83.1% 89.4% 96.1%

Skco -1.19 -1.13 -1.11 -1.10 -1.13 -1.20 -1.31 -1.46 -1.46 -1.50 -1.69

Kuco 5.19 4.80 4.56 4.35 4.36 4.51 4.55 5.81 6.61 7.17 8.38

Legend: q.99:= 99th quantile; q.95:= 95th quantile; q.5:= median; M(IRR):= average of the simulated3 generalized Teichroew ] : standardized skewIRRs; std(IRR):= standard deviation of the simulated generalized Teichroew IRRs; Skco = μσ33 = E [(x−μ) σ3 4 ness coefficient according to Irving Fisher; Kuco = μσ4 := standardized kurtosis coefficient, a value of 3 indicates a normal, less than 3 a platicurtic, more than 3 a leptocurtic; Pr(CF < 0 Vt):= probability of all negative cash flows for passively managed projects or of a never taken investment for actively managed ones; Pr(IRR = n.a.):= probability of non existence of Teichroew IRR.

In this paper we focus on IRR and PBP which were computed on the same CFt series both passively and actively managed with the optimal exercise of real options. Table 1 reports results about internal rate of return computed according to Teichroew, while tables 2 and 3, to be read together with table 4, report results about the payback period, the recovery rate when the investment is not completely repaid and the life of the project in the case in which it is actively managed. As a matter of fact, passively managed projects last just as much as their technical life.

From table 1 we can conclude that real options 14 are effective not only in increasing expected return from investment but also in reducing downside risk. As a matter of fact, both expected values and medians for the actively managed project are definetly higher than those of the passively managed ones. This is true for all the levels of θ0 but those above the initial investment threshold θ1→2 = .59. Hence, we can conclude that the option most effective in increasing expected value is the option to wait. Instead, the options to mothball and to restart are the least effective. No conclusion we can reach for the option to abandon being this thoroughly used for all the initial θ0.

Moreover, downside risk in actively managed projects is definetly reduced. Not only the probability of having a negative return is decreased many times, see first two columns in table 1, but also, V aR .99 and V aR .95 are definetly reduced. In essence, real options reduce negative skewness of IRR distributions, decreasing occurrences in which IRR is below 100%, i.e. occurrences in which investment projects not only burn all the initially invested capital, namely c1,2, but also require additional capital to be maintained over the project horizon. This dovetails with the results found by (Alesii, 2003) in which expanded NPV is rarely found to be lower than minus

the initially invested sum. The fact that we have used here a version of IRR which ex ante does not always exist does not prevent us to draw this parallelism since it was not possible to get an IRR in a negligible number of cases, see last column in table 1.

It is worth noting that these results are obtained following a very specific dynamically optimal behavior. To limit ourselves to the exercise of the option to wait, higher IRRs are obtained because investment is implemented in slightly more than 10% of the cases for θ0 = 0 or more than 70% for θ0 = .50. Obviously, investment project is always implemented for levels above the investment threshold. 15 In this way, representative agent discards also all the occurrences in which every cash flow from investment is negative, see last but one column in panel A in table 1. Payback Period is definetly reduced by real options, see mean and median columns in table 2. Although that is true, the most effective option in reducing PBP seems to be the option to wait. As a matter of fact, for levels of θ0 above the investment threshold, mean and median are the same or not significantly different. From results reported in the same table, real options appear even more effective in reducing downside risk for PBP, see q.99 and q.95 columns. Probabilities of not finding a PBP are drastically reduced because investment is not taken in a high percentage of cases as already noticed above for IRR results. As a matter of fact, probabilities that investment does not pay back are very low in the neighborhood of the investment threshold while are nil for the other initial levels of θ0. Hence, probabilities that investment pays back when and if it is taken are very high.

Even when investment project does not pay back completely, real options allow to recover a higher percentage than in the case of a passively managed project, see table 3. For instance for θ0 = .6 in almost 20% of the cases investment does pay back initially invested capital only fractionally, see table 2.

Recovery rates that can be read in table 3 are much higher expecially in the lower tail. The beneficial effects of real options on both IRR and PBP, both in terms of improved expected values and reduced downside risk, are due to an investment behavior that delays investing until the state variable reaches the threshold level for exercising the option to wait and abandons the investment as soon as the abandonment threshold is reached. This shortens the expected life of the project when it is implemented, see table 4. For instance, for levels lower than the investment threshold θ1→2 = .59, investment projects have a very short expected life being implemented late and abandoned early.

Considered all together, tables 1 - 4 give a whole string of parameters on which negotiation can take place between headquarters and division managers or between the shipowner and her banker. For instance, for a level of θ = .6, the expected level of net present value is E(NPVp) = 27.18 for the passively managed project while it is E(NPVe) = 30.51 for the actively managed one, being the difference the value of the switching options to mothball, to restart and to abandon.16 This present value has been translated into PBP and IRR. The levels of expected IRR are respectively E(IRRT)p = −5% and E(IRRT)e = 15%. It is worth noting that rates of return are more effective in underlining the difference between an active and a passive management. Instead, the corresponding expected levels of PBP are not significantly different since the option to wait has been exercised for θ0 > .59. Although that is true, the option to abandon together with the switching options, to mothball and to restart, are effective in increasing the recovery rates. As a matter of fact, not only occurrences in which capital is not recovered are drastically reduced, from 7.8% to

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Rules of Thumb in Real Options Analysis • 13

fsrforum • jaargang 12 • editie #5

Table 2: PBP Variability

Table 4: Life of the Actively Managed Project Variability

Passive management Table 2

Table 4

Ð0

q.01

q.05

q.50

M(PBP)

std

Skco

Kuco

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10.0 10.0 10.0 9.9 9.9 9.8 9.6 9.0 7.6 4.6 2.7

9.9 9.8 9.8 9.6 9.4 9.0 8.2 6.6 4.5 3.0 2.1

8.6 8.2 7.6 6.9 6.0 4.9 3.8 3.0 2.4 1.9 1.7

8.4 8.0 7.5 6.9 6.2 5.3 4.4 3.5 2.7 2.1 1.7

1.2 1.3 1.5 1.7 1.8 1.8 1.7 1.4 1.0 0.6 0.3

-0.69 -0.50 -0.27 0.00 0.31 0.71 1.25 2.02 3.24 5.25 8.86

2.84 2.43 2.16 2.01 2.11 2.62 4.00 7.43 16.81 45.73 159.02

Pr(Not taken) 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

Pr(No PB)

Pr(PB)

Pr(Frac Rec)

79.6% 69.8% 56.2% 40.8% 26.6% 15.2% 7.8% 3.4% 1.2% 0.3% 0.0%

6.3% 11.3% 20.5% 33.7% 49.2% 65.7% 80.2% 90.7% 96.6% 99.1% 99.9%

14.1% 18.8% 23.3% 25.5% 24.2% 19.1% 12.0% 5.9% 2.2% 0.6% 0.1%

Active management Table 2 Ð0

q.01

q.05

q.50

M(PBP)

std

Skco

Kuco

Pr(Not taken)

Pr(No PB)

Pr(PB)

Pr(Frac Rec)

0.0

6.3

5.1

3.0

3.2

1.0

1.17

4.44

88.86%

0.0%

8.3%

2.8%

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

6.8 7.2 7.6 8.1 8.6 9.5 8.8 7.3 4.5 2.7

5.5 5.8 6.1 6.6 7.1 8.0 6.5 4.5 3.0 2.1

3.1 3.2 3.3 3.4 3.6 3.8 3.0 2.4 1.9 1.7

3.3 3.5 3.6 3.8 4.0 4.3 3.4 2.7 2.1 1.7

1.1 1.1 1.2 1.4 1.5 1.7 1.4 1.0 0.6 0.3

1.27 1.26 1.26 1.26 1.23 1.25 1.99 3.14 4.93 7.64

4.85 4.72 4.60 4.43 4.21 4.04 7.39 16.25 41.52 122.96

84.44% 77.00% 65.25% 49.62% 29.36% 0.00% 0.00% 0.00% 0.00% 0.00%

0.0% 0.0% 0.0% 0.1% 0.2% 0.5% 0.0% 0.0% 0.0% 0.0%

11.7% 17.3% 26.5% 38.5% 55.2% 79.7% 90.3% 96.4% 99.1% 99.8%

3.9% 5.7% 8.2% 11.8% 15.2% 19.8% 9.6% 3.6% 0.9% 0.2%

Legend: q.01: first centile; q.05:= first ventile; q.5: median; M(PBP): mean of computed payback periods; std: standard devia3 ] : standardized skewness coefficient according to Irving Fisher; tion of computed payback periods; Skco = μσ33 = E [(x−μ) σ3 μ4 σ 4 := standardized kurtosis coefficient, a value of 3 indicates a normal, less than 3 a platicurtic, more than 3 a leptocurtic; Pr(Not taken):= probability of not taking the investment project; Pr(No PB):= probability that the investment project does not pay back at all; Pr(PB):= probability that the investment pays back completely and more; Pr(Frac Rec):= probability that the investment project yields back only a fraction of the initially invested lump sum, see table 3 for the distribution of these fractional recovery.

Kuco =

Table 3: Recovery Rate Variability Passive management Table 3 Ð0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Min 00.0% 00.0% 00.0% 00.0% 00.0% 00.0% 00.0% 00.0% 00.0% 01.0% 00.0%

q.99 00.5% 00.5% 00.5% 01.0% 01.0% 01.0% 01.0% 01.0% 00.5% 02.3% 00.0%

q.95 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 5.5% 6.0% 5.0% 7.3% 6.5%

q.5 39.0% 41.5% 45.5% 47.5% 50.0% 51.5% 52.0% 52.5% 53.0% 54.5% 51.8%

M(ReRa) 42.2% 44.2% 46.5% 48.1% 49.7% 50.8% 51.1% 51.5% 51.6% 52.2% 51.6%

std 28.0% 28.4% 28.3% 28.4% 28.3% 28.0% 28.0% 27.8% 27.6% 27.3% 26.3%

max S 99.5% 99.5% 99.5% 99.5% 99.5% 99.5% 99.5% 99.5% 99.5% 99.5% 96.5%

kco 0.30 0.23 0.13 0.06 -0.02 -0.05 -0.07 -0.09 -0.13 -0.12 -0.11

Kuco 1.95 1.89 1.84 1.83 1.84 1.86 1.86 1.88 1.92 1.91 2.05

Active management Table 3 Ð0

Min

q.99

q.95

q.5

M(ReRa)

std

max

Skco

Kuco

0.0

01.5%

15.6%

28.5%

72.3%

68.3%

22.3%

99.8%

-0.55

2.34

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

00.3% 01.3% 00.3% 00.3% 00.0% 00.0% 00.0% 01.3% 24.8% 52.8%

13.8% 10.3% 08.5% 07.1% 04.8% 03.4% 10.8% 21.3% 35.3% 53.3%

25.5% 22.8% 19.3% 17.8% 14.8% 11.8% 22.8% 36.8% 48.0% 63.3%

70.3% 69.3% 66.0% 63.8% 60.8% 55.1% 65.8% 75.3% 79.8% 81.8%

66.9% 65.5% 62.9% 61.2% 58.5% 54.5% 63.4% 72.0% 77.4% 81.1%

23.1% 23.8% 24.5% 25.3% 25.9% 26.5% 22.9% 19.0% 15.2% 11.6%

99.8% 99.8% 99.8% 99.8% 99.8% 99.8% 99.8% 99.8% 99.8% 99.3%

-0.50 -0.48 -0.39 -0.32 -0.24 -0.09 -0.38 -0.71 -0.77 -0.26

2.27 2.23 2.13 2.03 1.98 1.89 2.25 3.01 3.16 2.16

Legend: The table reports descriptive statistics for recovery rates of fractionary recovering investments. The probability of these occurrences are those reported in the last column of table 2. q.99:= 99th quantile; q.95:= 95th quantile; M(ReRa)= 3 ] : standardized average of the recovery rates; std: standard deviation computed on recovery rates; Skco = μσ33 = E [(x−μ) 4 σ3 skewness coefficient according to Irving Fisher; Kuco = μσ4 := standardized kurtosis coefficient, a value of 3 indicates a normal, less than 3 a platicurtic, more than 3 a leptocurtic. Values of -1 in both coefficients indicate degenerate distributions on which it was not possible to compute them.

14 • Rules of Thumb in Real Options Analysis

Ð0

Min

q.99

q.95

q.5

M(Project Life) std

max

Skco

Kuco

0.0

0.5

1.0

3.0

5.0

5.1

1.4

9.0

-0.24

3.05

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.5 0.5 0.5 0.5 0.5 2.0 2.0 2.5 3.0 3.0

2.0 2.5 2.5 3.0 3.0 3.5 4.5 5.0 6.0 6.5

3.0 3.0 3.5 4.0 4.5 5.5 6.5 7.5 8.0 8.5

5.5 6.0 6.5 7.5 8.5 10.0 10.0 10.0 10.0 10.0

5.4 5.9 6.4 7.1 7.9 9.3 9.5 9.7 9.8 9.8

1.5 1.6 1.7 1.8 1.8 1.5 1.2 1.0 0.8 0.6

9.0 9.5 9.5 9.5 9.5 10.0 10.0 10.0 10.0 10.0

-0.24 -0.30 -0.40 -0.63 -1.08 -2.20 -2.72 -3.29 -3.89 -4.36

2.80 2.65 2.53 2.62 3.29 7.00 10.00 14.04 19.33 23.87

Legend: The table reports descriptive statistics for actively managed projects lives. The probability of these occurrences are the complement to one of Pr(Not taken) reported in table 2. q.99:= 99th quantile; q.95:= 95th quantile; M(Project Life)= expected life of the actually taken projects; 3 ] : std: standard deviation computed on recovery rates; Skco = μσ33 = E [(x−μ) σ3 standardized skewness coefficient according to Irving Fisher; 4 Kuco = μσ 4 := standardized kurtosis coefficient, a value of 3 indicates a normal, less than 3 a platicurtic, more than 3 a leptocurtic. Values of -1 in both coefficients indicate degenerate distributions on which it was not possible to compute them.

0.5%, but also cases in which only a fractional recovery takes place are increased, from 12.0% to 19.8%. Even in those cases, capital recovered in the worst 5% occurrences is 11.8% if real options are used to manage the project, while it is 5.5% in cases in which it is passively managed.

In conclusion, in this paragraph we apply the methods previously devised to a sketched case study in shipping finance deriving the distributions of IRR, PBP together with recovery rates and life of the project. This allows us to reach conclusions not only about the value enhancing properties of real options but also about the risk reducing ones. As a matter of fact, IRR is not only increased in its expected values but also its downside risk is definetly trimmed down by real options. To the same token, PBP is reduced in its expected values and its upside risk is definitely decreased. Computing these thumb rules has allowed us to give a more intuitive insight into the optimal behavior endogenous to real options valuation. As a matter of fact we have shown what is the probability of exercising the option to wait and how the life of the project is affected by optimal investment delaying and abandoning strategies.

4 Conclusions In this paper it is devised a new method to translate into two of the most used thumb rules, namely PBP and IRR, the effect of the optimal exercise of real options. Using a numerical method already used by (Alesii, 2003) to derive project at risk in the presence of real options, we were able to derive the whole distribution of the Teichroew version of the internal rate of return and of simple payback period together with

It is impressive that in both active and passive management results, convergence in standard deviation follows the same proportions for the same increases in the number of trials.

fractional recovery rate and life of the project in the presence of the option to wait, to mothball, to restart and to abandon.

The effect of these downside risk options on the computed thumb rules is not only in improving their expected value but also in reducing their shortfall. Moreover, this paper explores when real options are actually exercised giving a more intuitive insight into the endogeneity of real options valuations models with respect to actual optimally dynamic behavior. This in turn could help to consider real options models not as a black box but simply as a way of quantifying optimal management of an industrial project. A possible extension of this paper would be in delving into the description of this dynamic optimal behavior giving practitioners a probable sequence of optimal actions they should conform to in order to create value with the optimal exercise of real options, e.g. in this case how many time the options to mothball and to restart are exercised. This would allow to compare current practice with optimal project management according with real options and to conclude whether these are a simple pure abstraction or a realistic model.

A Convergence Tests The gist of the present paper stays in the convergence of the expected Running Present Value in the DP procedure and the average of the Markov chain Monte Carlo simulations. This fact guarantees that these two procedures model the same dynamic optimal behavior although the former is based on backward induction while the latter is based on a forward computation of the expanded net present value. Therefore, this appendix reports some results about convergence of expected value of Markov chain Monte Carlo simulations towards the values produced by the Dynamic Programming procedure. We have checked both the results based on active and passive management. Moreover, we have checked whether the initial level has some influence on convergence choosing a level in the middle of the discretized state variable space 0 = .5 together with the minimum and the maximum, 0 = 0 and 0 = 1. To test convergence we have computed for 500 experiments expected values of the Markov chain Monte Carlo simulations for several number of trials n, namely 1000, 5000, 10000, 20000, 40000 and 80000. Changing the step allowed us to test the variability of the results in different neighborhoods of n. Because of this, experiments have a higher degree of granularity in the hundreds while this decreases as the number of experiments increases. This experiment design was chosen to show that results variability decreases as the number of experiments increases. We have considered the relative difference between the averages as computed above

and the corresponding RPV, see expression (9), where the exponents f and b indicate respectively forward and backward computation. ⎧ ⎪ ⎨

Relative

⎫ ⎪ ⎬

⎪ ⎩ Dif f erence ⎪ ⎭

=

M1 N P Vθf0 − E ∗ RP Vθb0

E ∗ RP Vθb0

(9)

Results are reported in table 5. Observing means and medians together with skewness and kurtosis coefficients, we can conclude that results distributions fairly approximate normal distributions especially for high number of trials. Because of this we can reasonably detect convergence from mean and standard deviations of (9).

Passive management results show a complete convergence in the mean and a fast convergence in the standard deviation of the results. This is true for all the three levels of 0 that have been chosen. Instead, active management results depend on the initial level of 0. While for low levels relative difference converges to around 1.5%, for middle and high levels this is definitely good, between 0.06% and 0.003%. However it should be stressed that absolute differences are really meaningless when compared to the initial investment. It is impressive that in both active and passive management results, convergence in standard deviation follows the same proportions for the same increases in the number of trials. Convergence in standard deviation does follow the usual square of the number of trials rule. As a matter of fact, to double the accuracy of a simulation we must quadruple the number of trials. In conclusion, passive management results convergence is not dependent on the initial value of 0, while active management results depend, to a certain extent, on the initial value. However, averages of forward computed NPV from the Markov chain Monte Carlo simulations converge to the Running present values computed in the backward induction process with a relative difference that at most, on the average, reaches 1.5% and is never lower than 0.003%. A possible explanation of this difference, which would be hardly acceptable for financial derivatives models, stays in the short life of the project and in the coarse time discretization chosen.

Notes 1

The original intuition of a (real) growth option is already in (Myers, 1977) but the expression “real option” was coined by (Myers, 1984).

We should be wary in defining criteria some decision rules that simply do not have a representative agent criterion function to maximize or which do not respect the Value Additivity Principle, see for instance (Rao, 1992) page 240.

3 See for instance (Binder and Chaput, 1996) for a description of trends from the ’50s till the ’80s in capital budgeting practices in the US. 4 See for instance references in (Boyle and Guthrie, 1997) for survey results about Australia, Canada and New Zealand. See (Scapens and Sale, 1981) for two parallel surveys about UK and US firms, (Runyon, 1983) and (Coulthurst and McIntyre, 1987) for a survey about small-medium sized firms respectively in the US and the UK. See (Segelod, 2000) and (Sandahl and Sjogren, 2003) for evidence about Swedish professional service groups and large companies.

5 In essence, we generate scenarios that can be used for both risk management and pricing purposes. In this case, the natural probability measure used for pricing is shown to be also an EMM, see appendix of (Kulatilaka, 1993). Although that is true, it is considered correct to use in general an EMM for risk management purposes, while it is not right to use a natural measure of probability for pricing purposes, see (Tavella, 2002) page 77.

6 The definition of a general model of real option is in (Kulatilaka, 1995) in (Trigeorgis, 1995). The acronym is ours.

7 Bellman’s Optimum Principle has been applied to the pricing or real options also by (Dixit and Pindyck, 1994) page 95.

8 For details of the numerical procedure followed see (Alesii, 2000).

9 We wish to thank Professor Richard Stapleton for pointing out this issue.

10 See page 87 in (Tavella, 2002). In practice we simulate the Markov Chain that has a one step transition probability matrix given for each level t by the discretization of the normal distribution of the arithmetic Ornstein Uhlenbeck additive shocks. Itis worth noting that although a solution to the OU SDE does exist, see equation 4, the Euler scheme discretization has been adopted to maintain the same metrics on which results were derived in the backward induction procedure. We wish to thank Christian Schlag for pointing out this issue.

11 For some analytical proofs about existence of internal rates of returns and computational feasibility of NPV see (Saak and Hennessy, 2001) and (Oehmke, 2000).

12 We consider the wreckage profit instead of the second hand ship price because the former can be considered deterministic while the latter varies considerably in the horizon chosen for the model. Hence, a better specification to take into due account second hand ship price would be as a second state variable. We save this extension for another paper.

13 It is worth noting that this specification of the model does not consider that different real options have different lags between their exercise and their effect. For instance, the option to wait is effective at least after two or three years the ship has been ordered to the shipyard. This would require a new specification of the model. We save this extension for another paper.

14 In this case the option to wait, to mothball, to restart and to abandon, typically downside risk real options.

15 To give a thorough understanding of this optimal behavior, the probability of exercising the option to mothball once, twice etc should be included.

This would help to understand whether the optimal behavior endogenous to the expanded net present values is actually feasible or it is a simple abstraction. We save this extension for another paper.

16 Results about NPV are provided in (Alesii, 2003).

2

Rules of Thumb in Real Options Analysis • 15

fsrforum • jaargang 12 • editie#4

Being on the winning team

Bedrijspresentatie Flow Traders Schrijver

Flow Traders offered an interview to the FSR. A full description of the company was given by Folkert Joling, a young and energetic man of thirty years. He has been working for Flow Traders for three years now and is still very enthusiastic about his job.

Personal Folkert Joling was born in Ede. He went to high school in Harderwijk and he studied applied mathematics in Enschede. He took part in some committees during his studies and followed an internship as a trader in Amsterdam in his final year. Since high school he has been very enthusiastic about the stock market and he decided that he wanted to work at Flow Traders.

It didn’t take long before they noticed his enthusiasm and up till now, he has been working as a trader at Flow Traders. Joling indicates that his analytical skills are most useful in his job.

These skills help him to oversee various situations and quick calculations are made by heart.

Company Profile Flow Traders was set up five years ago by four talented traders. At this moment, the company counts over a hundred employees inside and outside the Netherlands. Flow Traders has its main office in Amsterdam and has also offices in New York and Singapore. The company encourages the exchange of employees between all the different offices. Joling remarks that he also wanted to work in New York, but they probably needed him more in Amsterdam. The language policy of the company concentrates on English; however, Dutch is being spoken in Amsterdam.

The atmosphere The atmosphere at the desks depends on the markets. Usually everyone is focusing on his quotes and communicating with others. Because of some serious decisions, all employees are very direct to each other and point at mistakes without hesitation. Unfortunately, some emotional applicants can’t take the pressure and the directness. When you aren’t trading, you make a chat with colleagues at the bar in the office or play a game in the arcade room.

Furthermore, all employees may join the company trips to (for instance) a ski resort, Iceland and Milan. Joling enjoys the informal time with his colleagues.

“A regular day at work begins with analysing the results and checking the positions of the previous day for one hour. Thereafter, I enter the market as a market maker. I offer as well as buy products on the market. I aim for the best ‘quotes’ and search for maximal spreads.

Trading takes about eighty per cent of my time. The rest of the day, I am looking for new arbitrages, better hedges, analysing historical data, et cetera. Furthermore, keeping an eye on my positions is essential, because of the shifting markets.” Besides all of this, he is also responsible for the training of new employees, as stated below. After a drink in the bar, he usually leaves the office at about seven o‘clock. In his spare time, Joling plays hockey and likes to read the newspapers. “If you don’t keep up, you will fall behind.” He also studies some mathematics every now and then. Up till now, he enjoys his work at Flow Traders and seldom takes a day off.

The division of tasks among traders is mainly based on personal interests and on the basis of required activity on a segment of the market. The office in Amsterdam consists of seven desks in total: each with their own specialisation. An average trader switches desks a few times a year. This also holds for Joling, who switched almost five times in three years. In these years, he learned techniques like pair-trading and index-trading. Pair-trading is based on the fact that two products with almost the same characteristics can be used to trade with minimal

16 • Bedrijfspresentatie Flow Traders

risk. Index-trading is the technique of trading portfolios that are slightly different from an index, like the AEX, with the index itself.

Since Flow Traders is not the only company in its kind, competitors are counteracting the profit or “spread” that can be made on certain transactions. The size of the arbitrage thus decreases. However, Joling explains: “Whenever the spread is low, a large profit can be made by increasing the volume. More risky products are made with a substantial lower volume, but yield a higher spread.” Because of the complexity of the financial markets, Joling reflects on arbitrage-trading as a sustainable profession. “If you are able to keep the overview and be quick, you can outperform the other traders and make a nice profit.” A good trader is always

The division of tasks among traders is mainly based on personal interests and on the basis of required activity on a segment of the market.

the first one to sell or buy a product at an optimal price at an optimal time. “A trader needs to be agile, clever and decisive.” The reason will be obvious: seconds matter in the world of trading. In order to be able to achieve this agility, Flow Traders maintains a team of developers and information technologists of about thirty experts. “This office contains a lot of financial software and a very quick connection to the Internet.” Next to being quick, you should also act wisely. In case of a loss, you ought to be able to explain your choices on rational grounds. The chief, also called the head of trading, can check their transactions anytime.

After talking about the credit crunch and the nice profit this meant for Flow Traders, Joling assured us that you shouldn’t worry about your future here. The structure of the company is flat and everyone gets rewarded for his achievements. Therefore, setting up his own trading company isn’t Joling’s ambition. According to him, you need software, partners and money.

“These times are too rough for a small company; even banks are having serious problems with setting up trading projects.” Therefore, Joling is very proud of the achievements of Flow Traders: “It’s always fun to be on the winning team.”

Bedrijfspresentatie Flow Traders • 17

fsrforum • jaargang 12 • editie #5

Discrete Space-Time Options Pricing

Ilya Gikhman

This paper presents a formal approach to the derivatives pricing. In this paper we will study derivatives pricing in a discrete space-time approximation. The primary principle of the pricing theory we introduce in the paper is the notion of equality of investment which based on the investors goal: ‘investing in a greater return’.

We say that two investments are equal at a moment of time if their instantaneous rates of return at this moment at are equal. If equality of two investments holds any moment of time over [ t , T ] then these investments are equal on [ t , T ]. This definition represents investment equality, IE law or principle which will be applied throughout of the paper for definition of a derivative price. Next we use cash flow notion that implies a series of transactions as a specification of somewhat broad notion of investment.

It is not difficult to see that this concept of the investment equality is a perfect and more accurate than the present value, PV concept. Indeed, if two investments are equal in IE sense then they are equal for any possible scenario or simply to say always. On the other hand, it is clear that if two investments are equal in IE sense then they are equal in the PV. The inverse statement is incorrect. More accurately, if two investments are equal in the PV sense then it is easy to present example of a scenario that demonstrates arbitrage opportunity. In this example, the present value of two investments are equal while one investment has a higher rate of return over a time subinterval and lower over another subinterval than other investment. As far as PV suggests equal price for two investments one can sell a lower rate instrument over the correspondent subinterval and buy the instrument with higher rate of return instrument. Then at the end of this period investor

We say that two investments are equal at a moment of time if their instantaneous rates of return at this moment are equal. would sell short higher priced instrument which promises lower rate of return over the next period and buy for lower price other instrument which promises higher rate of return. At the end of the period the investor has pure profit for the scenario though two investments have equal PV. This type of the example illustrates the fact that PV reduction of cash flows insufficient to be used as a definition of the equality of two cash flows. Nevertheless, PV reduction might be helpful for construction of the market estimates of the spot or future prices. Bearing in mind that price definition depends on a scenario we should be aware that any spot price calculated with the help of PV or other rule implies risk. This risk for buyer is measured by the probability of the events for which scenario’s price is bellow than spot price.

1. Plain Vanilla options valuation.

Let us introduce the definition of the plain vanilla option contracts which is a class option covered European and American types. An option is a right to buy or sell an asset at a known price, within a given period of time. The known price, K is called exercise or strike price. The last date, T of the lifetime of the option is called maturity. The right to buy is known as the call option, while the right to sell is the put option. The price of the option also referred to as premium.

European options can only be exercised at maturity, whereas American type of the options can be exercised at any moment up to maturity.

18 • Discrete Space-Time Options Pricing

As far as PV suggests equal price for two investments one can sell a lower rate instrument over the correspondent subinterval and buy the instrument with higher rate of return instrument.

Let S ( t ) denote an asset spot price at date t, t * 0. Formally an European option contract is deďŹ ned by its payoff at expiration. The call and put values at expiration T are deďŹ ned by formulas C ( T , S ( T ) ) = max { S ( T ) â&#x20AC;&#x201C; K , 0 } P ( T , S ( T ) ) = max { K â&#x20AC;&#x201C; S ( T ) , 0 }

(1.1)

Thus, a buyer of the call option would agree to exercise the right to buy the underlying asset in case if the value of the call option at maturity T is positive, i.e. if C ( T , S ( T ) ) > 0. It is clear that there is no sense in realization of the right when C ( T , S ( T ) ) ) 0. On the other hand a buyer of the put option would exercise the right to sell the underlying asset in case if S ( T ) < K, i.e. a put holder is interested to sell asset with price S ( T ) for K when S ( T ) < K. That is a holder of the put option can exercise the right to sell the option when P ( T , S ( T ) ) > 0. Otherwise, if P ( T , S ( T ) ) ) 0 the right will not be exercised.

The option pricing problem is to determine the call ( put ) option price at any moment of time t before the expiration date T. To illustrate pricing methodology we begin with a simple example. Next we use the terms asset, stock, or security as synonyms.

Example 1. Let a stock price at t = 0 be S ( 0 ) = 2 and at T = 1 stock take the values S ( 1 ) = { 5 , 1 }. Introduce the probability space of scenarios of the problem. Denote Ď&#x2030; = { u , }, where denotes the scenario { S ( 0 ) = 2, S d u ( 1 ) = 5 }, and d = { S ( 0 ) = 2, S ( 1 ) = 1 }. Putting strike price K = 2 we enable to deďŹ ne the call option price for each scenario. Denote C ( t , x ; T , K , ) the value of the call option for ďŹ xed scenario at the moment t , given S ( t ) = x. Here t , x are variables of the function C ( ), while T, K, are interpreted as parameters. The value of parameters T and K are assumed to be ďŹ xed and for the writing simplicity we will omit them next. Let us specify the value of the option along the scenario u . Applying IE principle we arrive at the equation with respect to unknown C ( t = 0 , S ( 0 ) = 2 ; u )

of the two scenarios: P u = P ( u ), P d = P ( d ). This distribution is assigned then to the option premium. Thus

J I - P

9 Let us consider two stocks that have probabilities P 1 ( u ) = P 2 ( d ) = 0.99 and P 1 ( d ) = P 2 ( u ) = 0.01 correspondingly. The call option price is the random variable taking values : C i ( 0 , 2 ; u ) = 1, C i ( 0 , 2 ; d ) = 0 , i = 1, 2. Then *) The average rate of return on 1st stock is equal to 1.48 and â&#x20AC;&#x201C; 0.43 on 2nd stock. **) The average rate of return on call option written on 1st stock

The difference between possible spot prices is the value of risk taken by investors.

while the average rate of return on call option written on the 2nd stock is equal to 1.5 %. Assume that market price at t = 0 is equal to the mean of the option at this moment.

The expectations of the options price on the 1st and the 2nd stocks at t = 0 are c 1 ( 0 , 2 ) = E C 1 ( 0 , 2 ; ) = 1.2 Ă&#x2014; 0.99 = 1.188

The solution of the equation is C ( 0 , 2 ; u ) = 1.2. Then as far as the option payoff for the scenario u is max { 1 2 , 0 } = 0 we put by deďŹ nition C ( 0 , 2 ; d ) = 0. There is no sense to pay for the option a sum if option value in the future moment is 0. Therefore, by deďŹ nition we put C ( 0 , 2 ; d ) = 0. The security distribution is the probabilities

c 2 ( 0 , 2 ) = E C 2 ( 0 , 2 ; ) = 1.2 Ă&#x2014; 0.01 = 0.012 Other possible estimate of the spot option prices can be based on the PV concept. Assuming that the risk free interest is equal to 0 we see that

Âť Discrete Space-Time Options Pricing â&#x20AC;˘ 19

fsrforum • jaargang 12 • editie #5

c 1 ( 0 , 2 ) = E C 1 ( t = 1 ; ) = 3 × 0.99 = 2.97 c 2 ( 0 , 2 ) = E C 2 ( t = 1 ; ) = 3 × 0.01 = 0.03 We can see that there no unique rule to define a ‘fair’ price to the option as far as any number used as a spot price implies the risk. The difference between possible spot prices is the value of risk taken by investors. This risk is the market risk which specified by the future behavior of the underlying asset. The risk management problem is a calculation of the market risk.

Here, the last row C ( 0 , S ( 0 ) ) = C ( 0 , 2 ) represents the option price at time 0. Each entry in the third row has probability of 1 / 6. The situation represented by the Table is the simplest in sense that the option’s return perfectly replicates the stock return. To illustrate more a general case in which the possibility perfectly replicates the stock return by the call option is impossible we assume, for instance, that K = $ 2.5.

The correspondent option payoff at maturity is the row C ( 1, S ( 1 )) and the price of the option is defined by the third row.

Its values can be calculated applying IE concept. Thus S (1) C ( 1, S ( 1 )) C ( 0, S ( 0 ))

Risk management. *) Consider for example call option written on stock 1. An investor pays premium A for the option at t = 0 takes risk associated with the scenarios G>H@ 7JN:G P

Q

This is the set of scenarios for which rate of return on call option will be lower than the rate of return on underlying stock. Indeed, buying call option for A and receiving S ( T ) K at T implies rate of return equal to the left hand side of the latter inequality while the right hand side is the rate of return on stock over the same period. In other words this

1 0 0

2 0 0

3 0.5 1/3

4 1.5 3/4

5 2.5 1

6 3.5 7/6

Risk management. Mean of the option price at t = 0 is 0.5417. Hence, if the market price of the option is A = 0.5417 then if outcome is 1 or 2 investor’s losses are its premium, i.e. 0.5417. Therefore the loss of 0.5417 occurs with probability 1/3. Then the probability 1/6 is assign to each of the next profit-loss outcomes: [ 0.5 – 1/3 ] = 0.17, [ 1.5 – 3/4 ] = 0.75, [ 2.5 – 1 ] = 1.5, [ 3.5 – 7/6 ] = 2.3 that correspond to stock values 3, 4, 5, 6. We introduce some useful risk characteristics. These are average profit of the option defined by the formula

The American option can be exercised at any time up to maturity T. Therefore, its payoff depends on time interval during which the option can be exercised.

Assuming for simplicity that risk free rate equal to 0 it looks reasonable to exercise option at the date when payoff reaches its maximum. risk set of scenarios forms buyer risk when investors pay higher price for stock that implies by the market. Thus the probability P { ω risk-buyer ( A ) } is a measure of the buyer risk. Let us consider are more complex case when random stock admits multiple values.

Example 2. Let us study the rolling dice example to illustrate the multiple values stochastic security in the option pricing problem. Let again assume that time takes two value t = 0, 1 which are the initial and expiration dates of the option. The set 1, 2..., 6 represents possible values of the stock and probabilities of the events { S ( 1 ) = j }, j = 1, 2, ... 6 are equal to 1 / 6. The payoff at the maturity is defined C ( 1 , S ( 1 ) ) = max { S ( 1 ) – K , 0 } and let K = $ 0.8. The value S ( 0 ) = $2 can be interpreted as a price to roll the dice . Applying the IE concept we arrive at the definition of the call option price.

The option price is a random variable taking different values j - 0. 8, j = 1, 2, ... 6. We express the theoretical price of the game with the help of the table S(1) C ( 1, S ( 1 )) C ( 0, S ( 0 ))

20 • Discrete Space-Time Options Pricing

1 0.2 0.4

2 1.2 1.2

3 2.2 1.47

4 3.2 1.6

5 4.2 1.68

6 5.2 1.73

< Profit ( 0 , T ; K ) > = E C ( 0, S ( 0 )) χ { C ( 1, S ( 1 )) > K } and average losses < Loss ( 0 , T ; K ) > = E C ( 0, S ( 0 )) χ { C ( 1, S ( 1 )) ) K } The profit-loss ratio is < Profit ( 0 , T ; K ) > / < Loss ( 0 , T ; K ) >. These are primary risk characteristics of the option.

Let us highlight the difference between European and American option prices. Consider American option written on stock. The IE rule applying for the European call or put options at t, t < T in either discrete or continuous time is a solution of the equation

P - . % Q

(1.2)

P - . % Q

0 ) t ) T. Here . 2 . % B6M P 2 % Q

* . 2 . % B6M P % 2 Q

are payoffs on call and put options at maturity T. The American option can be exercised at any time up to maturity T. Therefore, its payoff depends on time interval during which the option can be exercised. Assuming for simplicity that risk free rate equal to 0 it looks reasonable to exercise option at the date when payoff reaches its maximum. Hence, the exercise price of the American option is { S ( t ) - K , 0 }.

Applying IE for the American call pricing leads to the equation for American call option value at t = 0

P - % Q

where Ď&#x201E; ( Ď&#x2030; ) = { t ) T : = max }. Given distribution S ( t ) an investor can establish the level L such that the American option would be exercised before T if = L for t ) T. Otherwise the option would be exercised at T if S ( T ) > K. Denote C A ( t , S ( t ) ; T ) American option price at t. Then C A ( 0 , S ( 0 ) ; T = 2 ) = C E ( 0 , S ( 0 ) ; T = 1 ) )Ď&#x2021;{ S ( 1 ) 3 S ( 2 ) } + + C E ( 0 , S ( 0 ) ; T = 2 )Ď&#x2021;{ S ( 2 ) > S ( 1 ) }

P F . % Q

Then the $-value of the call option contract at date t is (1.3) This formula holds regardless whether the exchange rate q ( t ) is supposed to be stochastic or deterministic. For instance, let N = ÂŁ 31,250, K = $/ ÂŁ 1.50, q ( T ) = $ / ÂŁ1.55.

Then payoff at maturity T is equal to N max { q ( T ) - K , 0 } = ÂŁ 31,250 Ă&#x2014; $ / ÂŁ ( 1.55 â&#x20AC;&#x201C; 1.5 ) = $ 1,562.5 Now we apply for more complex option problem that involves intermediate moment of time with more than 2 states at expiration. Assume that the value of 100 British pounds over three dates 0, 1, 2 are given as follow t=0

3

Note, for example, that when the event { S ( 1 ) S ( 2 ) } is true when annualized rate of return on stock over the period [0, 1] is higher than over the period [0, 2]. The latter formula expresses American option price through European option price and the conclusion that follows from this formula does not coincide with a well-known statement that the current prices of American and European options on no dividend asset are identical. The last formula can be easily extended on multiple steps economy.

In one-step economy, let us brieďŹ&#x201A;y outline the construction of the call option price. Let t and T denote initial moment and option maturity and let underlying values at T and strike price satisfy inequalities: S 1 < S 2 < â&#x20AC;Ś. < S p ) K ) S p + 1 < â&#x20AC;Ś < S n . Then call option premium is deďŹ ned as a random variable

I M . % P

P - .

P - .

j = p + 1 , â&#x20AC;Ś , n. If c 0 is a market price of the option then the risk connected to the price is a chance that realized scenario belongs to the set G>H@ 8 P I M . % 8 Q P - .

where -

is

Q

a solution of the equation

P - % Q. This equation speciďŹ es one- to- one correspondence between S to the option price c. If the value of the underlying at maturity T will be below than S then this scenario is an element of the risky set 1 risk ( c ) associated with the investorâ&#x20AC;&#x2122;s market risk.

We consider now options written on exchange rate. This problem is similar to the problems studied above nevertheless some peculiarities are needed to be speciďŹ ed. We will use cross currency exchange as underlying of the option contracts. Let K denote strike price measured in $ / ÂŁ and q ( t ) denotes $ / ÂŁ - exchange rate at time t. That is ÂŁ1 ( t ) = $ q ( t ) and therefore a ÂŁ1 can be interpreted as a portion of asset that can be sold or bought on $-market. All contracts are settled by delivery of the underlying currency. By deďŹ nition, the contract payoff at maturity T is N max { Q ( T ) â&#x20AC;&#x201C; K , 0 }, where N denotes a contract size. For instance, the size of a British pound call option contract traded on PLHX is N = ÂŁ 31,250. The call option equation (1.2) can be rewritten in the form

t=1 q(1) = 185,

p(180, 185) = 2/3

q(1) = 178,

p(180, 178) = 1/3

q(0) = 180

q(2) = 186 q(2) = 182 q(2) = 181 q(2) = 179 q(2) = 176

t=2 p(185, 186 ) = 1/4 p(178, 182 ) = 1/8 p(178, 181 ) = 1/4 p(185, 179 ) = 3/4 p(178, 176 ) = 5/8

where p ( a, b ) denotes transition probability from the state â&#x20AC;&#x2DC;aâ&#x20AC;&#x2122; to state â&#x20AC;&#x2DC;bâ&#x20AC;&#x2122;. Assume that all transitions are mutually independent. Consider European call option with the strike price K = 180. We begin with calculations of the option price by moving backward in time. Applying the method that we used above over period it is easy to see that P

E

and P

E

E

E

Then

P

E

E

E

E P Q P Q

Here, p (a, b, c) = P { q (0) = a, q (1) = b, q (2) = c }and {a} {b} is the union of two states â&#x20AC;&#x2DC;aâ&#x20AC;&#x2122; and â&#x20AC;&#x2DC;bâ&#x20AC;&#x2122;. We summarize calculations in the table C( 0, 180 )

C ( 1, 5.968 1.956 0.983 0

5.807 1.978 0.994 0

Ď&#x2030;)

C ( 2, 6 2 1 0

Ď&#x2030;)

Ď&#x2030;

p( ) 1/6 1/24 1/12 17/24

The probabilities in the fourth column related to the events in each cell in the row. Now let us investigate a possible investorâ&#x20AC;&#x2122;s strategy. The average return on the exchange rate over 3 4 >H :FJ6A ID

An investor who might interested in calculation of the value of the option price which expected return would be not worse then 1.0148. This price is a solution of the equation E C ( 1 , Ď&#x2030; ) / x = 1.0148. Solving this equation for x yields M

Hence, the premium of 1.14 on call option with strike K = 180 approximately in average promises the return of 1.48% .

The risk of buying option for $1.14 is the probability

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* P Q * 3P

Q P Q4

This might be a high risk for an investor. Note that one can reach an arbitrary high average return by chosen the option price sufďŹ ciently small but the risk of any price will be not less than 17/24. We can use data provided by the latter Table to present calculation for the put option. Consider European put option with the strike K = 182. Then

E

* P

E

E

E

Then

E

*

P

E

EP Q

EP Q P Q

The mean and standard deviation of the put premium are 2.8697, 2.0598 correspondingly. Let for example, investor pays $1 premium for the put option then the risk to receive at expiration less return than invested is 7/24. This loss is associated with the scenario { q (2) = 186, 182, or 181 } If the put premium is $4 then the risk is P [ q ( 2 ) = { 186, 182, 181, 179 }] = 19/24.

2. Exotics options.

In this section, we introduce option pricing formulas for some popular exotic classes. The exotics or non-standard options are those which payoff cannot be reduced to American or European options. They are divided onto two primary classes referred to as to path-dependent and path-independent. Exotic options are generic name of these derivatives. Exotic options are referred to as pathindependent if their payoff does not depend on the path during the lifetime of the option.

Cash-or-nothing options also known as digital or binary options. The call and put digital options are defined by their payoff at maturity as Ccn ( T , q ( T )) = X Ď&#x2021; { q ( T ) > K } Pcn ( T , q ( T )) = X Ď&#x2021; { q ( T ) < K } where X is a predetermined constant and q ( t ) can be interpreted as a spot exchange rate in dollars per unit of foreign currency at time t, t ) T. Note, that in contrast to the continuous payoff of the European or American options the digital options have discontinuous payoff. The constant X is usually assumed equal to 1. The valuation of the options contracts can be represented by the formula

Ď&#x2030;)

Ccn ( 2, 1 1 1 0

Ď&#x2030;)

Ď&#x2030;

p( ) 1/6 1/24 1/12 17/24

Each raw in this table represents a path of the call option for a ďŹ xed scenario Ď&#x2030; 0 = { q ( 0, Ď&#x2030; 0 ), q ( 1, Ď&#x2030; 0 ), q ( 2, Ď&#x2030; 0 ) } and therefore for the ďŹ xed scenario Ď&#x2030; 0 the optionâ&#x20AC;&#x2122;s rates of return coincide with the correspondent rates of return of the underlying exchange rate. Similar class of exotics is assets-or-nothing call and put options payoff at maturity are deďŹ ned as

Pan ( T , q ( T )) = q ( T ) Ď&#x2021; { q ( T ) < K } The pricing formulas can be derived from the general pricing formulas Can ( t , q ( t )) = N q ( t ) Ď&#x2021; { q ( T ) > K } Pan ( t , q ( t )) = N q ( t ) Ď&#x2021; { q ( T ) < K } Gap options are contacts for which European call payoff is be written in the form Cg ( T , q ( T )) = ( q ( T ) â&#x20AC;&#x201C; R ) Ď&#x2021; { q ( T ) > K } where K, R are known constants and K > R. The value of the contracts can be represented by the cash-or-nothing option solution where X = q ( T ) â&#x20AC;&#x201C; R. The gap-put payoff is Pg ( T , q ( T )) = ( R - q ( T )) Ď&#x2021; { q ( T ) < K } where K < R. Then the gap-put pricing formula can be perform by the second formula (2.1) where X = R â&#x20AC;&#x201C; q ( T ).

Paylater options call and put payoffs are deďŹ ned by formulas Cpl ( T , q ( T )) = [ q ( T ) - K - Cpl ( t , q ( t )) ] Ď&#x2021; { q ( T ) > K } (2.2) Ppl ( T , q ( T )) = [ K - q ( T ) - Ppl ( t , q ( t )) ] Ď&#x2021; { q ( T ) < K } where Cpl ( t , q ( t )) , Ppl ( t , q ( t )) are the values of the options at their date of origination date t and paid only on the exercise of the options. These are up-front payments paid at date t. We show that the paylater payoff can be negative. To produce the valuation of the problem one needs to use the benchmark formula (1.3). The solution of this equation when N = 1 can be presented in the form Cpl ( t , q ( t )) =

X Ď&#x2021;{ q ( T ) < K }

Here N is the contract size expressed in foreign currency, K is the strike price, q ( T ) is the currency exchange rate at date T. Let us the numeric example. Assume that the underlying security data is given by the Table on page 7 and N = X = 1. Then using the same algebra one arrives at the table

Cpl ( T , q ( T ))

Bearing in mind formula (1.3) the above equation can be represented in the form Cpl ( t , q ( t )) =

X Ď&#x2021;{ q ( T ) > K } (2.1)

22 â&#x20AC;˘ Discrete Space-Time Options Pricing

Ccn ( 1, 0.9946 0.978 0.9834 0

Can ( T , q ( T )) = q ( T ) Ď&#x2021; { q ( T ) > K }

* P

P cn ( T , q ( T )) = N

0.9677 0.989 0.9945 0

E

and

Ccn ( T , q ( T )) = N

Ccn ( 0, 180 )

[ q ( T ) - K - Cpl ( t , q ( t )) ] Ď&#x2021; { q ( T ) > K }

Solving the equation for Cpl ( t , q ( T )) we arrive at the call paylater option price Cpl ( t , q ( t )) =

(q(T) - K)

Ď&#x2021;{ q ( T ) > K } = =

(2.3) ( q ( T ) - K ) Ď&#x2021;{ q ( T ) > K }

The exotics or non-standard options are those which payoff cannot be reduced to American or European options. They are divided into two primary classes referred to as path-dependent and path-independent.

Ď&#x2021; { q ( T ) ) K 1 } = 1 -Ď&#x2021;{ q ( T ) > K 1 }

Similarly, ( K - q ( T ) ) Ď&#x2021;{ q ( T ) > K }

Ď&#x2021; { q ( T ) ( K1 , K2 ] } = Ď&#x2021; { q ( T ) > K1 } - Ď&#x2021; { q ( T ) > K2 }

In the next Table we enclose the valuation of the paylater call option when underlying is the value of foreign currency unit which value given by the Table on page 7.

one can see that payoff of the collar can be presented as following

Ppl ( t , q ( t )) =

C pl ( 0, 180 ) 0.2.911 0.9944 0.4978 0

C pl ( 1, 2.9919 0.9889 0.4958 0

Ď&#x2030;)

C pl ( 2, 3.089 1.0056 0.5022 0

Ď&#x2030;)

- q ( T ) Ď&#x2021; { q ( T ) > K2 } + K2 Ď&#x2021; { q ( T ) > K2 } = K1 +

Ď&#x2030;

p( ) 1/6 1/24 1/12 17/24

+ [ q ( T ) - K1 ] Ď&#x2021; { q ( T ) > K1 } - [ q ( T ) - K2 ] Ď&#x2021; { q ( T ) > K2 }

Indeed, applying formula (2.3) we see that

P

P

I ( T ) = K1 - K1 Ď&#x2021; { q ( T ) > K1 } + q ( T ) Ď&#x2021; { q ( T ) > K1 } -

T 3 4 E

E 3 4 T

E 3 4 T

E

E

E

E F P

P

E

E

E

E

F P

E

E

T T

F 3 F T % T

4 P F % Q P T T

T T

Note that we omitted for writing simplicity index â&#x20AC;&#x2DC;plâ&#x20AC;&#x2122; that speciďŹ es paylater option. One might note that the risk characteristics of the paylater call option as well as other exotics call option with the same strike price have been introduced above coincide with the correspondent risk characteristics of the standard European option with the same strike price. All these options offered the same return though their premiums and payoffs are different.

A collar contract payoff at maturity T is deďŹ ned by a formula

The right hand side of this equality is equal to a portfolio holding $K1 cash, long European call with the strike price K1 , and short European call with the strike price K2. This decomposition of the collar payoff is not unique. Indeed, one can be easily verify other payoffâ&#x20AC;&#x2122;s representation I ( T ) = K1 + K2 - q ( T ) + [ q ( T ) - K1 ] Ď&#x2021; { q ( T ) > K1 } - [ K2 - q ( T ) ] Ď&#x2021; { q ( T ) < K2 } Thus collar payoff is equivalent now to the value of the portfolio that contains $( K1 + K2 ) cash , short stock , long European call, and short European put. The price of a collar contract at any time prior expiration coincides with the value of the portfolio.

We introduce direct evaluation of the collar contract applying formula (2.4). It follows that the collar payoff (2.4) is the basket of the three hypothetical ďŹ nancial instruments with payoffs at I1 ( T ) = K 1 Ď&#x2021; { q ( T ) ) K1 } I2 ( T ) = q ( T ) Ď&#x2021; { q ( T ) ( K 1 , K 2 ] } I3 ( T ) = K 2 Ď&#x2021; { q ( T ) > K 2 } with the same maturity T. Then the collar contract price at t is I(t) = I1(t) +I2(t) + I3(t),

I ( T ) = min { max { q ( T ) , K1 } , K 2 }}.

Note that this payoff can be rewritten in a more comprehensive form

where

I ( T ) = K1 Ď&#x2021; { q ( T ) ) K1 } + q ( T ) Ď&#x2021; { q ( T ) ) ( K1 , K2 ] } + + K2 Ď&#x2021; { q ( T ) > K2 }

(2.4)

Below we will introduce standard arguments that perform the valuation of the collar contract. Using identities

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A cliquet or ratchet option is a series of at the money options, with periodic settlement, resetting of the strike price at the reset date spot price level, at which the option locks in the difference between the old and new strike prices and pays that difference out as the proďŹ t.

A chooser or as-you-like option is other exotic option type. A holder of this option can choose whether the option is a call or put after specified period of time. An interesting point is that the chooser option payoff does not specify it as call or put options. More accurately, this type of derivatives could be named as a forward-choice options contract. Consider a chooser option that matures at moment Tch , the maturity of the underlying call and put denote Tc , Tp respectively min (Tc , Tp ) > Tch . Thus, the values of underlying call and put at the date Tch are

Therefore, the return to the chooser option can be represented in the form

P

Q

P

Q

P

Q

C ( Tch , q (Tch ) ; Tc , Kc ) and P ( Tch , q (Tch ) ; Tp, Kp ) correspondingly, q ( t ) is the underlying security of the call and put options, and Kc , Kp are the correspondent strike prices. The payoff to the chooser option at maturity Tch is

Note that the payoff can be expressed in the form

+ P ( Tch , q ( Tch ) ; Tp , Kp ) Ď&#x2021; {C ( Tch , q ( Tch ) ; Tc , Kc ) < P ( Tch , q (Tch ) ; Tp , Kp )} Using explicit representation of the call and put prices given by (1.2.1) it is easy to verify equalities

P .8= F .8= .8 %8 * .8= F .8= .E %E Q

P

4 B>C 3

Q P

4 Q

Q

P .8= F .8= .8 %8 * .8= F .8= .E %E Q

P B>C 3

4 B>C 3

4 Q

Q

P

P

Q

Q

Solving this equation, we ďŹ gure out that the value of the chooser option is

Ă&#x2014; Ď&#x2021; { C ( Tch , q ( Tch ) ; Tc , Kc ) * P ( Tch , q (Tch ) ; Tp , Kp ) } +

P B>C 3

Recall that exchange rate q ( * ) is the underlying process for admissible scenarios. Therefore the equation for co ( t , q ) could be presented in the form

co (Tch , q (Tch )) = max { C ( Tch , q (Tch ) ; Tc , Kc ) , P ( Tch , q (Tch ) ; Tp, Kp ) }

co (Tch , q (Tch )) = C ( Tch , q (Tch ) ; Tc , Kc ) Ă&#x2014;

P

8D I F I F I P

P

Q

P

Q

A cliquet or ratchet option is a series of at the money options, with periodic settlement, resetting of the strike price at the reset date spot price level, at which the option locks in the difference between the old and new strike prices and pays that difference out as the proďŹ t. This proďŹ t might be paid out at each reset date or could be accumulated until maturity.

Thus, a cliquet option can be thought, as a series of options that settles periodically at the reset dates is an example of the path-dependent class of options.

Let us introduce a n-years cliquet option with k-resets annually. Let t j i be reset moments of time j = 0, 1, â&#x20AC;Ś , n ; i = 0, 1, â&#x20AC;Śk â&#x20AC;&#x201C; 1 and T denotes maturity. The payoff over the period [ t j i , t j i + 1 ] that is due to paid at the t j i + 1 is max { q ( t j i + 1 ) - q ( t j i ) , 0 }

P F .E %E

Q

Q

P

24 â&#x20AC;˘ Discrete Space-Time Options Pricing

This formula corresponds to the case when option writer pays out periodically at the reset dates. Denote the underlying of the cliquet option q ( s ) = q ( s ; t , x ), s * t and C ( t , x ; T , Q ) the value of the European cliquet call option at date t with strike price Q and expiration date T. Applying IE valuation we arrive at the pricing equation

Hence, C ( tji, q ( t ji ) ; t ji + 1 , q ( t ji ) ) =

3 F I ? > F I ? > 4 P F I ? > F I ? > Q

Using this formula, we can calculate cliquet call value recursively. On the other hand, the present value at t of the stochastic cash flows generated by the series of 1) the initial values of the forward start options and 2) payoff of these options are equal to *0 3 I M . 4

I I ? > I ? > F I ? > I ? > F I ? >

*0 3 * I M . 4

I I ? > I ? > F I ? > I ? > F I ? > T

exchange rates q ( s ) , s * t. An investor buys the ladder option with a strike price Q = Q 0. Thus a ladder start with the height Q and going upwards in the step interval of ε > 0 until the maximum rung of Q N , Q j = Q + j ε , j = 0, 1, … , N. At maturity, T buyer of the ladder call option would receive payoff B6M P F . + + ? + Q P + ?

B6M P F . + + ( + Q P

+ ? Q

+ ( Q

From this formula, follows that the ladder payoff takes into account the maximum value of the underlying price over lifetime of the option. To construct ladder call option price assume that ω { ω : Q j ) < Q j + 1 } for some j .

Then ladder call option payoff realized for this scenario will be equal to C lad ( T , q ( T )) = max { q ( T ) - Q , Q j - Q }

correspondingly. If c is spot price of the option at t then buyer and seller risk can be estimated by probabilities P { C ( t , x , ω ) < c } , P { PC ( t , x , ω ) < c } which are the measure of scenarios when these counterparties pays more than the scenarios provide for.

A Couple option is a similar type of the cliquet options. As for cliquet option, payoff to a holder could take place either at specified reset dates or at maturity. The only distinction between couple and cliquet is that the couple options at reset dates switch its value to the smaller of the current spot level and the initial strike price. The cash flow generated by the couple call option is B6M P F I ? B>C 3 F I ? % 4 Q P I I ? Q

where t = t 0 < t 1 < … < t N = T are reset dates, and min [ q ( t j ) , K ] is reset strike price. The price of the call and put couple options are C cp ( t j , q ( t j ) ; t j + 1 , min [ q ( t j ) , K ] ) = 3 F I ? > B>C 3 F I ? % 4 4 P F I ? > B>C 3 F I ? % 4 Q

P cp ( t j , q ( t j ) ; t j + 1 , min [ q ( t j ) , K ] ) = 3 F I ? > B>C 3 F I ? % 4 4 P F I ? > B>C 3 F I ? % 4 Q

A ladder option payoff is also similar to a cliquet payoff with exception that the gains are locked in when the asset price breaks through certain predetermine rung. The strike price is then intermittently reset. Consider a ladder option on

Therefore the IE rule brings us to the valuation equation

P + ?

+ ? Q

?

S ( T 6C9

P

>; P

+ ( Q

+ ( Q

Thus A69 I F I

P + ?

3

B6M P F . + + ? + Q

+ ? Q B6M P F . + + ( + Q P

+ ( Q4

Remark. Other modification of the ladder call option can be introduced by assuming that call option payoff is defined as following + ? + P + ?

+ ? Q + ( + P + (

Q

In this case in which the payoff is similar to the ladder which admits a finite number of values 0, Q 1 - Q, … , Q N – Q with probabilities P j = P{ Q j ) Q j + 1 }, j = 0, 1, … , N - 1, and P N = P{ > Q N }. The valuation formula in this case can be obtained from the above formula by replacing payoff in the brackets by its modification.

The purchaser of the ladder put will receive at maturity payoff of

» Discrete Space-Time Options Pricing • 25

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* A69 . F .

B6M P + F . + + T ? Q P + T ? T

B6M P + F . + + T ' Q P

+ T ? Q

+ T ' Q

Here, Q – M < Q – M + 1 < …< Q – 1 < Q is a rung sequence.

The pricing equation for the ladder put is

extendible expiration date. The additional premium of $d is paid by the holder in case when extension feature is chosen to exercise at Te . There are new factors involved to the problem. Valuation equation of the call extendible can be represented in the form

P + T ? T

+ T ? Q

P F . : + . : F . : . % 9 Q

? ' ' S 6C9

P

if ω { ω : is then

+ T ' Q

< Q – M }. The value of the put ladder option

* A69 I F I

B6M P + F . + + T ? Q P + T ? T

The indicator on the right hand side of the equality contains union of two events, which signify that at least one of the possibilities at Te should be strictly positive. Otherwise, the value of C eh ( Te , q ( Te )) and C eh ( t , q ( t )) for this particular scenario is 0.

Taking this into account and solving call option price equation we arrive at the premium formula

+ T ? Q := I F I

B6M P + F . + + T ' Q P

+ T ' Q

Extendible options have become popular over recent time for volatile underlying. There are two types of the extendible options: holder and writer extendible. A holder extendible option is an option that can be extended by the holder at option maturity Te. This possibility is required an additional premium. The holder of the extendible option on call or put has a choice to get an ordinary call option payoff or by paying a predetermine premium $d to the writer at time Te to get call option with extended maturity.

Assume that the holder’s choice is based on the maximum value of the option payoffs at Te . That is C eh ( Te , q ( Te )) = max {{ q ( Te ) - Q , 0 } , C ( Te , q ( Te ) ; T , K) -d}= = max { q ( Te ) - Q , C ( Te , q ( Te ) ; T , K ) - d , 0 } , P eh ( Te , q ( Te )) = max { Q - q ( Te ) , P ( Te , q ( Te ) ; T , K ) -d,0} Though for example option buyer at Te may expect over [ Te , T ] to get higher overall return by exercise extendibility than to get call option payoff at Te and investing it at risk free rate.

We do not analyse such possibility.

In above formulas C ( Te , x ; K , T ) , P ( Te , x ; K , T ) denote the price of the European call or put options at date Te with a strike price K that might be equal to Q, and T , T > Te is the

26 • Discrete Space-Time Options Pricing

B6M P F .: + .: F .: . % 9 Q

P 3 F .: + 4 P F .: + B6M 3 .: F .: . % 9 4 Q

3 .: F .: . % T 9 4

P .: F .: . % 9 B6M 3 F .: + 4 Q

The formula for the holder extendible put option can be perform in the similar way * := I F I

B6M P + F .: * .: F .: . % 9 Q

P 3 + F .: 4 P + F .: B6M 3 * .: F .: . % 9 4 Q

3 * .: F .: . % T 9 4

P * .: F .: . % 9 B6M 3 + F .: 4 Q

A reciprocal problem given option price to estimate the value of the premium d is important too.

A writer extendible option allows a seller of the option, option writer to extend the option either call or put with zero cost at the maturity Te if the option is out-of-money. Recall that option call ( put ) is out-of-money at date t if its value at this moment is less ( larger) or equal to the strike price.

Thus, if option have a negative value its can be exercise later at a date T. Therefore, writer extendible payoff of the call and put are equal to C ew ( Te , q ( Te )) = [ q ( Te ) - Q ] χ { q ( Te ) * Q } + [ q ( T ) - Q ] χ { q ( Te ) < Q }

P ew ( Te , q ( Te )) = [ Q - q ( Te )] χ { Q * q ( Te ) } + [ Q - q ( T ) ] χ { Q * q ( Te )} correspondingly. These payoff types give additional benefit to buyers of the call or put options. The first term on the right hand side of the call and put option payoffs, correspond to “in-the-money” scenarios at Te while the second term implies “out-of-the-money” scenarios. By using extended feature does not cost or imply more losses for counterparties. The valuation equation of writer extendible call and put options can be presented in the form =

These derivative contracts can be interpreted as derivatives having variable strikes in contrast to a constant strike used in the previous examples. The pricing formulas to the contracts can be obtained using standard IE pricing rule. Indeed, only two possibilities are available underlying exchange and derivatives. If a scenario ω is such that C e ( T ; Δ ,T 0 ) = 0 then there is no sense to invest in extreme call. If C e ( T ; Δ ,T 0 ) > 0 then there is the unique price to avoid arbitrage. This price is defined as a solution of the equation

P : . I .

Q

Hence, : I .

B6M P

F K

F J Q

Similarly,

> : I .

* : I .

B6M P

F J

B6M P

F K

F K Q

F J Q

* > : I .

These pricing equations bring us to the valuation formulas

Note that European type of the underlying options can also be replaced by American options.

The Extreme or Reverse Extreme exotic options was introduced in 1996. Call extreme options payoff at maturity T is determined by the difference between maximum values on compliment subintervals constituted the lifetime of an underlying asset. Let t < T 0 < T and denote Δ = [ t , T ]. Then payoffs to the call option at maturity for the extreme and inverse extreme options are C e ( T ; Δ ,T 0 ) = max {

q(v) -

q(u), 0 }

C i e ( T ; Δ , T 0 ) = max {

q(u) -

q(v), 0}

The payoffs to put extreme and put inverse extreme options at maturity get the spread value between minimum over adjacent periods, i.e.

P e ( T ; Δ , T 0 ) = max {

q(v) -

q(u), 0 }

P i e ( T ; Δ , T 0 ) = max {

q(u) -

q(v), 0}

B6M P

F J

F K Q

Other path-dependent option class is Lookback options. The extreme exotic options introduced above sometimes are considered as subclass of lookback options and called it extrema lookback options. Two primary forms of the lookback options exist based on strike price definition. First form is defined as lookback options with fixed strike price. The payoffs of the call and put options are A7 . % B6M P

* A7 . % B6M P %

F J % Q F J Q

respectively. Applying the same arguments as for extreme options pricing we arrive at the formulas A7 I %

* A7 I .

B6M P

B6M P %

F J % Q

F J Q

The lookback options with floating strike price can be settled in cash or assets in contrast with the fixed strike options in which cash settlement is only admitted. The payoff of the lookback call and put options with floating strike price are defined as following

» Discrete Space-Time Options Pricing • 27

fsrforum • jaargang 12 • editie #5

A7; . % B6M P F .

F J Q

* A;7 . % B6M P

F J F .

Q

An attractive peculiarity of the lookback options with floating strike price is that they are never out-of-the-money. The formulae representing current options price are A7 I %

* A7 I .

3 F .

3

In the first line, the arithmetic average is used, as the underlying while in the second line formulas the arithmetic mean is used as a strike price. Sometimes, in currency market one applies the inverse mean as underlying. In this case, call - put options payoffs are defined by formulas max { a – 1 ( T ) - K , 0 }

max { K - a – 1 ( T ) , 0 }

,

F J 4

where a – 1 ( T ) is expressed in the same currency as the a ( T ) itself. The pricing formulas are

F J F . 4

Asian options is a popular class of exotics. Underlying of an Asian option is the average price of asset. In many cases, underlying of Asian option has lower volatility than the asset itself. There are three main subclasses of the Asian options which underlying are formed with the help of arithmetic, geometric, or weighted averages of asset. Next specification of the options is that the average can be used for either security or as the strike price. Thus, the payoff for Asian call options can be represented as

% I .

B6M P

F I ? % Q

F I .

B6M P F .

F I ?

Q

* % I .

B6M P %

F I ?

Q

* F I .

B6M P

F I ? F .

Q

Asian put payoff using the arithmetic mean as underlying or strike price can be presented in the form

correspondingly. The American style of the Asian options is also available for trade. The pricing formulas are

The Compound options is a class of derivatives in which underlying securities are options or other type of contingent claims. Consider examples when underlying instruments are options. This class is called compound or split free options. Possible specifications are call options on a call or put options, and put on call or put. Let C ( t , q ( t )) = C ( t , q ( t ); T, K ) denote a value of an European call option at date t with the maturity T and the strike price K written on rate q (*). Consider an option on call option.

Denote C c ( t , q ( t )) = C c ( t , C ( t , q ( t ) ; T c , K c ) ; T, K ) the compound call option price at t written on the European call option at date t with maturity T c , T c ) T with strike price K c . Then the payoff of the compound call option is C c ( T c , C ( T c , q ( T c ) ; Tc , Kc ) ; T, K ) = max { C ( T c , q ( Tc);T,K)– Kc,0}

where q ( t j ) = q ( t j ; t , x ) , j = 0, 1, … , n. For the Asian options with that involve the geometric or weighted averages to obtain valuation formulae one needs replace arithmetic average in the above formulae by their geometric or weighted average counterparts.

The Asian options of the European or American types are path dependent class of exotic options. Underlying of an Asian option is an average price of an asset. The security average price can be used as a strike price too. By comparison with other options its values are less volatile during its life and this is quite attractive for investors. Three types of mean are generally applied for Asian option payoff. These are payoffs form by either arithmetic, weighted arithmetic, or geometric averages 6 .

F I ? L6 .

? F I ? L=:G:

The valuation formula of the call on call option is

Applying twice the IE rule we present the valuation of this equation

?

< . P

- I ? Q

Note that these types of mean can be used as underlying securities as well as a strike price. For example, Asian call and put options payoffs with arithmetic mean could be defined as following max { a ( T ) - K , 0 } , max { K - a ( T ) , 0 } ,

28 • Discrete Space-Time Options Pricing

max { q ( T ) - a ( T ) , 0 } max { a ( T ) - S ( T ) , 0 }

Note that the price of the compound option at t depends on the underlying rate q ( t ) at two future dates Tc < T. Let us consider other types of compound option. The pricing equation of the put written on European call option is

contains N × ₤ ( t ) and other q ( t ) N × $ ( t ). At a moment T , T > t the values of the portfolios will be changed. The initial constant q ( t ) remains unchanged while N $ ( t ) will be transformed in N $ ( T ). The value of the second portfolio will be equal to ₤ N ( T ). As far as these portfolios are not equal at T it looks reasonable to hedge the exchange rate with the help of the derivatives contract which gives an investor the option to choose at maturity T the maximum between $ N ( T ) and ₤N(T)/q(t) = $ N ( T ). For this contract one can specify payoff at T

The solution of this equation can be written as

The pricing of the compound call or put written on European put option can be obtained in a similar way an can be represented in the form

Let us consider a derivative contract that admits a choice between two or more foreign bonds at a future moment of time. This type of the contract called options on maximum or minimum of several risky assets . This class of options is related to the rainbow or chooser options. Rainbow options get their name from the fact that more than one exchange rate. Assume that at maturity T a holder of the contract has the right to choose a bond domestic or foreign. Assume that at initial moment t, the size of two contracts is the same. Let q ( t ) denote indirect quotation of an exchange rate between two currencies at t 1 unit of foreign currency ( at date t ) = q ( t ) units of the domestic currency ( at date t ) The indirect quotation value q – 1 ( t ) shows a number foreign currency units per domestic currency. The value of a government a 0-defualt and 0-coupon bond at t with unit face value is defined by the relationship 1 unit currency ( at date T ) = B ( t , T ) 1 unit currency (at date t ) Recall, that bond value can also be interpreted as a relationship between future and current values of the currency. For example, let us domestic currency is USD and foreign is GBP. Let at initial moment t we have two equal portfolios. One

max { 1 , ₤ 1 ( T ) / ₤ 1 ( t ) } = max { 1 ,

}

For writing simplicity put N = 1. The price of the rainbow contract can be derived as following. For a scenario ω 1 ) 1 , where 1 ) 1 = { ω : ) 1 } the value of the payoff is $1 at T. It implies that at the initial moment t the contract value is B ( t , T ). On the other hand, if a scenario ω belongs to the complimentary set of scenarios 1 >1 = {ω: > 1 } the value of the contract at t can be derived from the condition offered equal return on USA T-bond and the chooser option, i.e. = I

ch ( t ) denotes the max-chooser option value at t. The solution of the equation is Ch ( t , ω ) = . Therefore, we can present the premium value of the option contract = I

P

Q

P

Q

B6M P

Q

A spot market price ch 0 ( t ) implies risk. Buyer risk is connected to scenarios, which imply lower price than paid by the buyer, i.e. { ω : Ch ( t , ω ) < ch 0 ( t ) }, while seller risk is the complementary set of scenarios. Theoretically, the value ch 0 ( t ) is constructed with “cash-and-carry” strategy. This strategy defines a spot price as the present value of the face value at maturity and therefore ch 0 ( t ) = B ( t , T ). Unfortunately, we should remark that in stochastic setting we could not ignore market fluctuations otherwise we will ignore the market risk. In other hand if market has an explicit trend with respect to risk-free then this estimate of the price could be biased too.

The rainbow option class is somewhat similar to the call option. To highlight this similarity one can define a variation of the contract which payoff is

» Discrete Space-Time Options Pricing • 29

fsrforum â&#x20AC;˘ jaargang 12 â&#x20AC;˘ editie #5

B6M P 3 . I 4 Q B6M P

Q

which leads to the contract price = I

P

Q

Generalization of the rainbow options on three or more underlying currencies is straightforward. Denote q i ( t ) the direct quotation of i-th currency, i = 1, 2, â&#x20AC;Ś with respect to domestic US dollar. If the payoff to the option at maturity is chosen as B6M P . F . F I S F C . F C I Q

Then the price in USD of the ( n + 1)-max-rainbow option at date t is

Let us consider the case when n assets and cash are involved in payoff. We will see that the pricing formulas for the contract that deals with the maximum of several stocks differ from the one presented above. It follows from the fact that foreign exchange market instruments can be compared if they have the same currency format and therefore the exchange rates play a signiďŹ cant role in valuations.

Let SĂ ( t ) , â&#x20AC;Ś, S n ( t ) denote price of n â&#x20AC;&#x201C; assets. Assume that the payoff is given at maturity T

B6M P - . S - C . % Q

where K 30 is a constant cash. The reasonable choice at T for the contract is one that suggests the maximum return.

Therefore, the pricing equation can be presented as following

That is Q Ă&#x2014; [ S ( T ; t , S ( t )) ] = Q S ( T ; t , Q S ( t )) It does not depend on a form in which this low is given. The low in particular can be represented by a stochastic or deterministic equation. The solution the pricing equation can be perform in the form

The rainbow with minimum of n risky assets payoff is similar to the best of n assets studied above. In order to present formal valuation of a contract one should replace the â&#x20AC;&#x2DC;maxâ&#x20AC;&#x2122; on â&#x20AC;&#x2DC;minâ&#x20AC;&#x2122; operations in the above formulas.

Other type of the exotic contracts is spread options. The payoff for European calls and puts at maturity T with the strike price K can be written as C sd ( T , S 1 ( T ) , S 2 ( T ) ) = max { S 1 ( T ) - S 2 ( T ) - K , 0 } P sd ( T , S 1 ( T ) , S 2 ( T ) ) = max { K - S 1 ( T ) + S 2 ( T ) , 0 } respectively. The generalization on n-assets in long position and m-assets in short is straightforward. For example H9 . - . - . B6M P

- > .

- ? . % Q

As far as the underlying of the spread option is a linear combination of the assets the price of the call on spread can be expressed by equation

P - . - . % Q

From which it follows that where rainbow ( t ) denotes the contract price at date t. The right hand side of the equation can be simpliďŹ ed. Indeed, putting S i ( T ) = S i ( T ; t , S i ( t )) and taking into account the linear dependence of S i ( T ) on initial value S i ( t ) one can see that the right hand side of the equation above can be rewritten in the form max { S 1 ( T ; t , 1 ) , â&#x20AC;Ś, S n ( T ; t , 1 ) , B â&#x20AC;&#x201C; 1 ( t , T ) } The linear dependence of an asset on the initial value follows from the fact that any portion Q of assets price Q S ( * ) over a period of time is governed by the same low as the single asset.

30 â&#x20AC;˘ Discrete Space-Time Options Pricing

H9 I M N B6M P - . - . % Q

P - . - . % Q

The price of the put spread option can be derived similarly * H9 I M N B6M P % - . - .

Q

P - . - . % Q

Now, we will look at a popular type of exotics options called barrier option. This is a family of the path-dependent. The

value of the barrier option is speciďŹ ed by an event whether the underlying rate crosses a given barrier. There are two different ways of intersections regarded as â&#x20AC;&#x2DC;inâ&#x20AC;&#x2122; or â&#x20AC;&#x2DC;outâ&#x20AC;&#x2122; and two types of the level â&#x20AC;&#x2DC;upâ&#x20AC;&#x2122; or â&#x20AC;&#x2DC;downâ&#x20AC;&#x2122; with respect to the initial value of the spot rate. A double barrier option is a barrier option with two â&#x20AC;&#x2DC;upâ&#x20AC;&#x2122; and â&#x20AC;&#x2DC;downâ&#x20AC;&#x2122; barriers.

The down-and-out ( knock-out) option speciďŹ es a low barrier.

If the spot exchange rate breaches this barrier during the lifetime of the option then the option payoff is equal to 0. In some cases a rebate can also be provided if the barrier is crossed.

Denote d a barrier level, K a strike price, and d < K. The payoff to the down-and-out call option at maturity T is deďŹ ned as 9D . F . B6M P F . T % Q P

F A 9 Q

Let Ď&#x201E; d denote the ďŹ rst moment when process q ( l ), l * t attains the level d. Then with probability 1

P

F A 9 Q 9 .

Let us present for example simple calculations, which illustrate the options pricing. Assume that exchange rate data is deďŹ ned by the Table on page 7 and let K = 180, d = 178. Ď&#x2030;

C do ( 0, ) 5.8064 0 0 0 0

Ď&#x2030;

P do ( 0, ) 0 1.0056 0 0 0

Ď&#x2030;

C di ( 0, ) 0 0 1.978 0.9945 0

Ď&#x2030;

Therefore 9D . F . B6M P F . T % Q 9 .

The down-and-out option price can be constructed applying the standard IE equation

P F . % Q 9 .

which leads to the solution 9D I F I

B6M P F . T % Q 9 .

The payoff to the down-and-out put option is given by the formula * 9D . F . B6M P % F .

Q 9 .

which deďŹ ne its price at the date t * 9D I F I

P di ( 0, ) 0 0 0 0 4.0909

Ď&#x2030;

C do ( 1, ) 5.9677 0 0 0 0

Ď&#x2030;

P do ( 1, ) 0 1.0335 0 0 0

Ď&#x2030;

C di ( 1, ) 0 0 1.956 0.9834 0

Ď&#x2030;

P di ( 1, ) 0 0 0 0 4.0455

C do ( 2, 6 0 0 0 0 P do ( 2, 0 1 0 0 0 C di ( 2, 0 0 2 1 0 P di ( 2, 0 0 0 0 4

Ď&#x2030;)

Ď&#x2030; {180, 185, 186} {180, 185, 179} {180, 178, 182} {180, 178, 181} {180, 178, 176}

Ď&#x2030;)

Ď&#x2030; {180, 185, 186} {180, 185, 179} {180, 178, 182} {180, 178, 181} {180, 178, 176}

Ď&#x2030;)

Ď&#x2030; {180, 185, 186} {180, 185, 179} {180, 178, 182} {180, 178, 181} {180, 178, 176}

Ď&#x2030;)

Ď&#x2030; {180, 185, 186} {180, 185, 179} {180, 178, 182} {180, 178, 181} {180, 178, 176}

Ď&#x2030;

p( ) 1/6 1/2 1/24 1/12 5/24

Ď&#x2030;

p( ) 1/6 1/2 1/24 1/12 5/24

Ď&#x2030;

p( ) 1/6 1/2 1/24 1/12 5/24

Ď&#x2030;

p( ) 1/6 1/2 1/24 1/12 5/24

Let us present a risk analysis of the investment in down-and-in call option. The average option price at date 0 is 1.978 Ă&#x2014; (1/24)+ 0.9945 Ă&#x2014; (1/12) = 0.16529 multiplied by a size of the contract.

If contract size is 1000 units of foreign currency ( ÂŁ ) , then the value of the one contract is $165.29. Assume that investor compares two scenarios in which the prices C di ( 0, Ď&#x2030; ) are a $100 or $200. That is 0.1 or 0.2 per ÂŁ-pound. If the option price is 0.1 then the chance to exercise the option at expiration implies one of two scenarios is realized: {180, 178, 182} or {180, 178, 181}.

The probability of the union of these two favourable events is 1/24 + 1/12 = 1/8. The expected rate of return is equal to [ ( 1 / 24 ) Ă&#x2014; 1.978 + ( 1 / 12 ) Ă&#x2014; 0.9945 â&#x20AC;&#x201C; 0.1 ] / 0.1 = 0.6529

B6M P % T F .

Q 9 .

If the option price is 0.2 then the expected rate of return is about The down-and-in ( knock-in ) call and put options exercise price at maturity T are deďŹ ned as following C di ( T, q ( T )) = max { q ( T ) â&#x20AC;&#x201C; K , 0 } Ď&#x2021; ( Ď&#x201E; d ) T ) P di ( T, q ( T )) = max { K â&#x20AC;&#x201C; q ( T ) , 0 } Ď&#x2021; ( Ď&#x201E; d ) T ) These payouts imply the options price 9> . F .

B6M P F . T % Q 9 .

* 9> . F .

B6M P % T F .

Q 9 .

[ ( 1 / 24 ) Ă&#x2014; 1.978 + ( 1 / 12 ) Ă&#x2014; 0.9945 â&#x20AC;&#x201C; 0.2 ] / 0.2 = - 0.17354 The mean-value analysis does not cover risk exposure. Given spot option price at initial moment t risk is a random variable that represents loss of the investment. The value of the risk is measured by the cumulative loss-distribution. In case when the down-and-in call option price is 0.1 per ÂŁ-contract

P

*

*

*

Âť

Discrete Space-Time Options Pricing â&#x20AC;˘ 31

fsrforum â&#x20AC;˘ jaargang 12 â&#x20AC;˘ editie #5

One can transform this pricing representation of the risk into equivalent presentation forms by the rate of return.

Now let us look at the next type of barrier option in which the â&#x20AC;&#x2DC;upâ&#x20AC;&#x2122; barrier is speciďŹ ed. If the spot exchange rate goes above the â&#x20AC;&#x2DC;upâ&#x20AC;&#x2122; barrier the up-and-out option ceases to exist. A rebate that should be speciďŹ ed at initiation of the contract may also be provided as the barrier is crossed. The payoff to the up-andout call or put options at maturity T is deďŹ ned by the formulas C uo ( T , q ( T )) = max { q ( T ) â&#x20AC;&#x201C; K , 0 } Ď&#x2021; ( θ u > T ) P uo ( T , q ( T )) = max { K â&#x20AC;&#x201C; q ( T ) , 0 } Ď&#x2021; ( θ u > T )

The valuation of the up-and-in barrier option is similar to the represented above.

Let us consider a double barrier pricing scheme for the case when K = 180, u = 185, d = 178. The payoff to the double-out barrier call and put options at maturity are deďŹ ned as C dbo ( T, q ( T )) = max { q ( T ) â&#x20AC;&#x201C; K , 0 } Ď&#x2021; { d < q(l)<u}

q(l),

P dbo ( T, q ( T )) = max { K â&#x20AC;&#x201C; q ( T ) , 0 } Ď&#x2021; { d < q(l)<u}

q(l),

respectively. The random time θ u is deďŹ ned as following Then payoff to the double-in barrier call and put options at maturity is

θ u = min { l : q ( l ) * u , l [ t , T ] } The price of the up-and-out call or put options at t are C uo ( t , q ( t )) =

max { q ( T ) â&#x20AC;&#x201C; K , 0 } Ď&#x2021; ( θ u > T )

P uo ( t , q ( t )) =

max { K â&#x20AC;&#x201C; q ( T ) , 0 } Ď&#x2021; ( θ u > T )

The payoff to the up-and-in call, put options at maturity T can be represented in the form C ui ( T, q ( T )) = max { q ( T ) â&#x20AC;&#x201C; K , 0 } Ď&#x2021; ( θ u ) T ) P ui ( T, q ( T )) = max { K â&#x20AC;&#x201C; q ( T ) , 0 } Ď&#x2021; ( θ u ) T )

C dbi ( T, q ( T )) = max { q ( T ) â&#x20AC;&#x201C; K , 0 } Ď&#x2021; { d * q(l)*u}

q(l),

P dbi ( T, q ( T )) = max { K â&#x20AC;&#x201C; q ( T ) , 0 } Ď&#x2021; { d * q(l)*u }

q(l),

Pricing formulas for these barrier options are C dbo ( T , q ( T )) = q(l)< u}

max { q ( T ) â&#x20AC;&#x201C; K , 0 } Ď&#x2021; { d <

q(l),

P dbo ( T , q ( T )) = q(l)<u }

max { K â&#x20AC;&#x201C; q ( T ) , 0 } Ď&#x2021; { d <

q(l),

C dbi ( T , q ( T )) = q(l) * u}

max { q ( T ) â&#x20AC;&#x201C; K , 0 } Ď&#x2021; { d *

q(l),

P dbi ( T , q ( T )) = q(l) *u}

max { K â&#x20AC;&#x201C; q ( T ) , 0 } Ď&#x2021; { d *

q(l),

Therefore C ui ( t , q ( t )) =

max { q ( T ) â&#x20AC;&#x201C; K , 0 } Ď&#x2021; ( θ u ) T )

P ui ( t , q ( t )) =

max { K â&#x20AC;&#x201C; q ( T ) , 0 } Ď&#x2021; ( θ u ) T )

Assuming that the underlying exchange rate given in Table 7 and K = 180, u = 185 C uo ( 0, Ď&#x2030; ) C uo ( 1, Ď&#x2030; ) C uo ( 2, Ď&#x2030; ) Ď&#x2030; p(Ď&#x2030;)

E C uo ( 0, Ď&#x2030; ) Ď&#x2021; { C uo ( 0, Ď&#x2030; ) < option market price ( 0 ) }

Final Remark. Here we present a short comment to the binomial option pricing. This discrete space-time approach presents option price using a standard algebraic methods.

Technically, binomial scheme in one period cannot present straight forward solution of the pricing problem with arbitrary ďŹ nite set of states. On the other hand this formal deďŹ nition has obvious logical drawback.

Let stock price at date t = 1 be S ( 1 ) = $2 and by the end of the single period the security price is either S u ( 2 ) = $4 or S d ( 2 ) = $1 and strike price K = $2 and for simplicity let the risk-free interest rate r = 0. Let us recall construction of the binomial scheme. It determines option price with two steps. Consider a call option example. On the ďŹ rst step the hedge ratio h of the hypothetical portfolio in the form Đ&#x; ( t ) = S ( t ) â&#x20AC;&#x201C; h C ( t ), t = 1,2 is established. The condition that used for the solution of the problem is (3.1)

E C uo ( 0, Ď&#x2030; ) Ď&#x2021; { C uo ( 0, Ď&#x2030; ) > option market price ( 0 ) }

Đ&#x;( 2 ) = S u ( 2 ) â&#x20AC;&#x201C; h C u ( 2 ) = S d ( 2 ) â&#x20AC;&#x201C; h C d ( 2 )

Thus, the complete option price data should be supplied by the risk characteristics, which can be established assuming a particular distribution of the underlying. The higher order moments of the option pricing give us more details that are more accurate represent risk exposure. Analogously, one can see that

where C u and C d are the call option payoffs corresponding two outcomes S u and S d respectively. Thus

0 0 1.978 0.9945 0

0 0 1.956 0.9834 0

0 0 2 1 0

{180, 185, 186} {180, 185, 179} {180, 178, 182} {180, 178, 181} {180, 178, 176}

1/6 1/2 1/24 1/12 5/24

Hence,

JD P

E E E

If market price of the option is equal to E C uo ( 0, Ď&#x2030; ) = 0.1653 then the risk is described by the set of scenarios D = {180, 185, 186} {180, 185, 179} {180, 178, 176} which correspond to 0 payoff. The value of risk is 21/24, which is the probability of the D. In general the average loss and average proďŹ t are

Ď&#x2030;

P uo ( 0, ) 0 1.0056 0 0 4.0909

32 â&#x20AC;˘ Discrete Space-Time Options Pricing

Ď&#x2030;

P uo ( 1, ) 0 1.0335 0 0 4.0455

P uo ( 2, 0 1 0 0 4

Ď&#x2030;)

Ď&#x2030; {180, 185, 186} {180, 185, 179} {180, 178, 182} {180, 178, 181} {180, 178, 176}

Ď&#x2030;

p( ) 1/6 1/2 1/24 1/12 5/24

C u ( 2 ) = max { 4 â&#x20AC;&#x201C; 2 , 0 } = 2 C d ( 2 ) = max { 1 â&#x20AC;&#x201C; 2 , 0 } = 0 From (3.1) it follows that h = 1.5. The second step leads to the option price. It follows from (3.1) that the portfolio value at date 2 is deterministic and equal to

П ( 2 ) = 4 – 1.5 * 2 = 1 – 1.5 * 0 = 1 By construction it does not depend on scenarios “up” or “down” at maturity. Therefore, the change in value of the portfolio should follow the risk-free rate. That is applying simple interest rate we have П( 2 ) = ( 1 + r ) П ( 1 ) As far as the risk-free interest rate r was assumed to be equal to 0 then П ( 2 ) = П ( 1 ) = 1 = S ( 1 ) – 1.5 C ( 1 ) Hence C ( 1 ) = 2/3. This is the theoretical option price suggested by the binomial scheme.

Let us test the theoretical solution against particular scenarios. Assume that two securities differ by the probabilities of the states at maturity. Let the probability of the state ‘4’ is for the first security equal to 0.99 and for the second security the state ‘4’ probability is equal to 0.01. The securities expected rate of returns are [ 4 * 0.99 + 1* 0.01 – 2 ] : 2 = 98.5% [ 4 * 0.01 + 1* 0.99 – 2 ] : 2 = – 48.5% correspondingly. According to binomial scheme the unique option price for either of these securities is the same 2/3. In other words binomial scheme does not sensitive with respect to underlying security rates return. One can note that selling option on bad stock and buying option on good stock an investor starts with zero financing. Then at the end of the period the investor has 1 chance to loss premium while in 99% the investor receives a profit. The investment of this type one can call statistical arbitrage. A curious fact is that the same price is established on two option-investments with different expected rates of return. Indeed, buying first option for 2/3 and given expected rate of return is [ ( 4 * 0.99 + 1* 0.01 ) – 2/3 ] : 2/3 = 4.925 On the other hand, the second option suggests expected rate of return [ ( 4 * 0.01 + 1* 0.99 ) – 2/3 ] : 2/3 = 0.537 This observation contradicts the common sense and theoretical understanding of the price in Finance. The common argument usually used to justify the binomial option price is its non-arbitrage pricing. This argument is a necessary condition of the correct pricing and it can also be true if the option price is determined incorrectly. That is there is no arbitrage between binomial pricing of the options and risk free financing. The no-arbitrage fact does not sufficient to accept suggested construction as a theoretical definition of the price. Having theoretically correct the option price definition one can apply it for the solution of the real world pricing problems.

Discrete Space-Time Options Pricing • 33

fsrforum • jaargang 12 • editie #5

DN/SB; het vertrouwen voorbij

K(r)anttekening | Drs. Joost Groeneveld RA RV

De opdracht voor de Commissie, die naar haar voorzitter de Commissie Scheltema is genoemd, is gefocussed op “... de gang van zaken bij DSB Bank N.V., ... de handelwijze van DNB en de AFM ten aanzien van DSB Bank N.V., de onderlinge samenwerking tussen DNB en AFM, ...”. Het onderzoek heeft een rapport opgeleverd van bijna 300 bladzijden + 13 bijlagen. Dat zal best aan de DSB zijn besteed. Veel mensen zijn daar hun geld kwijtgeraakt, zowel vóór als door de teloorgang van de bank die geen bank had mogen zijn.

Je kunt DS niet kwalijk nemen dat hij bank wilde zijn. Het extra vertrouwen bij het publiek zou hem groei bezorgen en een risicopremie besparen. Wèl kwalijk te nemen is dat DS niet de prijs wilde betalen die daar tegenover staat: in de vorm van (extra) in acht te nemen voorwaarden. De Neder-

tegenwoordig ‘niemand’ meer lijkt te kunnen vertrouwen: aan de top van de vertrouwenspiramide zou - na de spreekwoordelijke Bank van Engeland - toch De Nederlandsche Bank moeten staan. Maar daar staat ze niet. En zo mogelijk nog erger: bestaand wantrouwen is terecht gebleken. Niet doordat het bankwezen tegenwoordig zo ingewikkeld is. Als dat de oorzaak van het falen zou zijn, zou De Nederlandsche Bank zich daarmee hebben kunnen verontschuldigen. Je moet er niet aan denken wat er dan wèl is mis gegaan. Domheid? Onoplettendheid? Samenzwering? Vriendelijkheid?

Lafheid? Slapheid? Het heeft tot gevolg gehad dat de B van bank geen waarborg meer biedt.

Het vertrouwen in de banken is weg. Niet alleen bij het publiek maar ook tussen de banken onderling. De Nederland-

Dat vertrouwen moet worden hersteld En dat schijnt steeds maar niet te lukken. Natuurlijk niet.

Drs. Joost G. Groeneveld RA RV is directeur van Wingman Business Valuators B.V. te Breda en voorzitter van de Stichting WBO (register van business valuators).

Hij was hoofddocent aan de Economische Faculteit van de Erasmus Universiteit te Rotterdam.

landsche Bank had op voldoening moeten toezien en dat is in onvoldoende mate gebeurd. DNB heeft DS die kans geboden. En DS heeft die gegrepen.

Het is boeiend om je voor te stellen hoe de wereld er zou hebben uitgezien wanneer het bij DS zonder B was gebleven. De B biedt het publiek bescherming, althans in opzet. Het is een predikaat. Zoiets als hofleverancier. Aan toekenning gaat een onderzoek vooraf en je zult wel aan voorwaarden moeten blijven voldoen omdat je anders je bankstatus kunt verliezen. Dus een bank is niet zomaar niets. Een bank heeft prestige. Met een bank kun je zaken doen. Het hoge woord moet er uit: je kunt een bank vertrouwen. Veel mensen hebben dat gedaan. De veronderstelling is gewettigd dat veel mensen dit zouden hebben nagelaten als DS geen bank zou zijn geweest. Zij zijn niet alleen de risicopremie misgelopen die nu door DSB is opgestreken, maar hebben ten slotte ook hun hoofdsommen moeten achterlaten.

Het aanzienlijke financiële verlies is ontwrichtend. Dat hakt er in. Niet alleen door het totale aantal miljarden Euro’s, maar ook door de spreiding daarvan over zoveel directbetrokkenen. Te meer ontwrichtend door het feit dat je

34 • DN/SB; het vertrouwen voorbij

sche Bank zou daarin een sleutelrol (hebben) moeten spelen. Die rol is er nog, maar de sleutel is kwijt. Die bestaat namelijk uit hetzelfde vertrouwen dat is verspeeld. Dat vertrouwen moet worden hersteld. En dat schijnt maar steeds niet te lukken. Natuurlijk niet. Welke maatregelen zijn nu echt genomen? En door wie? En wanneer? In een achterkamertje? Daar vindt het publiek alleen maar meer achterdocht. De redenering is denk ik – omgekeerd - geweest dat met herstel ook het vertrouwen komt. En dan hoef je niets te doen, behalve te wachten op dat herstel.

Dan gaat alles vanzelf? Het tegendeel is waar. Het is een krachtmeting. Inmiddels weet iedereen wie hebben gewonnen en wie verloren. De Nederlandsche Bank hoort bij de laatsten. Maar kennelijk is er volgens de Minister van Financiën nog geen sprake van een ‘knock out’. Er komt nog een volgende ronde. President Wellink krijgt nog een maand. Pijnlijk om te zien. De Minister zou zijn kampioen juist de hand boven het hoofd houden door diens handdoek in de ring te gooien. Maar kennelijk moeten we de gehele ontluistering meemaken. Hoopt de Minister dat er over een maand een nieuwe regering is?

Het is voor een beleidsmaker goed om de blik op de toekomst te richten. Dat wil niet zeggen dat aan het verleden

mag worden voorbij gegaan. Een planningcyclus kent immers drie fasen: beleid - uitvoering en beheer - verantwoording. Er moet op evenwichtige wijze aandacht zijn voor elke fase. Verantwoording houdt niet alleen in dat schuld wordt bekend; dat het hoofd ootmoedig wordt gebogen; dat spijt wordt betuigd, en ja, zo had het niet moeten gaan. Ja, er zijn fouten gemaakt; ja, toen al. Ja, eigenlijk hebben we daar steeds in volhard. Het was een bank en het bleef een bank. Ja, alles moet anders! Ja, dat is al jaren zo. Ja, sorry! Maar geef me een maand, en de bank zal gezond worden. Of woorden van gelijke strekking.

Dat kun je toch niet serieus menen/nemen? Volgens mij houdt verantwoordelijkheid in dat bij de verantwoordelijke gehoudenheid bestaat tot het beantwoorden van bepaalde vragen van bepaalde personen of instanties op bepaalde momenten. Serieuze verantwoording berust op systematisch onderzoek. De Commissie Scheltema heeft zulk onderzoek verricht.

Daarbij is gebleken dat de verantwoordelijke ernstige fouten heeft gemaakt. De antwoorden maakten dat duidelijk. En dan ontstaat een ragfijn spel. Zadel hem op met een onmogelijke opdracht en dan zal hij wel voor de eer bedanken. Maar dat doet hij niet! Hij belooft beter-

Maar geef me een maand, en de bank zal gezond worden. schap. En hij mag doorgaan. Daarmee nemen we als het ware het vertrouwensverlies voor lief. Want lief is het wèl. En ja, het gaat om de toekomst. Daarmee botst hier beleid met verantwoording. Beleid krijgt voorrang. Wat stelt verantwoording dan nog voor?

Er zal wel weer meer en beter toezicht komen. Dat wil niet zeggen dat de verantwoordelijke daarmee ons vertrouwen toch waard is. We leggen ons er bij neer dat dit niet het geval zal zijn. Meer toezicht wil alleen maar zeggen dat fouten eerder kunnen worden gesignaleerd en gerepareerd. De functionarissen blijven; het systeem wordt beveiligd. Het mooiste is natuurlijk preventief toezicht. Het beste preventieve toezicht is de zaak zo te organiseren dat de verantwoordelijke zelfs zonder de lange arm van de toezichthouder zou kunnen worden vertrouwd. Door hun bestaan maken al die toezichthouders duidelijk dat vertrouwen - waar het bestaat - echt niet genoeg is. En de toezichthouders dan? Kun je die vertrouwen? Moet dat dan? Daar zet je dan toch ook een toezichthouder bij. Geeft niets; zolang het een klein clubje is van mensen die elkaar goed kennen, is er ook met al dat toezicht niets aan de hand.

Het einde is een systeem waarin vertrouwen overbodig is gemaakt. Waar de noodzaak ontbreekt om vertrouwen te hebben in elkaar en in jezelf. Het vertrouwen voorbij. Dat lijkt me geen plezierige omgeving.

DN/SB; het vertrouwen voorbij • 35

A U D I T TA X A DV I S O R Y

komt eraan ga naar www.gaaan.nu

fsrforum • jaargang 12 • editie #5

Ook al vereenvoudig je de wereld, het blijft ingewikkeld Interview president van De Nederlandsche Bank Nout Wellink (9 juli 2010) Door: Karin Knegt en Kim de Vries

Jobhoppen is hot: de gemiddelde werknemer blijft steeds korter bij zijn werkgever. Bij elke switch worden nieuwe indrukken en ervaringen opgedaan en wordt het pallet van de carrièretijger weer met nieuwe kleuren aangevuld.

Creativiteit en oplossingsgerichtheid worden gestimuleerd met nieuwe ideeën geboren uit opgedane ervaringen.

Als er meerdere Nout Wellinks in Nederland rondliepen, zou de gemiddelde arbeidsduur ongetwijfeld een stuk hoger liggen. Maar er is maar één iemand in Nederland met een visitekaartje waar de functie van President der Nederlandsche Bank op prijkt. En dat al dertien jaar lang; Nout Wellink.

Begonnen met Nederlands Recht aan de Universiteit van Leiden en vervolgens gepromoveerd tot doctor in de economie aan de Erasmus Universiteit Rotterdam. Nadat hij twaalf jaar werkzaam is geweest bij het Ministerie van Financien, wordt hij in 1982 directielid bij De Nederlandsche Bank.

Als opvolger van Wim Duisenberg is Wellink per 1 juli 1997 president bij datzelfde instituut.

Op een zomerse vrijdagmiddag reizen wij af naar het Amsterdamse hoofdkantoor van DNB aan het Frederiksplein voor een vraaggesprek met de man die al geruime tijd het vuur aan de schenen krijgt gelegd.

Heeft Wellink lessen geleerd uit het echec rondom de bankencrisis, of houdt hij stug vast aan het gevoerde beleid? Vindt hij dat hij medeverantwoordelijk is, of is zijn bevoegdheid als president daarvoor te beperkt?

Wat als eerste opvalt wanneer wij zijn statige kantoor worden binnengeleid, is dat wij tegenover één van de belangrijkste mannen van de Nederlandse financiële wereld zitten. En dat straalt hij ook uit.

Rapport Scheltema De eerste vraag gaat natuurlijk over het rapport Scheltema1 dat afgelopen week is uitgebracht en waarin veel kritiek wordt geuit op de gedragingen van de toezichthouder.

Wellink benadrukt nogmaals dat hij de conclusies van het rapport genuanceerd onderschrijft. “Ik ben nog steeds van mening dat

de toezichthouder een vergunning had moeten verstrekken aan de DSB Bank, maar dat het misschien verstandig zou zijn geweest om nog wat additionele voorwaarden aan deze DGA-constructie toe te voegen”. Maar de vergunning was en is volgens hem een verbetering ten opzichte van de vorige situatie2.

“Ik denk nog steeds dat onze toezichthouder, Wellink distantieert zich hier even, in redelijkheid die vergunning heeft kunnen afgeven”. Hij is stellig. “Ik heb de verontschuldigingen aangeboden voor zover het handelen betreft dat boven een redelijk handelende toezichthouder uitgaat”.

De nuancering in zijn spijtbetuiging komt echter al snel aan het licht. Zijn houvast zoekt hij in het Vie d”Or arrest3, in een vrije vertaling legt hij ons uit wat daarin wordt gezegd door de Hoge Raad over een redelijk handelend toezichthouder: “En als je dan in redelijkheid tot die beslissing bent gekomen, ja dan doet het er eigenlijk niet meer zo vreselijk veel toe hoe het afloopt”.

De koppeling in de media – met de achterliggende gedachte een verkeerde werkwijze en/of cultuur bij de bank aan te tonen – van het Rapport Scheltema met de zaken ABN Amro en IceSave vindt Wellink volstrekt onterecht.

Buitenlandse collega”s reageren volgens de centrale bank president geschokt als hij ze vertelt over de beschuldigingen die aan zijn adres plaatsvinden. In een gesprek met zijn Engelse collega, zo deelt hij ons mede, wordt dan ook verontwaardigd gereageerd als hij vertelt dat hij in Nederland verantwoordelijk wordt gehouden voor de perikelen rondom IceSave4. Zijn Engelse collega zou dan ook gereageerd hebben met: “Wat zeg je nou? Dat was toch een branche? Daar ben je niet verantwoordelijk voor!” “Ja, dat moet je hier maar eens komen vertellen”, zei Wellink.

1

In haar rapport doet de commissie Scheltema verslag van haar onderzoek naar de ondergang van de DSB Bank.

2

De vergunning waar het in het rapport Scheltema om gaat, stond toe de onderliggende bedrijven te fuseren met de DSB Bank opdat DNB toezicht zou

hebben op het geheel, aldus Wellink.

3

HR 13 oktober 2006, LJN: AW2077, C04/279HR (De Nederlandsche Bank/Stichting Vie d”Or).

4

DNB was verantwoordelijk voor het toezicht op het Amsterdamse bijkantoor van de IJslandse bank Landsbanki, dat via internet onder de naam IceSave

hoge spaarrentes aanbood.

5

Hier wordt gerefereerd aan de overnamestrijd rondom ABN Amro, dat uiteindelijk in de handen viel van Santander, Fortis en Royal Bank of Schotland.

38 • Interview president van De Nederlandsche Bank Nout Wellink

In de zaak ABN Amro5, dat vindt hij wel “een beetje gek”, waren zij de enige die tegen waren. “Ons is door de hele wereld verteld dat je een vergunning moet afgeven tenzij je kunt bewijzen dat het onverantwoord is. Wij voelden ons ongemakkelijk en wij

hebben ontzettend zware voorwaarden aan die vergunning opgelegd. Ik vind dat men dat hier fantastisch gedaan heeft”.

Poldermodel of niet, De Nederlandsche Bank pleit wel voor een beperking van haar aansprakelijkheid.

Dat de voorwaarden uiteindelijk nog niet zwaar genoeg waren, wijt Wellink aan de kredietcrisis. Volledig conform de huidig geldende Balkenende-norm voor (verkapte) excuses: “Maar als

6 The Basel II Framework describes a

je toen geweten had dat er een kredietcrisis was gekomen, had je

systeem dat hij samen met het Basel Comité aan het ontwikkelen is. “Dus voordat we kijken of er ruimte is voor de belasting en of het zinvol is, laten we nu eerst eens het regelgevingsysteem goedmaken”.

het niet gedaan”.

“Wel kan de kredietverlening in gevaar komen wanneer je te ver

cy that national supervisory authorities

gaat met het belasten van banken”. Wellink illustreert dit door

are now working to implement through

Wellink lijkt op meer begrip van buiten de landsgrenzen dan in Nederland te kunnen rekenen. Hij lijkt zijn beleid in het buitenland harder te verdedigen dan in eigen land: “Zij herinneren zich nog dat ik de hele wereld over ben getrokken om steun voor ons standpunt te krijgen en dat ook zij tot degenen behoorden die zeiden: “laat de markt zijn werk doen”. Terwijl ik in de Financial Times en andere bladen heb gezegd: “Jongens, pas nu op! Als het niet goed gaat, is het tax payers money.

Verabsoluteer nu niet de rol van de aandeelhouder”!

Hij concludeert zijn betoog: “Het zijn dus volstrekt verschillende

een systeembank met een werknemer te vergelijken. “Je gaat ook niet, als iemand door eigen schuld werkloos is geworden, voor straf zijn belasting verhogen. Dan ga je eerst proberen om hem weer aan een baan te helpen en zo moet je eerst zorgen dat banken weer behoorlijk gaan functioneren”.

We worden er tevens aan herinnerd dat geen organisatie perfect is en dat de discussie die wij nu voeren uniek is in de wereld. Kijken wij wel op de juiste manier naar dit soort zaken? Je moet hier lessen uit leren. Zowel de toezichthouders als de politici.

“Als men zegt “jullie zijn niet daadkrachtig genoeg geweest”, dan zeg ik “waar waren dan de politici toen wij vroegen om afschaffing van de tophypotheken? DSB had nooit bestaan als tophypotheken verboden waren geweest”.

“Nu wil ik de politici helemaal niet beschuldigen, ik noem een hele hoop dingen die zij hadden moeten doen. Ik vind het flauw om ze daar te zeer voor verantwoordelijk te stellen, want het leven is veel ingewikkelder. Maar je moet je wel realiseren dat je met z’n allen verantwoordelijk bent in deze samenleving”.

minimum standard for capital adequa-

domestic rule-making and adoption procedures. It seeks to improve on the existing rules by aligning regulatory capital requirements more closely to the underlying risks that banks face. In addition, the Basel II Framework is intended to promote a more forwardlooking approach to capital super-

Wellink meent dat er voor een toezichthouder niets zo lastig is om te

vision, one that encourages banks to

besluiten dat je een bank toch laat omvallen. Want zo zegt hij: “als

identify the risks they may face, today and in the future, and to develop or

DSB niet was omgevallen, waren we van een hoop gedonder af

improve their ability to manage those

geweest”.

risks (definition taken from BIS website: http://www.bis.org/publ/bcbsca.htm).

gevallen, je kunt daaruit geen lijn trekken. Ik vind dan ook dat men met grote zevenmijlslaarzen aan het rondlopen is”.

more comprehensive measure and

“Banken zullen in onze economische orde altijd failliet blijven gaan. Er moet altijd een kans blijven dat een bank het niet overleefd”.

7 Wellink legt ons de regels van Basel III zelf als volgt uit: I) Het minimum kapitaal dat een bank moet aanhouden

“Eigenlijk zie je dat na elk ongeluk er meer regelgeving komt en regelgeving kan weer leiden tot het ontwijken van die regelgeving en zo blijf je maar tegen elkaar opbieden om elkaar dan het leven zuur te maken. Te veel regelgeving leidt er toe dat dingen buiten het officiële toezichtcircuit plaats gaan vinden en dat moet je niet hebben. Dat is het risico van dit alles”.

komt hoger te liggen dan in het verleden en we zorgen ook dat het kapitaal van betere kwaliteit is; II) daarnaast moet er een additionele buffer worden aangehouden die is gekoppeld aan de situatie bij een specifieke bank. In de periode dat die buffer wordt opgebouwd mag die bank minder vrijgevig zijn met z’n dividend en beloningsbeleid; III) en

Basel II en III

daarboven komt nog een buffer die

Naast zijn voorzitterschap van De Nederlandsche Bank en het

met de conjunctuur samenhangt. Met

daaraan verbonden lidmaatschap van de raad van de Europese Centrale Bank, is Nout Wellink ook voorzitter van het Basel Comité.

Wij willen dan graag ook weten of hij denkt dat Basel II6 meer effect zou hebben gehad wanneer het eerder was ingevoerd.

de nieuwe regels van Basel III worden de banken volgens hem dan zo gedwongen om, als het heel hard loopt in de economie, een extra kapitaalbuffer op te bouwen.

“De brokken waren in elk geval minder groot geweest”, zo stelt hij.

Poldermodel of niet, De Nederlandsche Bank pleit wel voor een beperking van haar aansprakelijkheid. “Heel eenvoudig, om de

doodgewone reden dat ik niet eens kan zeggen wat ik er van vind, want voor ik het weet heb ik een claim aan de broek hangen omdat ik heb toegegeven dat wij fouten hebben gemaakt”.

Daarnaast beaamt hij dat de regels van Basel III7 wel hun gewenste effect zullen hebben. “Zij dwingen namelijk een hoger kapitaal over de hele linie af. En dat betekent dat ze toch meer buffers zullen hebben”.

Bij het vaststellen van een invoeringsdatum voor Basel III is wel rekening gehouden met de druk die op de kredietverlening komt te

Bankenbelasting

liggen. “We hebben gekeken naar de impact die de verhoogde

Hoewel er vanuit De Nederlandsche Bank al geluiden kwamen dat de bankenbelasting niet geschikt zou zijn voor de Nederlandse economie, wil Wellink daar niet echt een mening over geven. Een alternatief ervoor heeft hij echter ook niet. Maar de bankenbelasting past naar zijn mening niet goed in het

kapitaal en liquiditeitseisen hebben op de economie. Uiteindelijk zijn we tot de conclusie gekomen dat die in de overgangsperiode een half procent lager zal komen te liggen, en in de uiteindelijke evenwichtssituatie 0,2 procent lager. Dat betekent dat de invloed op de economie van dit alles nog wel meevalt”.

»

Interview president van De Nederlandsche Bank Nout Wellink • 39

fsrforum • jaargang 12 • editie #5

De tweede termijn van Wellink loopt in 2011 af en er gaan geruchten dat hij de compromis kandidaat zou zijn in de opvolging van Trichet.

Europees financieel toezicht Wanneer het onderwerp Europees toezicht wordt aangekaart, valt er een verandering in de houding van Wellink te bespeuren. Waar hij tijdens het eerste kwartier nog redelijk terughoudend was en veel met zijn wenkbrauwen fronste, toont hij zich nu actief en enthousiast.

“Binnen het Europees financieel toezicht wordt een aantal veranderingen8 doorgevoerd. Zo wordt er een macro-prudentiële toezichthouder in het leven geroepen die zich bezighoudt met de relatie tussen de instelling en het financiële stelsel en de relatie tussen het financiële stelsel en de macro economie.

Daarnaast komt er nog meer micro toezicht op individuele instellingen”.

Krijgt de nieuwe Europese toezichthouder een preventieve rol of treedt deze eigenlijk pas in werking tijdens een crisis? “De ervaring die tijdens de crisis is opgedaan, is eigenlijk geweest dat er toch te weinig informatie uitwisseling tussen de toezichthouders in verschillende landen is geweest. Maar in het geval van een crisis ligt de verantwoordelijkheid volgens Wellink toch primair bij de overheid. “Want overheden nemen dat stuk van verantwoordelijkheden dan over”.

“Onze ervaring naar aanleiding van de Fortis affaire en IceSave was om meer voortgang te maken met het Europees toezicht.

Men zegt wel eens: “jullie willen grote instellingen, want dan blijven jullie belangrijk”. Je kunt alles verzinnen in deze wereld, daar trappen mensen dan in. Wij waren de eerste voorstander in de EU van wat je noemt een lead supervisor. Deze zou dan bevoegdheden krijgen die over je eigen bevoegdheden heen gaan. Wij zijn altijd bereid geweest om mee te werken aan het overdragen van soevereiniteit”.

“Als bank hebben wij hier al langer een duidelijk standpunt over ingenomen. Ik vind het een beetje jammer dat er niet meer voortgang is geboekt. Het had van mij verder mogen gaan”.

Euro en de Europese Monetaire Unie Ondanks alle perikelen is Wellink van mening dat we de Euro in elk geval niet te vroeg hebben ingevoerd. De koersen van de kern-

landen waren volgens hem al sinds 1986 aan elkaar gekoppeld.

“Je zou kunnen zeggen, dat was al een semi-Euro. Er was de facto al een monetaire unie tussen de zes9“.

8

Wellink heeft het hier over de invoering van een European Systematic Risk Board (ESRB) en

European System of Financial Supervisors (ESFS).

9

Het gaat hier om de zes landen (te weten België, Duitsland, Frankrijk, Italië, Luxemburg en

Nederland) die samen de Europese Economische Gemeenschap vormden, de voorganger van de Europese Unie.

40 • Interview president van De Nederlandsche Bank Nout Wellink

“Je bent dag en nacht bezig. Ik ben permanent aan het reizen en dat begint wel een beetje te vervelen… Ik loop tegen de 68 aan wanneer mijn termijn afloopt. Het kan dus ook zijn dat ik zeg van: nou joh, het is genoeg geweest”.

Maar op de vraag of we sommige landen daarna niet te snel tot de EU

financiële markten die een crisis willen voorkomen vooral wil mee-

hebben toegelaten zou het antwoord volgens Wellink best wel eens

geven, is “om altijd kritisch te blijven”. “Dat betekent overigens

“ja” kunnen zijn. Deze landen nu nog de Europese Unie uitzetten

landen tot de EU. De veronderstelling was dat ze zich zouden

niet dat je de wereld altijd in je vingers hebt. Maar ik denk dat een van de belangrijkste dingen toch gewoon is dat je naar eer en geweten moet blijven handelen en dat je moet blijven staan achter wat je zelf vindt”.

Een van de moeilijkste dingen volgens Wellink is om jezelf niet te snel door de waan van de dag in verwarring te laten brengen. “Kenmerk van de waan van de dag is dat mensen die zich

gedragen”.

hiertegen verzetten, daarvan zegt men dat je niet meer meedraait,

is volgens hem echter in het belang van geen enkele partij.

Hadden we dan strenger moeten zijn? “Men zegt wel eens dat toezichthouders strenger moeten zijn. Ik denk het wel. De afgelopen jaren heb ik geleerd dat de wereld toch anders draait dan we vaak dachten en dit geldt ook voor het toelaten van nieuwe

dat je bent achterhaald bent”.

“Maar Griekenland heeft maar een jaar dat het in de EU zat aan het vereiste van 3% voldaan. En Griekenland heeft vanaf 1980 een begrotingstekort van 7,8% gehad. Dat is niet wat bij een monetaire unie hoort”.

“Maar je bent onderdeel van de waan van de dag. Als wij als toezichthouder fouten hebben gemaakt, is het onder meer omdat de hele wereld euforisch was voor de kredietcrisis; the sky was the limit”.

“Dat Griekenland ons twee keer heeft voorgelogen mag natuurlijk niet, maar het echte probleem is eigenlijk veel ernstiger. Dat land is alle jaren dat het in de EMU zat, boven zijn stand blijven leven. De loonstijging in Griekenland is alle jaren dat er een Europese Monetaire Unie was dik uitgegaan boven de loonstijging van de rest van Europa. Dat betekent dat jaar in jaar uit de concurrentiepositie verslechterde. Griekenland had

“Door dingen heen kijken in een wereld die steeds haastiger wordt, is lastig. Al haastend moeten we toch het gezond verstand erbij houden en zeggen “jongens, pas nu even op, denk nu even na”.

Want de toekomst is meer onvoorspelbaar dan je denkt.

Toezichthouders zijn een onderdeel van die wereld en die zijn misschien te weinig kritisch geweest.

toen de crisis losbarstte een begrotingstekort van zo”n 15%. En dat is gewoon onhoudbaar!”

Of de common sense dan ook boven de econometrische modellen moet prevaleren, wil Wellink niet zeggen. Wel vindt hij dat je heel

Toekomstvisies

voorzichtig moet zijn met modellen. “Het is wikken en wegen omdat

De tweede termijn van Wellink loopt in 2011 af en er gaan geruchten

je de toekomst niet kent”.

dat hij de compromis kandidaat 10 zou zijn in de opvolging van Trichet.

Op de vraag of Wellink een taak als president van de Europese Centrale Bank zelf ambieert antwoordt hij echter: “Ik ga pas bepalen wat ik wil doen, gewoon aan het einde van het jaar”.

“Je kunt de wereld heel sterk vereenvoudigen en dan weet je wat je moet doen, maar vermoedelijk zit je er dan naast. Want ook al vereenvoudig je de wereld, het blijft ingewikkeld”.

Wel constateert hij dat hij al sinds 1965 aan het werk is en vanaf 1982 in de directie zit – en daarmee overigens waarschijnlijk ook het langst dienende centrale directie lid ter wereld is. En hoewel hij dit werk altijd met veel plezier heeft gedaan, is het heel druk. Of het gerucht van een tweede Nederlandse bankier in Frankfurt de waarheid zal worden, moeten we dus nog even afwachten. “Je bent dag en nacht bezig. Ik ben permanent aan het reizen en dat begint wel een beetje te vervelen…

Ik loop tegen de 68 aan wanneer mijn termijn afloopt. Het kan dus ook zijn dat ik zeg van: nou joh, het is genoeg geweest”.

Ook is nog niet bekend wie dan zijn eigen opvolger zal worden.

Maar wat Wellink hem of haar en de toekomstige spelers op de

10

Financial Times, 8 juli 2010

Interview president van De Nederlandsche Bank Nout Wellink • 41

fsrforum • jaargang 12 • editie #5

Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation

Peter Carr (Bloomberg LP and Courant Institute, New York University) and Liuren Wu (Zicklin School of Business, Baruch College)

Markets for both stock options and credit derivatives have experienced dramatic growth in the past few years. Along with the rapid growth, it has become increasingly clear to market participants that stock option implied volatilities and credit default swap (CDS) spreads are inherently linked.

Many academic studies have also empirically documented the positive link between credit spreads and stock volatility at both the firm level and the aggregate level.1 Interestingly, this empirical relationship has been presaged by classical asset pricing theory. According to the classical structural model of Merton (1974), corporate bond credit spreads are functions of financial leverage and firm asset volatility, which both contribute to volatility in the underlying company’s stock and hence to stock option implied volatilities.

Furthermore, when a company defaults, the company’s stock price inevitably drops by a sizeable amount. As a result, the possibility of default on a corporate bond generates negative skewness in the risk-neutral probability distribution of stock returns. This negative skewness is manifested in the relative pricing of stock options across different strikes. When the Black and Scholes (1973) implied volatility is plotted against some measure of moneyness at a fixed maturity, the slope of the plot is positively related to the risk-neutral skewness of the stock return distribution. Dennis and Mayhew (2002) and Bakshi, Kapadia, and Madan (2003) examine the negative skew of the implied volatility plot for individual stock options. Recent empirical work, e.g., Cremers, Driessen, Maenhout, and Weinbaum (2004), shows that CDS spreads are positively correlated with both stock option implied volatility levels and the steepness of the negative slope of the implied volatility plot against moneyness.

In this paper, we propose a dynamically consistent frame-

42 •Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation

work that allows joint valuation and estimation of stock options and credit default swaps written on the same reference company. Wemodel company default as controlled by a Poisson process with a stochastic arrival rate. When default occurs, the stock price drops to zero. Prior to default, we model the stock price by a continuous process with stochastic volatility. The instantaneous default rate and instantaneous diffusion variance rate follow a bivariate continuous Markov process, with its joint dynamics specified to capture the empirical evidence on stock option prices and CDS spreads.

Under this joint specification, we derive tractable pricing solutions for stock options and credit default swaps. We estimate the joint dynamics of the default rate and the diffusion variance rate using stock option prices and CDS spreads for four actively traded companies. Our estimation shows that for all four companies, the default rate is more persistent than the diffusion variance rate under both statistical and risk-neutral measures. The statistical persistence difference suggests different degrees of predictability.

The risk-neutral difference suggests that the default rate has a more long-lasting impact on the term structure of option volatilities and CDS spreads than does the diffusion variance.

The estimation also highlights the interaction between market risk (diffusion variance) and credit risk (default arrival) in pricing stock options and credit default swaps.

We find that while credit risk dominates the CDS spreads at long maturities, diffusion variance can also affect CDS spreads at short maturities due to positive co-movements between diffusion variance and default arrival. On the other hand, the default arrival rate affects stock option pricing

through both its correlation with the diffusion variance rate and its direct impact on the risk-neutral drift of the return process. The impact of the diffusion variance rate on the implied volatility is relatively uniform across different moneyness levels, but the impact of the default arrival rate is mainly on options at low strikes. Furthermore, the impact of the credit risk factor on stock option prices increases with the option maturity. For options maturing in six months, the contribution of the credit risk factor to option pricing is comparable in magnitude to the contribution of the diffusion variance rate.

The positive empirical relation between CDS spreads and stock option implied volatilities has been recognized only very recently in the academic community. As a result, efforts to theoretically capture this link are only in an embryonic stage. In a recent working paper, Hull, Nelken, and White (2004) link CDS spreads and stock option prices by proposing a new implementation and estimation method for the classical structural model of Merton (1974). As is well known, this early model is highly stylized as it assumes that the only source of uncertainty is the ďŹ rmâ&#x20AC;&#x2122;s asset value.

As a result, stock option prices and CDS spreads have changes that are perfectly correlated locally. Thus, the empirical observation that implied volatilities and swap spreads sometimes move in opposite directions can only be accommodated by adding additional sources of uncertainty to the model. In this paper, we assume that prior to default, the stock price process is continuous. The drift and diffu-

1. Joint Valuation of Stock Options and Credit Default Swaps Consider a reference company which has positive probability of defaulting. Let Pt denote the time-t stock price for this company, which we assume falls to zero upon default. Let (Ί, F , (F W )Wâ&#x2030;Ľ , Q) be a complete stochastic basis and let Q be a risk-neutral probability measure. Prior to any default, the risk-neutral stock price dynamics are given by: â&#x2C6;&#x161; (1) G3W /3W = (UW â&#x2C6;&#x2019; TW + ÎťW ) GW + YW G:W3 , where rt and qt denote the instantaneous interest rate and dividend yield respectively, which we assume evolve deterministically over time. In Îť(W) denote the risk-neutral arrival rate of the default event and vt denotes the instantaneous variance rate for the stock diffusion return component. Both processes evolve stochastically over time.

The stock price P also evolves stochastically and is driven by standard Brownian motionWtP. The incorporation of ÎťW in the drift compensates for the possibility of a default, so that the stock price remains a martingale unconditionally under the risk-neutral measure. Thus, the drift and diffusion coefďŹ cients of this pre-default stock price process are both stochastic.

1.1. Joint dynamics of diffusion variance rate and default arrival rate We model the joint dynamics of the default arrival rate and the diffusion return variance rate under the risk-neutral

Thus, the empirical observation that implied volatilities and swap spreads sometimes move in opposite directions can only be accommodated by adding additional sources of uncertainty to the model. sion coefďŹ cients of this process are both stochastic as we assume that the default arrival rate and diffusion variance rate obey a bivariate stochastic process. As a result, we are able to capture the imperfect positive correlation between stock volatility and default risk. Thus, when compared to efforts based on the structural model of Merton (1974), our contribution amounts to adding consistent, inter-related, but separate dynamics to the relation between volatility and default. The CDS market and the stock options market contain overlapping information on the market and credit risk of the company. Our joint valuation and estimation framework exploits this overlapping informational structure to provide better identiďŹ cation of the dynamics of the stock return variance and default arrival rate. The estimation results highlight the inter-related and yet distinct impacts of the two risk factors on the two markets.

The rest of the paper is organized as follows. The next section proposes a joint valuation framework for stock options and credit default swaps. Section 2 describes the data set and summarizes the stylized evidence that motivates our speciďŹ cation. Section 3 describes the joint estimation procedure. Section 4 presents the results and discusses the implications. Section 5 concludes.

probability measure Q as follows: GYW ÎťW

â&#x2C6;&#x161; = (θY â&#x2C6;&#x2019; ÎşYYW ) GW + Ď&#x192;Y YW G:WY , = βYW + ]W ,

(3)

= (θ] â&#x2C6;&#x2019; Îş] ]W ) GW + Ď&#x192;]

Ď = E G: 3 G: Y /GW.

G]W

(2)

â&#x2C6;&#x161;

]W G:W] ,

E G: ] G: 3 = E [G: ] G: Y ] = (4) (5)

The above speciďŹ cation is motivated by the following empirical evidence and economic justiďŹ cation: â&#x20AC;˘ It is well-documented that stock return volatility is stochastic. We use a square-root process in equation (2) to model the dynamics of the instantaneous variance of the diffusion return component. â&#x20AC;˘ Cremers, Driessen, Maenhout, and Weinbaum (2004) ďŹ nd that implied volatilities covary with CDS spreads. Our own empirical analysis ďŹ nds similar evidence. Equation (3) captures the positive comovement via a positive loading coefďŹ cient β between the default arrival rate ÎťW and the diffusion return variance rate vt . â&#x20AC;˘ Although it is important to recognize the co-movement between the stock market and the credit market, it is also important to accommodate the fact that the credit market can show movements independent of the stock and stock

Âť

Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation â&#x20AC;˘ 43

fsrforum â&#x20AC;˘ jaargang 12 â&#x20AC;˘ editie #5

options market. We use zt to capture this independent credit risk component, with its dynamics controlled by an independent square-root process speciďŹ ed in (4). â&#x20AC;˘ When the stock price falls, its return volatility often increases. A traditional explanation that dates back to Black (1976) is the leverage effect. So long as the face value of debt is not adjusted, a falling stock price increases the companyâ&#x20AC;&#x2122;s leverage and hence its risk, which shows up in stock return volatility.2 Equation (5) captures this phenomenon via a negative correlation coefďŹ cient between diffusion shocks in return and diffusion shocks in return variance.

Hence, by equating the present values of the two legs, we can solve for the CDS spread as, R

R EW Z W7 ÎťV H[S (â&#x2C6;&#x2019; WV (UX + ÎťX )GX) GV (14) , R 6(W, 7 ) = RV 7 EW W H[S (â&#x2C6;&#x2019; W (UX + ÎťX )GX) GV which can be regarded as a weighted average of the expected default loss. Under the dynamics speciďŹ ed in (2) to (5), we can solve for the present values of the two legs of the CDS. The value of the premium leg is,

Z V

(15) EW H[S â&#x2C6;&#x2019; (UX + ÎťX )GX GV W W

Z V Z 7 = 6(W, 7 ) %(W, V)EW H[S â&#x2C6;&#x2019; E GV, Îť [X GX

3UHPLXP(W, 7 )

1.2. Pricing stock options

W

Consider the time-t value of a European call option c(Pt ,K,T) with strike price K and expiry date T. The terminal payoff of the option is (PT â&#x2C6;&#x2019;K)+ if the company has not defaulted by that time, and is zero otherwise. The value of the call option can be written as,

Z 7

F (3W , ., 7 ) = EW H[S â&#x2C6;&#x2019; (UV + ÎťV ) GV (37 â&#x2C6;&#x2019; .)+ (6) W

where the expectation operator EW [¡] is under the risk-neutral measure Q and conditional on the ďŹ ltration Ft . Given the deterministic interest rate assumption, we have,

Z 7

(7) F (3W , ., 7 ) = % (W, 7 ) EW H[S â&#x2C6;&#x2019; ÎťV GV (37 â&#x2C6;&#x2019; .)+ , W

with B(t,T) denoting the time-t value of a default-free zerocoupon bond paying one dollar at its maturity date T. The expectation can be solved by inverting the following discounted generalized

Fourier transform,

Ď&#x2020; (X) â&#x2030;Ą EW H[S â&#x2C6;&#x2019;

Z 7 W

ÎťV GV HLX OQ 37 /3W ,

X â&#x2C6;&#x2C6; D â&#x160;&#x201A; C,

(8)

where D denotes the subset of the complex plane under which the expectation is well-deďŹ ned. Under the dynamics speciďŹ ed in (1) to (5), the Fourier transform is exponential afďŹ ne in the bivariate risk factor [W â&#x2030;Ą [YW , ]W ] (9)

Ď&#x2020; (X) = H[S LX(U(W, 7 ) â&#x2C6;&#x2019; T(W, 7 ))Ď&#x201E; â&#x2C6;&#x2019; D (Ď&#x201E;) â&#x2C6;&#x2019; E (Ď&#x201E;) [W ,

Ď&#x201E; = 7 â&#x2C6;&#x2019; W,

where r(t,T ) and q(t,T) denote the continuously compounded spot interest rate and dividend yield at time t and maturity date T, respectively, and the time-homogeneous coefďŹ cients [D(Ď&#x201E;), E(Ď&#x201E;)] are given by,

Ρ Y â&#x2C6;&#x2019; ÎşM θY Y OQ â&#x2C6;&#x2019; â&#x2C6;&#x2019; Hâ&#x2C6;&#x2019;ΡY Ď&#x201E; + ΡY â&#x2C6;&#x2019; ÎşM Ď&#x201E; Y Ď&#x192; Y ΡY

Ρ ] â&#x2C6;&#x2019; Îş] θ] (10) â&#x2C6;&#x2019; Hâ&#x2C6;&#x2019;Ρ] Ď&#x201E; + (Ρ] â&#x2C6;&#x2019; Îş] ) Ď&#x201E; , + OQ â&#x2C6;&#x2019; Ď&#x192;] Ρ]

EY ( â&#x2C6;&#x2019; Hâ&#x2C6;&#x2019;ΡY Ď&#x201E; ) E] ( â&#x2C6;&#x2019; Hâ&#x2C6;&#x2019;Ρ] Ď&#x201E; ) E (Ď&#x201E;) = , , (11) â&#x2C6;&#x2019;ΡY Ď&#x201E; ) ΡY â&#x2C6;&#x2019; (Ρ â&#x2C6;&#x2019; ÎşM Ρ] â&#x2C6;&#x2019; (Ρ] â&#x2C6;&#x2019; Îş] ) ( â&#x2C6;&#x2019; Hâ&#x2C6;&#x2019;Ρ] Ď&#x201E; ) Y ) ( â&#x2C6;&#x2019; H D (Ď&#x201E;) =

M

with ÎşM (Îş] ) Y = ÎşY â&#x2C6;&#x2019; LXĎ&#x192;Y Ď ÎˇY = (ÎşY ) + Ď&#x192;Y EY Ρ] =

+ Ď&#x192;] E] EY = ( â&#x2C6;&#x2019; LX)β + LX + X and E] = â&#x2C6;&#x2019; LX Appendix A provides details of the derivation. Given Ď&#x2020; (X), option prices can be obtained via fast Fourier inversion (Carr and Wu (2004a)).

1.3. Pricing credit default swap spreads For a credit default swap initiated at time t and with maturity date T, we let S(t,T) denote the premium (the â&#x20AC;&#x153;CDS spreadâ&#x20AC;?) paid by the buyer of default protection. Assuming continuous payments for simplicity, the present value of the premium leg of the contract is,

Z V

Z 7 3UHPLXP(W, 7 ) = EW 6(W, 7 ) H[S â&#x2C6;&#x2019; (UX + ÎťX )GX GV . (12) W

W

Assuming that the fractional loss given default is constant at w, the present value of the protection leg is, Z 7 ZV

3URWHFWLRQ(W, 7 ) = EW Z ÎťV H[S â&#x2C6;&#x2019; (UX + ÎťX )GX GV . (13) W

Z 7

= 6(W, 7 )

W

44 â&#x20AC;˘ Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation

W

[β, ] .

with EÎť = The afďŹ ne dynamics for the bivariate risk factors x and the linear loading function EÎť dictate that the present value of the premium leg is an exponential afďŹ ne function of the state vector (DufďŹ e, Pan, and Singleton (2000)): 3UHPLXP(W, 7 )

= 6(W, 7 )

Z 7 W

%(W, V) H[S â&#x2C6;&#x2019;DÎť (V â&#x2C6;&#x2019; W) â&#x2C6;&#x2019; EÎť (V â&#x2C6;&#x2019; W) [W GV, (16)

where the afďŹ ne coefďŹ cients can be solved analytically:

Ρ Y â&#x2C6;&#x2019; ÎşY

θY OQ â&#x2C6;&#x2019; â&#x2C6;&#x2019; Hâ&#x2C6;&#x2019;ΡY Ď&#x201E; + (ΡY â&#x2C6;&#x2019; ÎşY ) Ď&#x201E;

Ď&#x192;Y ΡY

Ρ ] â&#x2C6;&#x2019; Îş]

θ] â&#x2C6;&#x2019; Hâ&#x2C6;&#x2019;Ρ] Ď&#x201E; + (Ρ] â&#x2C6;&#x2019; Îş] ) Ď&#x201E; , + OQ â&#x2C6;&#x2019; (17) Ď&#x192;] Ρ]

( â&#x2C6;&#x2019; Hâ&#x2C6;&#x2019;Ρ] Ď&#x201E; ) β ( â&#x2C6;&#x2019; Hâ&#x2C6;&#x2019;ΡY Ď&#x201E; ) , , EÎť (Ď&#x201E;) = (18) ΡY â&#x2C6;&#x2019; (Ρ â&#x2C6;&#x2019; ÎşY) ( â&#x2C6;&#x2019; Hâ&#x2C6;&#x2019;ΡY Ď&#x201E; ) Ρ] â&#x2C6;&#x2019; (Ρ] â&#x2C6;&#x2019; Îş] ) ( â&#x2C6;&#x2019; Hâ&#x2C6;&#x2019;Ρ] Ď&#x201E; )

DÎť (Ď&#x201E;) =

with ΡY =

(ÎşY ) + Ď&#x192; Y β DQG Ρ] =

(Îş] ) + Ď&#x192; ]

The present value of the protection leg is, 3URWHFWLRQ(W, 7 )

Z = EW Z

Z V

%(W, V)ÎťV H[S â&#x2C6;&#x2019; ÎťX GX GV (19) W

Z V Z 7

%(W, V)EW E E GV, = Z Îť [V H[S â&#x2C6;&#x2019; Îť [X GX 7

W

W

W

which also allows for an afďŹ ne solution: Z 7

3URWHFWLRQ(W, 7 ) = Z %(W, V) FÎť (V â&#x2C6;&#x2019; W) + GÎť (V â&#x2C6;&#x2019; W) [W H[S W

(20) â&#x2C6;&#x2019;DÎť (V â&#x2C6;&#x2019; W) â&#x2C6;&#x2019; EÎť (V â&#x2C6;&#x2019; W) [W GV, where the coefďŹ cients (DÎť (Ď&#x201E;), EÎť (Ď&#x201E;)) are the same as in (16), and the coefďŹ cients FÎť (Ď&#x201E;), GÎť (Ď&#x201E;) can also be solved analytically by taking partial derivatives against DÎť (Ď&#x201E;), EÎť (Ď&#x201E;) with respect to maturity Ď&#x201E;: (21) FÎť (Ď&#x201E;) = â&#x2C6;&#x201A;DÎť (Ď&#x201E;)/â&#x2C6;&#x201A;Ď&#x201E;, GÎť (Ď&#x201E;) = â&#x2C6;&#x201A;EÎť (Ď&#x201E;)/â&#x2C6;&#x201A;Ď&#x201E;.

Combining the solutions for the present values of the two legs in equations (15) and (20) leads to the CDS spread S(t,T).

When we estimate the model, we discretize the above equation so as to accommodate quarterly premium payments.

1.4. Market prices of risks and time-series dynamics Our joint estimation identiďŹ es both the time-series dynamics and the risk-neutral dynamics of the bivariate state vector [W = [YW , ]W ] . To derive the time-series dynamics for the bivariate vector xt under the statistical measure P, we assume that the market prices of risks are proportional to the corresponding risk level. Under this assumption, the time-series dynamics are,

â&#x2C6;&#x161; GYW = θY â&#x2C6;&#x2019; ÎşPY YW GW + Ď&#x192;Y YW G:WYP ,

(22) â&#x2C6;&#x161; G]W = θ] â&#x2C6;&#x2019; ÎşP] ]W GW + Ď&#x192;] ]W G:W]P , with ÎşPY = ÎşY â&#x2C6;&#x2019; Ď&#x192;Y ÎłY DQG ÎşP] = Îş] â&#x2C6;&#x2019; Ď&#x192;] ÎłY

2. Data and Evidence Both the stock option prices and the CDS spreads are functions of the two risk factors [W = [YW , ]W ] , which jointly deter-

The practice is to take the average of the two implied volatilities at each strike and convert them into out-of-the money European option prices using the Black-Scholes formula.

mine the stock diffusion variance and the default arrival rate.

Therefore, we can use data on stock option prices and CDS spreads to infer the joint dynamics.

2.1. Data description We estimate the model using CDS spreads and stock option prices on four reference companies. Bloomberg provides CDS spread quotes from several broker dealers. We use quotes from different broker dealers in order to cross-validate them. Then, we take the quotes on each series from the most reliable sources. We choose four companies for which CDS quotes have both a long history and frequent updates.

The four companies are: Ford (F), General Motors (GM), Altria Group Inc (MO), and Duke Energy Corp (DUK). For each company, we have CDS spread series at five fixed maturities of one, three, five, seven, and ten years.

The corresponding stock options data is from OptionMetrics.

Exchange-traded options on individual stocks are Americanstyle and hence the price reflects an early exercise premium.

OptionMetrics uses a binomial tree to back out the option implied volatility that explicitly accounts for this early exercise premium. For each stock, OptionMetrics provides a standardized implied volatility surface at fixed Black-Scholes forward deltas from 20 to 80 with a five-delta interval for both call and put options, and fixed option maturities of 30, 60, and 91 days.

OptionMetrics estimates the implied volatility surface via a kernel smoothing approach whenever the underlying quotes are available and leave as missing values when there are not enough quotes to make the smoothing estimation. Data at longer maturities are also available but only very sparsely. Hence we only use the first three maturities. The implied volatility estimates from OptionMetrics are often different from calls and puts at similar strikes, which is to be expected when options are American and the Black Scholes model is not holding. Our interest is in testing the validity of our model for European options and so we adopt a standard practice for estimating market prices of European options. The practice is to take the average of the two implied volatilities at each strike and convert them into out-of-themoney European option prices using the Black-Scholes formula.

To price the CDS contracts and to convert the implied volatility into option prices, we also need the underlying interest rate curve. Again following standard industry practice, we use the interest rate curve defined by the Eurodollar LIBOR and swap rates. We download LIBOR rates at maturities of one, two, three, six, nine, and 12 months and swap rates at two, three, four, five, seven, and ten years. We use a piece-wise constant forward function in bootstrapping the discount rate curve.

2.2. Summary statistics Our model estimation uses the common samples of the three

data sets from January 2, 2002 to April 30, 2004. The data are available on a daily basis, but we estimate the model using weekly-sampled data on every Wednesday to avoid the impacts of weekday effects. Table 1 reports the summary statistics of the CDS spreads on the four reference companies. The mean term structures of the spreads are relatively flat for all four companies, but the standard deviations of the spreads for all four companies decline with increasing maturities. The weekly autocorrelation estimates for the spreads range from 0.90 to 0.97, showing that the CDS spreads are highly persistent.

Table 1 Summary Statistics on credit default swap spreads Maturity Mean: F GM MO DUK Standard Deviation: F GM MO DUK Autocorrelation: F GM MO DUK

1

3

5

7

10

2.19 1.51 1.79 2.31

2.89 2.03 1.78 2.14

2.97 2.19 1.75 1.99

2.95 2.28 1.69 1.92

2.87 2.15 1.79 1.27

1.31 0.89 1.15 1.93

1.38 0.82 0.84 1.60

1.16 0.72 0.72 1.31

1.06 0.69 0.62 1.17

0.96 0.67 0.32 0.31

0.97 0.96 0.91 0.96

0.97 0.95 0.92 0.97

0.96 0.94 0.92 0.96

0.95 0.92 0.90 0.96

0.95 0.93 0.94 0.96

Entries report the sample estimates of the mean, standard deviation, and weekly autocorrelation on the credit default swap spreads (in percentages) at five fixed maturities for each of the four reference companies. The statistics are based on weekly sampled data from January 2, 2002 to April 28, 2004.

Table 2 Summary statistics on stock option impled volatilities Delta Mean: F

20

25

30

35

40

45

50

55

60

65

70

75

80

1m 50.07 49.09 48.02 46.82 45.84 44.74 43.68 43.12 42.52 42.25 41.95 41.98 42.28

F

2m 50.49 48.87 47.45 46.18 45.19 44.14 43.19 42.62 42.13 41.63 41.45 41.47 41.67

F

3m 49.98 47.97 46.64 45.46 44.53 43.61 42.73 42.14 41.46 40.93 40.63 40.46 40.53

GM

1m 40.87 39.39 38.16 37.14 36.33 35.60 34.95 34.42 33.96 33.54 33.19 32.94 32.95

GM

2m 41.45 39.92 38.64 37.49 36.51 35.70 34.97 34.35 33.81 33.29 32.81 32.41 32.17

GM

3m 41.58 39.84 38.46 37.29 36.24 35.34 34.58 33.92 33.31 32.71 32.14 31.64 31.27

MO

1m 33.73 32.07 30.84 29.89 29.24 28.69 28.36 28.06 27.74 27.50 27.42 27.55 28.07

MO

2m 33.23 31.79 30.71 29.85 29.17 28.63 28.21 27.81 27.42 27.09 26.86 26.78 26.91

MO

3m 33.09 31.78 30.80 29.97 29.25 28.63 28.12 27.66 27.24 26.85 26.50 26.23 26.04

DUK

1m 46.40 44.28 42.65 41.37 39.94 38.66 37.67 37.04 36.56 36.12 35.94 35.79 36.14

DUK

2m 46.09 43.96 42.26 40.85 39.48 38.22 37.25 36.58 35.94 35.30 34.69 34.30 34.38

DUK

3m 44.97 43.08 41.42 39.94 38.65 37.39 36.42 35.70 34.92 34.17 33.46 32.97 32.80

Standard Deviation: F

1m 15.93 15.29 14.71 14.31 13.88 13.40 12.95 12.39 11.90 11.72 11.47 11.17 10.59

F

2m 15.55 15.03 14.39 13.57 12.97 12.40 11.86 11.64 11.37 10.62 10.33 10.15 10.03

F

3m 15.24 14.41 13.69 12.85 12.27 11.79 11.31 11.14 10.63 10.04

9.75

9.49

9.16

GM

1m 15.37 14.64 13.95 13.39 12.86 12.23 11.68 11.26 10.83 10.39 10.00

9.61

9.20

GM

2m 14.60 13.88 13.14 12.51 11.95 11.35 10.79 10.30

8.61

8.14

GM

3m 13.97 13.16 12.37 11.68 11.08 10.51

9.98

9.49

9.85 9.05

9.46 8.63

9.04 8.20

7.76

7.29

MO

1m 10.99 10.38

9.96

9.62

9.25

8.98

8.71

8.46

8.18

7.93

7.77

7.70

7.74

MO

2m

9.65

9.10

8.71

8.39

8.07

7.82

7.60

7.36

7.12

6.95

6.81

6.66

6.52

3m

9.18

8.65

8.20

7.85

7.57

7.33

7.11

6.89

6.69

6.56

6.42

6.26

6.13

MO DUK

1m 19.25 18.36 17.76 17.05 16.44 15.82 15.20 14.65 14.23 13.91 13.47 13.04 12.54

DUK

2m 17.43 16.52 15.94 15.36 14.72 14.06 13.47 12.94 12.55 12.15 11.64 11.16 10.64

DUK

3m 16.54 15.61 14.80 14.16 13.50 12.86 12.29 11.77 11.38 10.96 10.50 10.05

9.63

Autocorrelation: F

1m

0.83

0.86

0.87

0.89

0.87

0.84

F

2m

0.89

0.87

0.88

0.91

0.88

0.86

0.87

0.86

0.88

0.90

0.91

0.91

0.90

F

3m

0.93

0.91

0.91

0.94

0.91

0.89

0.89

0.89

0.91

0.93

0.92

0.91

0.91

GM

1m

0.92

0.93

0.93

0.93

0.93

0.93

0.84

0.93

0.85

0.93

0.84

0.92

0.85

0.92

0.85

0.92

0.85

0.92

0.87

0.90

GM

2m

0.95

0.95

0.95

0.95

0.95

0.95

0.95

0.95

0.95

0.94

0.94

0.93

0.92

GM

3m

0.96

0.96

0.96

0.96

0.96

0.96

0.95

0.95

0.95

0.95

0.95

0.95

0.94

MO

1m

0.78

0.78

0.79

0.80

0.81

0.82

0.82

0.81

0.81

0.81

0.80

0.76

0.69

MO

2m

0.84

0.84

0.85

0.85

0.86

0.86

0.86

0.86

0.87

0.87

0.86

0.84

0.80

MO

3m

0.87

0.88

0.88

0.88

0.89

0.89

0.89

0.89

0.89

0.89

0.89

0.89

0.88

DUK

1m

0.90

0.90

0.90

0.89

0.90

0.90

0.91

0.91

0.90

0.90

0.89

0.89

0.88

DUK

2m

0.93

0.93

0.93

0.92

0.92

0.93

0.93

0.92

0.92

0.91

0.91

0.91

0.91

DUK

3m

0.95

0.94

0.94

0.94

0.94

0.94

0.93

0.93

0.93

0.93

0.93

0.93

0.93

»

Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation • 45

fsrforum â&#x20AC;˘ jaargang 12 â&#x20AC;˘ editie #5

Table 2 reports the summary statistics of stock option implied volatilities at the three ďŹ xed maturities and 13 ďŹ xed put-option deltas for each of the four reference companies. For each company and at each option maturity, the implied volatilities at low strikes (low put deltas) are on average higher than the implied volatilities at high strikes, generating a negatively sloped average implied volatility smirk across moneyness. The standard deviations of the implied volatility series are also larger for out-of-the-money puts than for out-of-the-money calls, but the difference is smaller than the difference in the mean estimates. The weekly autocorrelation for the volatility series range from 0.69 to 0.93, indicating that the implied volatilities are persistent, but less so than the CDS spreads.

Figure 1 plots the average implied volatility smirk at the three ďŹ xed maturities as a function of the put option delta. For all four reference companies and for all three ďŹ xed maturities, the average implied volatility smirk is negatively skewed, corresponding to a negatively skewed risk-neutral stock return distribution. The three lines in each panel, which correspond to the three option maturities, stay closely to one another, suggesting that the conditional risk-neutral distribution of the stock return retains similar shapes at the three conditioning horizons.

Generically, our model speciďŹ cation can generate the negative skewness from two sources: (1) a positive probability of default Îť > and (2) a negative correlation between the return Brownian motion component and its instantaneous variance rate Ď < .

Figure 1 The average implied volatility smirk on stock options Lines are the average implied volatility plotted against put option delta at three ďŹ xed maturities: one month (solid lines), two months (dashed lines), and three months (dash-dotted lines). Each panel is for one company.

F

50

40

48

38

46 44

40 20

30 20

40

50 Put Option Delta

60

70

80

MO

44 Implied Volatlity, %

46

32

30 29

34 50 Put Option Delta

60

70

80

50 Put Option Delta

60

70

80

DUK

38 36

40

40

40

27 30

30

42

28

26 20

32 20

30

40

50 Put Option Delta

60

70

80

Figure 2 Time series of CDS spreads and at-the-money stock option implied volatilities.

The solid lines are the time series of CDS spreads at ďŹ xed maturities of one, three, ďŹ ve, seven, and ten years, with scales on the left hand size. The dashed lines are the time series of the at-the-money (50 delta) stock option implied volatilities at ďŹ xed maturities of 30, 60, and 91 days, with the scales on the right hand side.

100

0 0 Dec01 Mar02 Jul02 Oct02 Jan03 Apr03 Aug03 Nov03 Feb04 Jun04

CDS Spread, %

5

100

4

80

3

60

2

40

1

20

0 Dec01 Mar02 Jul02

MO

DUK 100

80

6

60

4

40

2

20

0 0 Dec01 Mar02 Jul02 Oct02 Jan03 Apr03 Aug03 Nov03 Feb04 Jun04

CDS Spread, %

8

10

ATM Implied Volatility, %

CDS Spread, %

10

0 Oct02 Jan03 Apr03 Aug03 Nov03 Feb04 Jun04

5

100

50

0 0 Dec01 Mar02 Jul02 Oct02 Jan03 Apr03 Aug03 Nov03 Feb04 Jun04

46 â&#x20AC;˘ Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation

ATM Implied Volatility, %

5

ATM Implied Volatility, %

CDS Spread, %

GM 200

ATM Implied Volatility, %

F 10

,9W (G) = DW + EW G + FW G ,

where the moneyness G is deďŹ ned as â&#x2C6;&#x161; G â&#x2030;Ą (OQ 3/. + (U â&#x2C6;&#x2019; T)Ď&#x201E; + ,9 Ď&#x201E;/ )/(,9 Ď&#x201E;). We use the intercept estimate D as a smoothed estimate for the at-themoney implied volatility (ATMVt) at d1 = 0, and use the norDW as a proxy for the riskEW / malized slope estimate 6.(:W = neutral skewness of the return distribution. Then, we use the ďŹ ve-year CDS spread to proxy the credit spread (CDSt ) and run restricted and unrestricted versions of the following regression: (23)

Table 3 Regressing CDS spreads on stock option implied volatility levels and skews

48

33

31

To quantify the co-movements, at each date we ďŹ t a second order polynomial on the three-month implied volatilities across moneyness (d1),

&'6W = D + E $7 09W + F 6.(:W + HW .

34 32

30

Figure 2 overlays the time series of the CDS spreads (solid lines) with the daily time series of at-the-money (50 delta) stock option implied volatilities at the three ďŹ xed option maturities (dashed lines) for the four chosen reference companies. We observe apparent common movements for the two types of time series for each company. The co-movements are the most obvious during periods of ďŹ nancial distress for the company, as the two sets of time series both spike up.

36

42

34

Implied Volatlity, %

GM

42

Implied Volatlity, %

Implied Volatlity, %

52

2.3. Co-movements between option implied volatilities and credit default swap spreads

Companies F GM MO DUK F GM MO DUK F GM MO DUK

-0.997 0.267 0.088 -1.231 0.996 0.900 0.978 -0.055 -0.928 0.289 -0.060 -1.226

a (-2.786) (1.139) (0.242) (-5.391) (4.796) (3.672) (3.282) (-0.143) (-2.523) (1.255) (-0.156) (-5.194)

b 0.092 0.055 0.059 0.088 â&#x20AC;&#x201D; â&#x20AC;&#x201D; â&#x20AC;&#x201D; â&#x20AC;&#x201D; 0.083 0.049 0.052 0.088

c (10.62) (9.46) (3.98) (10.63) â&#x20AC;&#x201D; â&#x20AC;&#x201D; â&#x20AC;&#x201D; â&#x20AC;&#x201D; (6.44) (4.68) (3.71) (7.09)

â&#x20AC;&#x201D; â&#x20AC;&#x201D; â&#x20AC;&#x201D; â&#x20AC;&#x201D; -15.989 -7.988 -5.465 -10.795 -2.584 -1.173 -2.491 0.081

â&#x20AC;&#x201D; â&#x20AC;&#x201D; â&#x20AC;&#x201D; â&#x20AC;&#x201D; (-8.03) (-5.75) (-2.43) (-4.27) (-1.30) (-0.65) (-1.55) (0.04)

R2 0.82 0.58 0.34 0.69 0.53 0.46 0.15 0.30 0.83 0.59 0.36 0.69

Entries report the parameter estimates, t-statistics (in parentheses), and R2 for different versions of the following regressions on four reference companies: CDSt = a + bATMVt + cSKEWt + et, where CDSt denotes the ďŹ ve-year credit default swap spreads in percentage points, ATMVt denotes a smoothed estimate of the three-month at-the-money implied volatility in percentage points, and SKEWt denotes a normalized slope estimate on the implied volatility skew against moneyness. Data are weekly from January 2, 2002 to April 28, 2004. To compute the t-statistics, we cast the regression into a GMM framework, and estimate the covariance matrix following Newey and West (1987) with four lags.

Table 3 reports the parameter estimates, t-statistics, and R2 of the regressions for the four reference companies. When the at-the-money volatility level is the only explanatory variable, the estimates for its slope coefďŹ cient are positive and highly signiďŹ cant for all four companies. When the skewness measure is the only explanatory variable, its slope coefďŹ cients are negative and highly signiďŹ cant for all four companies. Thus, an increase in credit spreads is often associated with an increase in the option volatility level and a steepening in the negative slope of the implied volatility smirk.

When we incorporate both the volatility level and the skewness measure as explanatory variables, the slope coefďŹ cient estimates on the skewness measure are no longer statistically signiďŹ cant, suggesting that the link with the credit market is driven by one source of risk. The R-squares of the joint regressions differ across different companies, as high as 83% for Ford, but as low as 36% for Altria Group. The variations across different companies and the low R2 estimates in some instances suggest that although return variance and default arrival share common movements, they also have their own independent movements. From a modeling per-

The practice is to take the average of the two implied volatilities at each strike and convert them into out-of-the money European option prices using the Black-Scholes formula.

spective, it is important to capture not only the common movements between the two markets, but also the idiosyncratic movements in each market. Our bivariate risk dynamics in equations (2) to (5) can accommodate different degrees of common and idiosyncratic movements.

Given the persistence of both CDS spreads and implied volatilities, we also study how the weekly changes of one series is correlated with the weekly changes of the other series. Figure 3 plots the crosscorrelation estimates at different leads and lags between weekly changes in the ďŹ ve-year CDS spread and the three-month at-the-money implied volatility. The dash-dotted lines in each panel denote the 95 percent conďŹ dence band. For all four reference companies, we identify signiďŹ cantly positive contemporaneous correlations between the weekly changes of the two series, with the estimates ranging from 0.52 to 0.61. The cross-correlation estimates at other leads and lags are largely insigniďŹ cant.

3. Joint Estimation of Return Variance and Default Arrival Dynamics We estimate the bivariate risk dynamics jointly using both CDS spreads and stock options. We cast the model into a state-space form and estimate the model using a quasi-maximum likelihood method. In the state-space form, we regard the bivariate risk vector as the unobservable states and specify the state propagation equation using an Euler approximation of the timeseries dynamics in equation (22): â&#x17D;Ą [W = â&#x17D;Ł

θY θ]

â&#x17D;¤

â&#x17D;Ą

â&#x17D;Ś Î&#x201D;W + â&#x17D;Ł

P

â&#x17D;Ą â&#x17D;¤

Ď&#x192; Y Î&#x201D;W

Y Wâ&#x2C6;&#x2019; â&#x17D;Ś [Wâ&#x2C6;&#x2019; + â&#x17D;Ł â&#x17D;Ś ÎľW ,

Ď&#x192; ] ]Wâ&#x2C6;&#x2019; Î&#x201D;W â&#x17D;¤

Hâ&#x2C6;&#x2019;ÎşY Î&#x201D;W

Hâ&#x2C6;&#x2019;Îş] Î&#x201D;W

P

(24)

where denotes an iid bivariate standard normal innovation and t =7/365 denotes the sampling frequency. We construct the measurement equations based on CDS spreads and stock options, assuming additive, normally-distributed measurement errors: \W = K([W Î&#x2DC;) + HW ,

(25)

Figure 3 Cross-correlations between weekly changes in the ďŹ ve-year CDS spread and the three-month at-the-money implied volatility.

The bars show the cross-correlation estimates between weekly changes in the ďŹ ve-year CDS spread and weekly changes in the three-month at-themoney implied volatility at different leads and lags. The two dash-dotted lines in each panel deďŹ ne the 95 percent conďŹ dence band. Î&#x201D; ATMV]

GM: [ Î&#x201D; CDS (Lag), 0.6

0.5

0.5

0.4

0.4

0.3

Sample Cross Correlation

Sample Cross Correlation

F: [ Î&#x201D; CDS (Lag), 0.6

0.2 0.1 0 â&#x2C6;&#x2019;0.1

0.3 0.2 0.1 0

â&#x2C6;&#x2019;0.2

â&#x2C6;&#x2019;0.1

â&#x2C6;&#x2019;0.3

â&#x2C6;&#x2019;0.2

â&#x2C6;&#x2019;0.4 â&#x2C6;&#x2019;20

â&#x2C6;&#x2019;15

â&#x2C6;&#x2019;10

â&#x2C6;&#x2019;5 0 5 Number of Lags in Weeks MO: [ Î&#x201D; CDS (Lag),

10

15

â&#x2C6;&#x2019;0.3 â&#x2C6;&#x2019;20

20

â&#x2C6;&#x2019;15

â&#x2C6;&#x2019;10

Î&#x201D; ATMV]

â&#x2C6;&#x2019;5 0 5 Number of Lags in Weeks DUK: [ Î&#x201D; CDS (Lag),

0.8

Î&#x201D; ATMV]

10

15

20

Î&#x201D; ATMV]

0.6 0.5

0.6

Sample Cross Correlation

Sample Cross Correlation

0.4 0.4 0.2 0 â&#x2C6;&#x2019;0.2 â&#x2C6;&#x2019;0.4 â&#x2C6;&#x2019;0.6 â&#x2C6;&#x2019;20

0.2

â&#x2C6;&#x2019;10

â&#x2C6;&#x2019;5 0 5 Number of Lags in Weeks

10

15

20

6([W ,W + Ď&#x201E;V Î&#x2DC;) 2 ([W ,W + Ď&#x201E;2 , δ Î&#x2DC;)

â&#x17D;¤

â&#x17D;Ś,

Ď&#x201E;V = , , , , \HDUV

(26)

Ď&#x201E;2 = , , GD\V δ = , , ¡ ¡ ¡ , ,

where 6([W ,W + Ď&#x201E;V ) denotes the model value of the CDS spreads at time t and maturity Ď&#x201E;V as a function of the state vector [W and model parameters Î&#x2DC; 2([W ,W + Ď&#x201E;2 , δ Î&#x2DC;), denotes the model value for out-of-the-money options at time t, time-tomaturity Ď&#x201E;2 , and delta δ, as a function of the state vector xt and model parameters Î&#x2DC;. To deal with the predictable variation in option premia across strikes and maturity, we followed the standard industry practice of dividing out-of-the-money option prices by their Black-Scholes vega. There are missing values on both the CDS data and the implied volatility surface.

Our estimation algorithm readily handles missing observations. The term et in (25) denotes the measurement errors. We assume that the ďŹ ve CDS series generate iid normal pricing errors with the same error variance Ď&#x192; V . We also assume that the pricing errors on all the options (scaled by their vega) are also iid normal with error variance Ď&#x192; V .

When both the state propagation equation and the measurement equations are Gaussian and linear, the Kalman (1960) ďŹ lter generates efďŹ cient forecasts and updates on the conditional mean and covariance of the state vector and the measurement series. In our application, the state propagation equation in (24) is Gaussian and linear, but the measurement equation in (25) is nonlinear. We use the unscented Kalman ďŹ lter (Wan and van der Merwe (2001)) to handle the nonlinearity. The unscented Kalman ďŹ lter approximates the posterior state density using a set of deterministically chosen sample points (sigma points).

These sample points completely capture the true mean and covariance of the Gaussian state variables, and when propagated through the nonlinear functions in the measurement equations, capture the posterior mean and covariance of the CDS spreads and option prices accurately to the second order for any nonlinearity. Let \W+ and 9 W+ denote the time-t ex ante forecasts of time-(t +1) values of the measurement series and the covariance of the measurement series, respectively obtained from the unscented Kalman ďŹ lter. We construct the log-likelihood value assuming normally distributed forecasting errors,

The model parameters are chosen to maximize the log likelihood of the data series,

0.1 0 â&#x2C6;&#x2019;0.1

1 Î&#x2DC; â&#x2030;Ą DUJ PD[ L (Î&#x2DC;, {\W }W= ), Î&#x2DC;

â&#x2C6;&#x2019;0.2 â&#x2C6;&#x2019;15

â&#x17D;Ą

K([W Î&#x2DC;) = â&#x17D;Ł

â&#x2C6;&#x2019;

OW+ (Î&#x2DC;) = â&#x2C6;&#x2019; ORJ 9 W+ â&#x2C6;&#x2019; (\W+ â&#x2C6;&#x2019; \W+ ) .(27) (\W+ â&#x2C6;&#x2019; \W+ ) 9 W+

0.3

â&#x2C6;&#x2019;0.3 â&#x2C6;&#x2019;20

where \W denotes the observed series and K([W Î&#x2DC;) denotes the corresponding model value as a function of the state vector [W and model parameters Î&#x2DC;. SpeciďŹ cally, the measurement equation contains ďŹ ve CDS spread series and 39 option series,

â&#x2C6;&#x2019;15

â&#x2C6;&#x2019;10

â&#x2C6;&#x2019;5 0 5 Number of Lags in Weeks

10

15

20

with

1 L (Î&#x2DC;, {\W }W= )=

1â&#x2C6;&#x2019;

(28)

â&#x2C6;&#x2018; OW+ (Î&#x2DC;),

W=

where N denotes the number of weeks in our sample.

Âť

Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation â&#x20AC;˘ 47

fsrforum • jaargang 12 • editie #5

4. Joint Dynamics and Pricing of Return Variance and Default Arrival Risks

Table 4 Summary statistics on the pricing errors in stock option impled volatilities

First, we summarize the performance of our joint valuation model on CDS spreads and stock options on the four reference companies. Then, from the structural parameter estimates, we discuss the joint dynamics and pricing of the diffusion variance risk and default arrival risk.

4.1. Performance analysis

20

F

-0.26

Table 5 Summary statistics of pricing errors on credit default swap spreads 1

3

5

7

10

-0.58 -0.20 -0.41 0.02

0.12 0.01 0.01 -0.04

0.17 0.05 0.09 -0.05

0.14 0.04 0.08 -0.01

0.13 -0.07 0.28 0.03

0.51 0.40 0.93 0.20

0.27 0.40 0.68 0.11

0.31 0.44 0.60 0.13

0.32 0.46 0.53 0.20

0.32 0.47 0.27 0.12

0.85 0.79 0.34 0.99

0.96 0.76 0.35 1.00

0.93 0.63 0.30 0.99

0.91 0.55 0.27 0.97

0.89 0.50 0.31 0.84

Entries report the sample mean and standard deviation of the pricing errors on credit default swap spreads, defined as the difference between observations and model-implied values in percentage points, at five fixed maturities for each of the four reference companies. The last panel reports the explained variation, defined as one minus the ratio of the pricing error variance to the variance of the original implied volatility series. The statistics are based on weekly sampled data from January 2, 2002 to April 28, 2004.

48 • Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation

30

35

40

55

60

65

70

75

80

0.68

0.86

2m

-1.05 -0.07

0.41

0.57

0.67

0.47

0.16

0.09 -0.04 -0.32 -0.37 -0.36 -0.31

-1.87 -0.91 -0.10

0.35

0.67

0.71

0.55

0.52

GM

1m

0.01

0.25

0.16

0.03 -0.12 -0.22 -0.31 -0.40 -0.45 -0.43 -0.18

0.02

0.32

0.15 -0.36 -0.83 -0.86 -0.90 -0.91 -0.71 -0.43 0.22 -0.09 -0.29 -0.50 -0.66

2m

-0.74

0.51

0.51

0.49

0.42

0.35

0.29

0.18

GM

3m

-1.56 -0.66 -0.13

0.14

0.23

0.24

0.23

0.21

0.13

0.01 -0.16 -0.31 -0.38

MO

1m

0.67

0.00 -0.11 -0.23 -0.21 -0.21 -0.28 -0.32 -0.22

0.48

0.40

0.60

50

3m

0.30

0.75

45

F

GM

1m

25

F

0.24

0.05 -0.04 -0.01 0.05

0.66

MO

2m

-0.70 -0.30 -0.10 -0.02

0.02

0.05

0.09

0.06 -0.02 -0.10 -0.13 -0.08

0.13

MO

3m

-1.38 -0.63 -0.16

0.22

0.27

0.28

0.24

-1.10

DUK

Table 4 reports the sample mean in the first panel and the standard deviation in the second panel of the pricing errors on the stock options. We define pricing errors as the difference between the implied volatility quotes (in percentage points) and the corresponding model values. The mean pricing errors are fairly small and show no obvious patterns across moneyness and maturities. The standard deviation ranges from one to four percentages points. Comparing these estimates to the mean implied volatility estimates in Table 2 points to an average pricing error of less than ten percent. The last panel of the table reports the explained variation, defined as one minus the variance of the pricing errors over the variance of the original implied volatility series. The explained variations are over 90 percent for most series, showing that the model is relatively successful in capturing the behavior of stock options on all four companies.

Table 5 reports the sample mean and standard deviation of the pricing errors as well as the explained variation on the CDS spreads. The pricing errors are larger on the swap spreads. The explained variations are over 80 percent for Ford and Duke Energy, but the model’s performance is relatively poor for General Motors and Altria Group. The model explains over 50 percent of variation in the CDS spreads on General Motors, and just about 30 percent on Altria Group.

Inspecting the time series plots in Figure 2, we observe that for General Motors, the five CDS spread series diverge dramatically after January 2003 to generate a very steep term structure from a virtually flat term structure before 2003. This dramatic term structure change either comes from economic forces or from the mere fact that there was more frequent quote updating in the second half of the data. Irrespective of the underlying reasons, our two-factor model seems to have difficulties fitting the whole term structure of CDS spreads and the options data. The model performs well on all the options series, and also reasonably well on short-term CDS spreads, but it performs poorly on the long-term CDS spreads.

For Altria Group, the CDS quotes are not updated as frequently before 2003, whereas the options data are actively quoted and traded. It is potentially due to this difference that the model parameters are geared to price the options market better than the CDS spreads.

Maturity Mean: F GM MO DUK Standard Deviation: F GM MO DUK Explained Variation: F GM MO DUK

Delta Mean:

0.11

1m

-0.53 -0.49 -0.45 -0.38 -0.65 -0.96

DUK

2m

-1.05 -0.55 -0.19

DUK

3m

-2.02

0.15

0.03 -0.12 -0.25 -0.35

-1.01 -0.85 -0.73 -0.46 -0.25

0.33

0.02

0.05

0.25

0.33

0.29

0.35

-1.11 -0.56 -0.24 -0.03 -0.04

0.05

0.23

0.21

0.08 -0.14 -0.28 -0.22

0.08

0.10

0.14

0.08

Standard Deviation: F

1m

3.92

3.87

3.60

3.74

3.57

3.30

2.93

2.76

2.65

2.57

2.53

2.45

2.73

F

2m

2.52

2.52

2.50

2.66

2.57

2.37

2.00

1.92

1.60

1.27

1.30

1.39

1.84

F

3m

1.90

1.88

1.87

2.00

2.08

2.03

1.88

1.81

1.60

1.54

1.56

1.58

1.72

GM

1m

1.76

1.77

1.68

1.61

1.47

1.20

0.98

0.85

0.76

0.84

0.95

1.14

1.63

GM

2m

1.25

0.99

1.08

1.16

1.17

1.06

0.96

0.89

0.87

0.90

0.95

1.12

1.40

GM

3m

1.37

0.66

0.69

0.87

1.00

1.04

1.06

1.08

1.07

1.07

1.13

1.23

1.40

MO

1m

2.57

2.37

2.14

2.01

1.84

1.56

1.36

1.30

1.19

1.02

1.18

1.72

2.65

MO

2m

1.39

1.27

1.17

1.11

1.08

1.02

1.01

1.07

1.09

1.09

1.14

1.41

1.82

MO

3m

1.08

0.93

0.94

0.97

1.07

1.10

1.14

1.19

1.22

1.23

1.24

1.28

1.38

DUK

1m

3.48

3.26

3.33

3.26

3.25

2.98

2.56

2.58

2.47

2.37

2.71

2.81

2.90

DUK

2m

2.86

2.45

2.29

2.19

2.04

1.87

1.62

1.49

1.54

1.58

1.66

1.81

2.20

DUK

3m

2.72

2.46

2.25

2.18

2.08

1.96

1.95

1.88

1.78

1.77

1.76

1.75

1.89

Autocorrelation: F

1m

0.94

0.94

0.94

0.93

0.93

0.94

F

2m

0.97

0.97

0.97

0.96

0.96

0.96

0.97

0.97

0.98

0.99

0.98

0.98

0.97

F

3m

0.98

0.98

0.98

0.98

0.97

0.97

0.97

0.97

0.98

0.98

0.97

0.97

0.96

GM

1m

0.99

0.99

0.99

0.99

0.99

0.99

0.99

0.99

1.00

0.99

0.99

0.99

0.97

GM

2m

0.99

0.99

0.99

0.99

0.99

0.99

0.95

0.99

0.95

0.99

0.95

0.99

0.95

0.99

0.95

0.99

0.95

0.98

0.93

0.97

GM

3m

0.99

1.00

1.00

0.99

0.99

0.99

0.99

0.99

0.99

0.98

0.98

0.97

0.96

MO

1m

0.95

0.95

0.95

0.96

0.96

0.97

0.98

0.98

0.98

0.98

0.98

0.95

0.88

MO

2m

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.97

0.96

0.92

MO

3m

0.99

0.99

0.99

0.98

0.98

0.98

0.97

0.97

0.97

0.96

0.96

0.96

0.95

DUK

1m

0.97

0.97

0.96

0.96

0.96

0.96

0.97

0.97

0.97

0.97

0.96

0.95

0.95

DUK

2m

0.97

0.98

0.98

0.98

0.98

0.98

0.99

0.99

0.99

0.98

0.98

0.97

0.96

DUK

3m

0.97

0.98

0.98

0.98

0.98

0.98

0.97

0.97

0.98

0.97

0.97

0.97

0.96

Entries report the sample mean and standard deviation of the pricing errors in stock option implied volatilities, defined as the difference between observations and model-implied values in percentage points, at 13 fixed deltas and three fixed maturities for four reference companies.

The last panel reports the explained variation, defined as one minus the ratio of the pricing error variance to the variance of the original implied volatility series. The statistics are based on weekly sampled data from January 2, 2002 to April 28, 2004.

Table 6 Maximum likelihood estimates of model parameters Companies

κY κ] κPY κP] θY θ] σY σ] β ρ −Z γY γ] EP [λ] EQ [λ]

F 4.0788 0.0067 1.1878 0.1745 0.4153 0.0050 1.3738 0.1740 0.3062 -0.1354 0.6417 2.1044 -0.9645 0.1355 0.7751

(47.34) (0.40) (1.52) (3.71) (33.03) (9.34) (84.84) (21.70) (21.58) (15.35) (85.54) (3.56) (4.20) (2.13) (0.42)

GM MO 7.8085 (121.24) 5.7515 (163.79) 0.0065 (0.12) 0.0067 (0.08) 1.6451 (25.48) 1.2558 (5.34) 1.8806 (1.81) 0.1811 (0.81) 0.5536 (83.33) 0.2604 (82.12) 0.0421 (19.15) 0.0068 (2.29) 0.8675 (36.05) 0.7512 (112.46) 0.5749 (23.64) 1.5864 (47.56) 0.3303 (22.88) 0.4776 (40.52) -0.2690 (53.27) -0.1804 (28.83) 0.8090 (134.09) 0.4688 (13.89) 7.1046 (35.35) 5.9851 (17.49) -3.2600 (1.80) -0.1099 (0.71) 0.1335 (11.59) 0.1367 (4.42) 6.4610 (0.12) 1.0416 (0.08)

DUK 6.5862 (69.90) 0.0485 (2.84) 3.4894 (2.46) 0.2966 (1.43) 0.6873 (96.49) 0.0058 (15.36) 2.0178 (38.30) 0.3685 (26.18) 0.0993 (14.34) -0.4256 (52.77) 0.5657 (126.65) 1.5347 (2.04) -0.6733 (1.20) 0.0392 (1.91) 0.1302 (3.13)

(in parentheses), estimated for each of the four reference companies. The estimation is based on weekly sampled data from January 2, 2002 to April 30, 2004. Panel B reports the estimates and t-statistics for the market price of risk for the two risk factors (z and v), computed from the model parameter estimates and covariance matrix.

4.2. The joint dynamics of return variance and default arrival rates Table 6 reports the maximum likelihood estimates and t-statistics of the structural parameters that control the joint dynamics of the diffusion variance rate and the default arrival rate.

The joint dynamics differ across different companies. Nevertheless, several common features emerge from the estimates.

First, the estimates for the risk-neutral mean-reverting coefficients κY , κ] and their statistical counterparts κPY , κP]

show that the default arrival rate is more persistent than the diffusion variance rate under both the risk-neutral measure Q and the statistical measure P . The difference in statistical persistence suggests that the diffusion return variance rates are strongly mean-reverting and hence predictable, but it is difficult to predict changes in the independent credit risk factor based on its past values. The difference in risk-neutral persistence dictates that the two factors have different impacts across the term structure of options and CDS spreads. Shocks on the diffusion variance rate affect the short-term options and CDS spreads, but dissipate quickly as

An increase in credit spreads is often associated with an increase in the option volatility level and a steepening in the negative slope of the implied volatility smirk.

γY = (κY − κPY )/σY ,

γ] = (κ] − κP] )/σ] .

(29)

We compute the market prices (γY , γ] based on the parameter estimates and report them in the bottom panel of Table 6.

The estimates for all four companies show positive market price for the diffusion variance risk, but negative market price for the independent credit risk.

Several studies, e.g., Bakshi and Kapadia (2003a,b) and Carr andWu (2004b), use stock and stock index options and the underlying time series returns to study the total return variance risk premia. They find that the risk premia are negative for some stocks, and highly negative for stock indexes. Our model decomposes the total risk on an individual stock into two components: risk in the diffusion variance rate and risk in the default arrival rate. By using both the CDS data and stock options data, we are able to separate the two sources of risks and identify their respective market prices. Our estimation suggests that for the four stocks, negative risk premia only come from the default arrival rate, but not from the diffusion variance rate.

Under our specification, market prices not only dictate the persistence difference of the risk factors under the two measures, but also create differences in the long-run means of the risk factors under the two measures. In particular, the statistical mean and the risk-neutral mean of the default arrival rate are given by, EP [λ] = EP [βY + ]] = β(κPY )− θY + (κP] )− θ] , EQ [λ] = EQ [βY + ]] = β(κY )− θY + (κ] )− θ] .

(30) (31)

where the solutions to Eλ (τ) and Dλ (τ) are given in equations (17) and (18). Hence, Eλ (τ)/τ measures the contemporaneous response of the credit spread term structure to unit shocks on the two risk factors. Figure 4 plots this response as a function of the credit spread maturity. The solid lines denote the response to the independent credit risk factor z and the dashed lines denote the response to the diffusion variance factor v. As the time to maturity approaches zero, the loading coefficient Eλ (τ)/τ converges to the instantaneous coefficient Eλ , which is normalized to unity for the credit risk factor z and is β for the diffusion variance rate v. The decay rate due to increases in time to maturity are controlled by the risk-neutral persistence of the two risk factors. The higher persistence in z dictates that its impact declines ore slowly as maturity increases than does the impact of the more transient factor v.

Another common finding among the four reference companies is that the default arrival rates all covary positively with the diffusion variance rate, as the estimates for the loading coefficient β are all positive. Furthermore, for all four companies, the estimates for the instantaneous correlation between stock return and return variance ρ are negative, consistent with the classic leverage effect. Finally, the literature has often found it difficult to separately identify the recovery rate and the default arrival rate using credit spread data alone (Houweling and Vorst (2005), Hull andWhite (2000), and Longstaff, Mithal, and Neis (2005)). As a result, researchers often assume a fixed recovery rate, usually between 30 to 50 percent, instead of estimating it along with other model parameters. By exploiting the overlapping information from the stock options market and the CDS market, we are able to separately identify the recovery rate (1−w) and the default arrival rate dynamics with high statistical significance. Our recovery rate estimates are between 47 and 81 percent, higher than the normally assumed values. Nevertheless, the estimates are in line with the high actual recovery rates during recent years reported in Altman (2006). They are also similar to the average recovery estimates by Das and Hanouna (2006) using corporate CDS spreads and sovereign recovery rate estimates by Pan and Singleton (2005) based on sovereign CDS term structures.

Figure 4 The contemporaneous response of the credit spread to unit shocks in the two risk factors Lines denote the contemporaneous response of the credit spread, defined as the difference between continuously compounded spot rate of a reference company and the corresponding spot rate for the libor/swap market, to unit shocks to the two sources of risks z (solid lines) and v (dashed lines).

F

1

0.9

0.8

0.8

0.7 0.6 0.5 0.4 0.3 0.2

0 0

0.7 0.6 0.5 0.4 0.3 0.2

0.1

0.1 2

4 6 Maturity, Years

8

0 0

10

MO

1

0.9

0.8

0.8

0.7 0.6 0.5 0.4 0.3 0.2

4 6 Maturity, Years

8

10

DUK

0.7 0.6 0.5 0.4 0.3 0.2

0.1 0 0

2

1

0.9 Credit Spread Response

The bottom panel of Table 6 also reports the two mean estimates based on the parameter estimates. The mean default arrival rate is much lower under the statistical measure P than under the risk-neutral Q for all four companies. These estimates are consistent with the empirical findings in the corporate bond literature that the historical average default probabilities are much lower than those implied from the corporate bond credit spreads.3 If we define the credit spread at a maturity τ as the difference between the continuously compounded spot rate on a reference company and the corresponding spot rate in the benchmark Eurodollar market, this spread is affine in the two risk factors under our model specification,

D (τ) E (τ) (32) &6(W, τ) = λ [W , + λ τ τ

GM

1

0.9 Credit Spread Response

the option and CDS maturity increases. Shocks on the more persistent credit risk factor last longer across the term structure of options and credit spreads.

For each risk factor, the difference in persistence under the two probability measures defines the market price of that factor’s risk:

0.1 2

4 6 Maturity, Years

8

10

0 0

2

4 6 Maturity, Years

8

» 10

Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation • 49

fsrforum • jaargang 12 • editie #5

Figure 5 The time series of return variance rates and default arrival rates.

Solid lines are the extracted time series of the instantaneous variance rate on the diffusion component of the stock return, with the scales on the left hand side. Dashed lines are the extracted time series of the default arrival rate on the reference companies, with the scales on the right hand side.

0.5

0 0 Dec01 Mar02 Jul02 Oct02 Jan03 Apr03 Aug03 Nov03 Feb04 Jun04 MO 0.2

0

vt

0.1

Jul02

Oct02

Jan03

0

Dec01 Mar02 Jul02 Oct02 Jan03 Apr03 Aug03 Nov03 Feb04 Jun04

0.2

0.5

0.1

0

Dec01 Mar02

Jul02

Oct02

Jan03

4.3. The term structure of credit default swap spreads

0

Apr03 Aug03 Nov03 Feb04 Jun04

DUK

1

zt

vt

0.2

0.2

Dec01 Mar02

0.4

0.4

zt

0.1

GM

zt

1

vt

vt

0.5

0.2

zt

F

1

0

Apr03 Aug03 Nov03 Feb04 Jun04

Figure 6 The term structure of credit default swap spreads.

The solid lines represent the mean term structures computed from the estimated model and the sample mean levels of the two risk factors.

Dashed lines are computed by setting vt to the sample average and zt to one standard deviation away from its sample mean. Dotted lines are computed by setting zt to the sample mean and varying vt one standard deviation away from its sample mean.

F

4.5 4

Credit Swap Spread, %

3 2.5 2

2 1.5 1 0.5

1.5 1 0

2

4 6 Maturity, Years MO

4

8

0 0

10

2

4 6 Maturity, Years DUK

4.5

8

10

4 3.5 3

Credit Swap Spread, %

Credit Swap Spread, %

3.5

2.5 2

3 2.5 2

1.5 0.5 1 0

2

4 6 Maturity, Years

8

0 0

10

2

4 6 Maturity, Years

8

10

Figure 7 The one-month√implied volatility smirks.

Moneyness is defined as ln(./6)/ Yτ. The solid lines are the mean implied volatility smirks at one-month maturity computed from the estimated model and the sample mean levels of the two risk factors. Dashed lines are computed by setting vt to its sample average and zt to one standard deviation away from its sample mean. Dotted lines are computed by setting zt to the sample mean and vt to one standard deviation away from its sample mean.

F

1

One−Month Implied Volatility, %

One−Month Implied Volatility, %

0.8

0.8 0.7 0.6 0.5 0.4

0.7 0.6 0.5 0.4 0.3

0.3 −1.5

−1

−0.5

0 Moneyness

0.5

1

1.5

0.2 −2

2

MO

0.65

−1

−0.5

0 Moneyness

0.5

1

1.5

2

DUK

0.9 One−Month Implied Volatility, %

0.55 0.5 0.45 0.4 0.35 0.3 0.25

0.8 0.7 0.6

−1.5

−1

−0.5

0 Moneyness

0.5

1

1.5

2

To understand how the two risk factors contribute to the pricing of stock options, we compute and plot the onemonth implied volatility smirks across different moneyness in Figures 7 at different risk levels. In computing the option values and constructing the implied volatility smirks, we assume zero interest rates and dividend yields, √ and define the moneyness as ln (./6)/ Yτ , which can be approximately interpreted as the number of standard deviations that log spot is below log strike. The solid lines are the mean implied volatility smirks evaluated at the sample means of the two risk factors. The two dashed lines in each panel are generated with the diffusion variance rate at its sample mean and the independent credit risk factor one standard deviation away from its sample mean. Hence, they capture the impact of shocks in the independent credit risk factor. The two dotted lines in each panel are generated by setting the independent credit risk factor at its sample mean and the diffusion variance rate at one standard deviation away from its sample mean. Hence, the dotted lines capture the impact of shocks in the diffusion variance rate.

0.5 0.4 0.3

0.2 −2

−1.5

1

0.6 One−Month Implied Volatility, %

GM

0.9

0.9

The estimated model parameters on the four companies generate different mean term structures on the CDS spreads. Nevertheless, the impacts of the two risk factors show similar patterns. First, a one standard deviation move of the independent credit risk factor has a much larger impact on the CDS spreads than a one standard deviation move of the diffusion variance rate, supporting the hypothesis that the CDS market is mainly a market for credit risk.

Furthermore, the impact of the diffusion variance rate is mainly at short maturities. Its impact declines rapidly as maturity increases. In contrast, the impact of the independent credit risk factor is much more persistent.

4.4. The implied volatility smirk and term structure

1.5 1

0.2 −2

Given the model parameter estimates in Table 6, we can compute the term structures of the CDS spreads at different levels of the risk factors (v, z). In Figure 6, we plot the model-implied mean term structure of the CDS spreads in solid lines, where we set the risk levels to their respective sample averages. The two dashed lines in each panel are constructed by setting the diffusion variance rate v to its sample mean and the independent credit risk factor to one standard deviation away from its sample mean. The two dotted lines in each panel reflect the impact of one standard deviation movements of the diffusion variance rate while holding the independent credit risk factor to its sample mean.

2.5

3.5 Credit Swap Spread, %

GM

3

Figure 5 plots the extracted time series on the variance rate (solid line) and the default arrival rate (dashed line), with scales on the left and right hand sides of the y-axis, respectively. The extracted time series show co-movements that match the time series plots of the CDS spreads and implied volatilities in Figure 2. The plots for all four companies show a spike for both the variance rate and the default arrival rate in late 2002, a reflection of the financial stress during that period.

0.2 −2

−1.5

−1

−0.5

0 Moneyness

0.5

50 • Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation

1

1.5

2

The implied volatility smirks show similar patterns across the four companies. Furthermore, variations in the diffusion variance rate level lead to relatively uniform shifts in

the implied volatility smirk across moneyness. In contrast, the impact of the independent credit risk factor is mainly at low strikes. The impact of the credit risk factor on far out-ofthe-money call option implied volatilities (at high strikes) is negligible. To see how the impact changes at different maturities, we also plot in Figure 8 the corresponding implied volatility smirk for six-month options. As for the one-month implied volatility smirk, the impacts of the diffusion variance rate (dotted lines) are relatively uniform across all moneyness levels, whereas the impacts of the independent credit risk factor (dotted lines) are stronger at lower strikes. Comparing Figures 7 and 8 also brings out visible differences: The impact of the independent credit risk factor is larger at longer maturities.

Figure 9 plots the term structure of the at-the-money implied volatilities at different risk levels. Again, we use the solid line to denote the mean term structure, the dashed lines to capture the impact of one standard deviation moves on the independent credit risk factor, and the dotted lines to capture the impact of the diffusion variance rate. At short option maturities, we find that for all four companies, the impact of the diffusion variance rate is much larger than the impact of the independent credit risk factor. However, as maturity increases, the influence of the diffusion variance rate declines, whereas the influence of the credit risk factor increases. For six-month options on GM, the impacts of the two risk factors become comparable in magnitude.

Figure 8 The six-month √ implied volatility smirks.

Moneyness is defined as ln(./6)/ Yτ. The solid lines are the mean implied volatility smirk at one-month maturity computed from the estimated model and the sample mean levels of the two risk factors. Dashed lines are computed by setting vt to its sample average and zt to one standard deviation away from its sample mean. Dotted lines are computed by setting zt to the sample mean and vt to one standard deviation away from its sample mean.

F

1

Six−Month Implied Volatility, %

Six−Month Implied Volatility, %

0.9 0.8 0.7 0.6 0.5 0.4

0.8 0.7 0.6 0.5 0.4 0.3

−2

−1.5

−1

−0.5

0

0.5 Moneyness

1

1.5

0.2 −2

2

MO

0.65

−1.5

−1

−0.5

0

0.5 Moneyness

1

1.5

2

1

1.5

2

DUK

1

0.6

0.9

0.55

Six−Month Implied Volatility, %

Six−Month Implied Volatility, %

GM

1 0.9

0.5 0.45 0.4 0.35 0.3

0.8 0.7 0.6 0.5 0.4 0.3

0.25 0.2 −2

−1.5

−1

−0.5

0 0.5 Moneyness

1

1.5

0.2 −2

2

−1.5

−1

−0.5

0 0.5 Moneyness

Figure 9 The term structure of at-the-money implied volatilities.

The solid lines are the mean term structure of the at-the-money forward implied volatility computed from the estimated model and the sample mean levels of the two risk factors. Dashed lines are computed by setting vt to the sample average and zt one standard deviation away from its sample mean. Dotted lines are computed by setting zt to the sample mean and vt to one standard deviation away from its sample mean.

F

0.65 0.6

0.45

0.55

ATM Implied Volatility, %

ATM Implied Volatility, %

GM

0.5

0.5 0.45 0.4

0.4 0.35 0.3 0.25

0.35

1

2

0.38

3 4 Maturity, Months MO

5

0.2

6

1

2

3 4 Maturity, Months DUK

5

6

1

2

3 4 Maturity, Months

5

6

0.55

0.36

0.5

0.32

ATM Implied Volatility, %

ATM Implied Volatility, %

0.34

0.3 0.28 0.26 0.24

0.45 0.4 0.35 0.3

0.22 0.25

0.2 0.18

1

2

3 4 Maturity, Months

5

6

0.2

»

Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation • 51

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fsrforum â&#x20AC;˘ jaargang 12 â&#x20AC;˘ editie #5

We ďŹ nd that the independent credit risk factor dominates CDS spreads at long maturities, but stock return volatility can also affect CDS spreads at short maturities, due to positive co-movements between diffusion variance and default arrival.

5. Summary and Conclusions Based on documented evidence on the joint movements between CDS spreads and stock option implied volatilities, we propose a dynamically consistent framework for the joint valuation and estimation of stock options and CDS spreads written on the same reference company. We model the possible default of a company by a Poisson process with stochastic arrival rate, and we assume that the stock price falls to zero upon default. We model the pre-default stock price as following a continuous process with stochastic volatility. We assume that the default arrival rate and diffusion variance rate follow a bivariate process with dynamic interactions that match the empirical evidence linking stock option implied volatilities and CDS spreads. Importantly, our dynamic speciďŹ cation allows both common movements and independent variations between the two markets. Under this joint speciďŹ cation, we derive tractable pricing solutions for stock options and credit default swaps. We then estimate the joint dynamics of the diffusion variance rate and the default arrival rate using data on stock option implied volatilities and CDS spreads for four of the most actively traded reference companies.

Estimation of the model parameters shows that the default arrival rate is much more persistent than the diffusion variance rate under both the statistical measure and the risk-neutral measure. The statistical persistence difference suggests different degrees of predictability. The risk-neutral difference in persistence suggests that the default arrival rate has a more long-lasting impact on the term structure of option volatilities and CDS spreads than does the diffusion variance.

The estimation also highlights the interaction between market and credit risk in pricing stock options and credit default swaps. We ďŹ nd that the independent credit risk factor dominates CDS spreads at long maturities, but stock return volatility can also affect CDS spreads at short maturities, due to positive co-movements between diffusion variance and default arrival. On the other hand, the default arrival rate affects stock option pricing through both its correlation with the diffusion variance rate and its direct effect on the risk-neutral drift of the return process. We ďŹ nd that the impact of the diffusion variance rate on the implied volatility is relatively uniform across different moneyness levels, while the impact of the credit risk factor is mainly on options at low strikes. Furthermore, the impact of the credit risk factor on stock options prices increases with the option maturity. When the option has about six months to maturity, the contribution of the credit risk factor to option pricing is comparable in magnitude to the contribution of the diffusion variance rate.

We conclude that one can learn more about the stock options and the CDS market by developing a model that integrates both markets, rather than having separate models for each market. In particular, one can identify the recovery rate on the bond insured by CDS much more effectively by adjoining stock option prices to CDS data.

References on request.

Appendix.

Generalized Fourier transform of stock returns To derive the generalized Fourier transform:

Z Ď&#x2020; (X) â&#x2030;Ą EW H[S â&#x2C6;&#x2019;

7

W

ÎťV GV HLX OQ 37 /3W ,

X â&#x2C6;&#x2C6; D â&#x160;&#x201A; C,

(A1)

we use the language of stochastic time change of Carr and Wu (2004a) and deďŹ ne

TW â&#x2030;Ą

Z 7 W

YV GV,

TW] â&#x2030;Ą

Z 7 W

]V GV,

TW Îť â&#x2030;Ą

Z 7 W

ΝV GV = T W ] + βT W .

Then, conditional on no default during the time horizon [t,T], with = T â&#x2C6;&#x2019;t, we can write the log stock return as

OQ (37 /3W ) = (U(W, 7 ) â&#x2C6;&#x2019; T(W, 7 )) Ď&#x201E; + T W Îť + :T3W â&#x2C6;&#x2019; T W ,

(A2)

where r(t,T) and q(t,T) denote the continuously compounded spot interest rates and dividend yields of the relevant maturity.

The discounted generalized Fourier transform becomes,

Ď&#x2020; (X) = EW H[S â&#x2C6;&#x2019;T W Îť + LX (U(W, 7 ) â&#x2C6;&#x2019; T(W, 7 )) Ď&#x201E; + LXT W Îť + LX:T3W â&#x2C6;&#x2019; LXT W

= EW H[S LX:T3W + X T W H[S â&#x2C6;&#x2019;T W Îť + LX (U(W, 7 ) â&#x2C6;&#x2019; T(W, 7 )) Ď&#x201E; + LXT W Îť â&#x2C6;&#x2019; LXT W â&#x2C6;&#x2019; X T W

= H[S (LX (U(W, 7 ) â&#x2C6;&#x2019; T(W, 7 )) Ď&#x201E;) EWM H[S â&#x2C6;&#x2019; ( â&#x2C6;&#x2019; LX) T W Îť â&#x2C6;&#x2019; LX + X T W

= H[S (LX (U(W, 7 ) â&#x2C6;&#x2019; T(W, 7 )) Ď&#x201E;) EWM H[S â&#x2C6;&#x2019; ( â&#x2C6;&#x2019; LX) T W ] â&#x2C6;&#x2019; ( â&#x2C6;&#x2019; LX)β + LX + X T W ,

where the new measureM is deďŹ ned by

GM

= H[S LX:T3W + X T W ,

GQ W

under which the drift of the two dynamic processes change to: Â&#x2014;M Y

= θY â&#x2C6;&#x2019; (ÎşY â&#x2C6;&#x2019; LXĎ&#x192;Y Ď )Y (W) = θY â&#x2C6;&#x2019; ÎşM Y Y (W) ,

Â&#x2014;M ]

= θ] â&#x2C6;&#x2019; Îş] ] (W) .

We have

Z Ď&#x2020; (X) = H[S (LX (U(W, 7 ) â&#x2C6;&#x2019; T(W, 7 )) Ď&#x201E;) EWM H[S â&#x2C6;&#x2019;

W

7

E , [V GV

with [W = [YW , ]W ] E = [EY , E] ] EY = ( â&#x2C6;&#x2019; LX)β + LX + X and E] = â&#x2C6;&#x2019; LX Since the risk factors x follow afďŹ ne dynamics, the solution is exponential afďŹ ne in xt ,

Ď&#x2020; (X) = H[S (LX (U(W, 7 ) â&#x2C6;&#x2019; T(W, 7 )) Ď&#x201E;) H[S(â&#x2C6;&#x2019;D(Ď&#x201E;) â&#x2C6;&#x2019; E(Ď&#x201E;) [W ),

where the coefďŹ cients can be solved analytically as in (10) and (11).

Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation â&#x20AC;˘ 53

fsrforum • jaargang 12 • editie#4

All Options

Bedrijspresentatie Schrijver

Op 1 September 2010, start er een nieuwe Traineeship bij All Options.

Jaap, trainee trader bij All Options licht een tipje van de sluier op.

Wat is je functie binnen All Options?

Twee maanden geleden ben ik hier in een Traineeship als handelaar begonnen. Je begint vanaf 0, leert hoe opties werken, hoe handelen werkt en gaat ook al een beetje handelen. Je leert steeds meer en groeit zo in het vak ‘trading’.

Hoe ziet over het algemeen het verloop van je carrière binnen All Options eruit?

Je begint aan een Traineeship, dat duurt officieel een half jaar, waar je in het begint ook wat theorie leert. Daarna krijg je steeds meer verantwoordelijkheid en groei je binnen trading door.

Je wordt Junior Trader, Trader en uiteindelijk Senior Trader. Eventueel kun je ook manager worden binnen een bepaalde sector later en groei je op die manier door. Plus dat je natuurlijk ook steeds meer risico kan nemen.

Waarom ben je als trader aan de slag gegaan bij All Options?

Ik wilde zelf heel graag handelen. Bij All Options vind ik het leuk dat er een Traineeship is en daar de nadruk ook echt op het trainen ligt. Je leert heel veel, dat zit hier goed in elkaar, dat is voor mij heel belangrijk. Verder is het ook een beetje gevoelsmatig, een leuke locatie, een hard groeiend bedrijf en leuke mensen. Dat speelt allemaal mee.

Hoe sluit de studie Econometrie aan op het traden? Is er veel voorkennis nodig of is het toch een kwestie van veel ervaring op doen in de praktijk?

Allebei, maar in mijn derde jaar heb ik de richting financieel gekozen, waarbij ik ook een vak ‘Option Pricing and Hedging’ kreeg en dat ging echt over optietheorie. Veel dingen die ik hier dus letterlijk terugzie. Zeker theoretisch sluit het wel heel direct aan op bepaalde vakken die ik heb gedaan, vooral Finance vakken. Maar er is natuurlijk ook heel veel te leren in de praktijk wat je niet van tevoren weet. Er zijn een hoop praktische vaardigheden die je echt alleen hier op de werkvloer leert.

54 • Bedrijfspresentatie All Options

Sommige mensen hebben het idee dat een trader toch iemand is die de hele dag achter de computerschermen zit. Klopt dit beeld of zou je dit anders omschrijven?

Zo ziet het er inderdaad uit, dat is ook echt zo. Er zijn hier heel veel schermen. Er gebeurt veel in de markt en dat zie je op je scherm. Je hebt bepaalde posities, je wint en verliest geld en dat moet je goed in de gaten houden. Dan is er ook nog het nieuws, alle informatie heb je voor je op het scherm.

Ondanks dat je ook contact hebt met je collega’s en veel met ze overlegt, maar uiteindelijk hou je altijd een oogje op je scherm.

Welke interesses en eigenschappen zoekt All Options in een toekomstige werknemer?

Zeker theoretisch sluit het wel heel direct aan op bepaalde vakken die ik heb gedaan, vooral Finance vakken.

Maar er is natuurlijk ook heel veel te leren in de praktijk wat je niet van tevoren weet.

Je moet het leuk vinden om bezig te zijn met het ‘kopenverkopen’ wereldje. Niet perse met aandelen, maar gewoon het principe. Als je dat spelletje leuk vindt en je hebt interesse in aandelenmarkten. Het is gewoon heel erg leuk werkt, als het je ligt. Je zit toch wel de hele dag achter de schermpjes. Je krijgt heel veel verantwoordelijk, je kan de hele dag beslissingen nemen. Als jij iets wilt, dan doe je dat. En aan het eind van de dag zie je wat je hebt gedaan. Je resultaten zijn erg goed meetbaar.

Wil je ook trader bij All Options worden? Stuur dan je CV en motivatie naar career@alloptions.nl of solliciteer via www. alloptions.nl

Bedrijfspresentatie All Options • 55

Wat belangrijk is, laat je niet los.

d Ik wil ruimte om te groeien. Waar zet ik

Waar je ook bent, belangrijke beslissingen zijn nooit ver weg. In je rol als accountant en bij het bepalen van je volgende carrièrestap.

Bij Grant Thornton begrijpen we dat je voortdurend bezig bent met je groei. Sterker nog, wij zijn er zelf ook mee bezig. Onder andere door jouw ambities alle ruimte te geven en door je talent te versterken met een goed doortimmerde opleidingsaanpak. Meer over ons op onze website.

www.carrierebijGT.nl

ol v e

g

e end

stap?

Grant Thornton bij jou in de buurt: Alphen aan den Rijn - Amsterdam Boskoop - Gouda - Leiden - Rijswijk Rotterdam - Woerden

Accountancy - Belastingen - Advies

fsrforum • jaargang 12 • editie #5

Woord van de voorzitter

Peter James

Waarde lezer, Inmiddels zit het collegejaar erop en kan ik samen met het XIIe FSR Bestuur terug kijken op een geslaagd academisch jaar 2009-2010. Het belangrijkste is dat het nieuwe bestuur bekend is en zij vol zelfvertrouwen het stokje over zullen nemen als FSR bestuurder. Volgend jaar bestaat het bestuur uit zes in plaats van zeven personen. En dat is niet omdat we geen zeven goede bestuurders kunnen vinden, maar omdat we deze talenten liever fulltime inzetten voor de FSR in plaats van semiparttime/semifulltime. Dus ondanks de enorme teleurstelling omtrent de WK finale, gaat het nog steeds goed met de Financiële Studievereniging Rotterdam.

Het blijft moeilijk om te voorspellen hoe de financiële economische situatie zich de komende weken zal gaan ontwikkelen. Aan de ene kant hebben de Europese markten weer het vertrouwen door te beleggen in Griekse staatsobligaties, maar aan de andere kant verlaagt Moody’s de kredietbeoordeling van Portugal. Gelukkig zijn het allebei geen bepalende landen in de Europese economie, zoals Duitsland, waardoor over het algemeen de financiële markten rustig zijn de afgelopen weken. Na deze zin is het overigens wel belangrijk om te zeggen dat dit voorzitterswoord op 13 juli jongstleden geschreven is, hoe de economische situatie er over vier weken uit ziet is onvoorspelbaar.

Komend jaar zullen weer vier voormalig actieve leden van de FSR het bestuur gaan vormen.

Dat heeft weer eens bevestigd hoe belangrijk de actieve leden zijn voor de FSR. Dus hebben wij hen op woensdag 23 juni op gepaste wijze bedankt voor hun diensten. Uiteraard was het actievenweekend in Milaan een bewijs dat wij graag met het hele team, het XIIe FSR Bestuur en alle actieve leden, een leuk uitje hebben, maar 23 juni was de afsluiting van het jaar. Speciaal voor deze gelegenheid hadden wij Professor dr. Casper de Vries uitgenodigd om iets te vertellen over de economische situatie. Interessant was dat Prof. Dr. Casper de Vries niks kon voorbereiden toen wij elkaar benadereden rond 8 juni, omdat hij simpelweg niet wist hoe de situatie eruit zou zien. Gelukkig heeft zijn ervaring en immense kennis er toe geleid dat de geplande presentatie en Q&A sessie van 20 minuten uit liep tot een uur. En voor de aanwezigen heeft hij een goed beeld geschetst van de belangen die op het spel staan. Economie en politiek gaan nu eenmaal niet altijd lekker samen.

Op een lager niveau, en wel die van de Erasmus Universiteit, maken wij ook een soort van politieke spellen mee. De FSR is de studievereniging voor alle financiële georiënteerde studenten.

Het maakt dus niet uit of dit een economie, bedrijfskund of een geneeskunde student. De kwaliteit van onze vereniging en de service die wij willen verlenen aan onze leden staat altijd voorop en iedereen die daar gebruik van wilt maken is meer dan welkom. Ondanks onze open houding kan er nog wel eens de gedachte bestaan dat wij alleen voor studenten aan de economische faculteit zijn. Dit is dus niet waar en helaas moeten wij ons soms politiek opstellen om dat te benadrukken bij andere studieverenigingen.

behoefte van haar leden voorop stellen in de planning van haar activiteiten.

Komend jaar zullen Luc, Eefke, Kim, Bart, Ellis en Sandhya het stokje gaan overnemen. In deze groep heb ik zeer veel vertrouwen, ook al staat er een uitdagend jaar op het programma.

Zoals eerder gezegd zal een team van zes de FSR gaan besturen. Dit team staat voor een aantal grote uidagingen, waarbij ze aan het eind van het jaar zich zo kunnen onderscheiden dat misschien weinigen meer praten over het goede academische jaar 2009-2010. De bestemmingen voor het IRP en de EFT zijn door het XIIIe bepaald, de participerende Investment Banks tijdens de IBC zijn uitgebreid, de Multinational Battle zal voor een grote uitdaging staan om het weer tot een succesvol te laten zijn en de grote accounting kantoren (na de Big Four) krijgen dit jaar de kans om zich in een volledige dag te laten zien aan de studenten aan de Erasmus. Ondanks dat ik nu niet in ga op de mogelijke nieuwe activiteiten, wil ik onze leden graag uitnodigen om volgend jaar wederom met alle enthousiasme deel te nemen aan onze activiteiten.

Laat de kans niet liggen!

De kans die ik persoonlijk afgelopen jaar heb gekregen als voorzitter van de FSR is onvergetelijk. En in een van de laatste zinnen van dit jaar wil ik het XIIIe bestuur der FSR heel veel succes en plezier wensen. Een jaar vol onvergetelijke leerzame momenten die waardevol zullen zijn. Als afsluiting nodig ik graag, namens het huidige bestuur, iedereen uit op de Wissel ALV op donderdag 2 september!

Met vriendelijke groet, Peter James Voorzitter FSR Bestuur 2009-2010

Het academisch jaar is over twee weken, na de laatste herkansingsweek, echt voorbij. Het jaar begon met de grote opkomst bij de WALV, daarna de geslaagde recruitment maand voor investment banking en accountants kantoren. De studiereizen resulteerde in blije gezichten, die logisch zijn na mooie reizen naar Istanbul en Mumbai. En de afsluiting zat hem in de Corporate Finance Competition in Duin & Kruidberg en de Investment Banking Masterclass. Waarbij de inhoudelijke eerste dag een zeer interessante was. Wij, van het XIIe FSR Bestuur hopen dat wij onze leden dat hebben aangeboden waar behoefte aan was. En ten alle tijden zal de FSR de

Verenigingsnieuws • 57

fsrforum • jaargang 12 • editie #5

International Research Project 2010 ‘India, you hate it or you love it, but you will never forget it’ Nienke Louwmans

Op 26 april 2010 stonden de deelnemers van het International Research Project (IRP) en onze begeleidende professor Ronald Huisman om 5 uur in de ochtend te trappelen op Schiphol, klaar voor vertrek naar Mumbai! Na een lange reis kwamen we maandagnacht aan op het vliegveld van Mumbai.

In die eerste dagen kregen we al snel een indruk van het enorm grote verschil tussen arm en rijk. Enerzijds is de wegeninfrastructuur, de elektriciteit- en watervoorzieningen bijvoorbeeld nog onderontwikkeld. Het stadsverkeer is in onze beleving vaak levensgevaarlijk (behalve voor de heilige koeien) en van sociale zekerheid hebben ze in India nog nooit gehoord. Geen geld hebben en kasteloos zijn betekent arm, ziek en achtergesteld blijven. Anderzijds is er eveneens een leven van hoge standaarden, luxe, de mooiste clubs, modellen, restaurants en hoogstaande internationaal georienteerde bedrijven.

Dinsdag 27 april vond ons eerste bedrijfsbezoek plaats bij TATA Consultancy Services (TCS). TCS is een van de indrukwekkende, snel groeiende bedrijven van TATA Group (onder andere eigenaar van Corus). Dit bezoek was erg indrukwekkend, het ontvangst was ontzettend gastvrij en de mensen waren geinteresseerd en hadden een interessant verhaal. Een belangrijke eerste les die meteen duidelijk werd is dat niet alleen Amerikaanse en Europese bedrijven domineren op het wereldtoneel maar familiebedrijven zoals Tata uit India enorm in opkomst zijn.

In de avond hadden we met de groep een welkomsdiner aan het strand van Mumbai. We merkten meteen hoe druk en chaotisch de stad kon zijn. Iedereen vertrok rond half 8 van het hotel met taxi´s en tuk tuks naar het restaurant waarna de eerste rond een uur of acht aankwamen en de laatste pas half tien! Het is dan ook een hele sport om op de goede locatie aan te komen met riksjarijders die minimaal engels spreken, constant getoeter om je heen en in een temperatuur rond de veertig graden celsius. Desalniettemin is iedereen om je heen altijd bereid om je te woord te staan en je te helpen om op de goede bestemming te arriveren!

De volgende dag bezochten we Suzlon Energy. Dit is een bedrijf dat zich specialiseert in het produceren, plaatsen en beheren van windmolens. Een ambitieus bedrijf wat net als TATA consultancy services indrukwekkende groeicijfers heeft. Dit bezoek was erg interactief en zeker interessant voor de deelnemers die onderzoek doen op het gebied van de energiemarkt.

Donderdag gingen we bij Ernst and Young op bezoek. Hier konden allerlei vragen worden gesteld op het gebied van accountancy. De verschillen tussen de praktijk in Nederland en in India kwamen hier goed naar voren. Aansluitend hebben we mogen genieten van een luxe diner in een van de mooiste restaurants van Mumbai samen met een aantal werknemers van E&Y Mumbai.

58 • Verenigingsnieuws

Vrijdag waren wij uitgenodigd door het Consulaat der Nederlanden om samen Koninginnedag te vieren in het luxe Trident Hotel. Naast Nederlanders die in Mumbai wonen en werken waren er voornamelijk veel zakelijke partners van het consulaat. Een ideale gelegenheid om leuke contacten op te doen!

Zaterdag gingen we met ATMA (een NGO in Mumbai) een van de sloppenwijken van de stad bezoeken. Het was verrassend om te zien hoe goed deze slopperwijken zijn gestructureerd en ingericht. Alle bewoners in de sloppenwijk hebben bepaalde taken (bijvoorbeeld op het gebied van recyclen). Uiteraard hebben we hier geen foto’s mogen maken. Het ging vooral om de ervaring. Deelnemers kregen op deze manier de kans om ook deze kant van de stad en India te zien. De opbrengsten die Atma hiermee verdient gaan allemaal naar de mensen in de slopperwijken.

Na een cultureel bezoek aan Elephanta Island op zondag, bezochten we maandag TATA Institute of Social Science (TISS). TISS is een universiteit waar de Erasmus Universiteit ook een samenwerkingsverband mee heeft. Wegens de regentijd die er zat aan te komen en de hittegolf waren alle studenten in deze periode vrij. Toch gaf de rondleiding een goed beeld van hoe het studeren daar is.

» Verenigingsnieuws • 59

fsrforum • jaargang 12 • editie #5

Deze laatste dagen hebben we vooral gezien dat het leven in rustigere gebieden niet te vergelijken is met de drukte en chaos van de grote stad.

Dinsdagochtend bezochten we Blue Frog, een groot platenlabel en club. Bekende artiesten van over de hele wereld treden op in deze club. Daarnaast hebben ze ook verschillende recording studios. Na een leuk welkom en een rondleiding kregen we de mogelijkheid om zelf een nummer op te nemen in de studio! Ronald en Ernst hebben daar een onvergetelijke optreden neergezet en opgenomen van Brown Eyed

Girl van Van Morrison! Later die middag hebben we een korte presentatie gehad bij de Bombay Stock Exchange (BSE).

Woensdag vond een bedrijfsbezoek plaats bij Special Economic Zones (SEZ). Dit bedrijf is bezig met het ontwikkelen en bouwen van een volledige stad (belastingparadijs) naast Mumbai. Het is niet voor te stellen hoeveel geld daar in om gaat, hoe groot het project is en hoeveel arbeidplekken dat creëert.

Donderdag zijn we met de hele groep naar Goa gevlogen om de laatste dagen met elkaar door te brengen. Goa is de kleinste deelstaat aan de westkust van India onder Maharashtra.

Deze deelstaat heeft een heel mooi landschap, varierend van goudgele stranden, baaien en rotspartijen tot mooie bamboebossen. In tegenstelling tot Mumbai is het verkeer in Goa een stuk rustiger en konden we hier scooters huren en lekker rond touren. Deze laatste dagen hebben we vooral gezien dat het leven in rustigere gebieden niet te vergelijken is met de drukte en chaos van de grote stad.

60 • Verenigingsnieuws

De reis naar India was een geweldige onvergetelijke ervaring.

In september zullen de deelnemers de resultaten van hun onderzoek toelichten en zal het boek, waarin alle onderzoeken zijn gebundeld, worden gepresenteerd. Mede dankzij de leuke groep deelnemers was het nu al een zeer geslaagd project! Ze zeggen ook wel: ‘India, you hate it or you love it, but you will never forget it.’ Dit klopt zeker en ik denk dat alle deelnemers het al met al ook een supermooie ervaring vonden, India is bijzonder!

Graag wil ik mijn commissie Ellis Heijboer, Joost Scholman, Onne Tjerkstra en Kim de Vries bedanken voor hun onmisbare input, doorzettingsvermogen en gezelligheid gedurende het hele jaar! Daarnaast wil ik de begeleidende docenten Karen Maas en Ronald Huisman bedanken voor hun input en ondersteuning door middel van diverse colleges en begeleiding aan de deelnemers vanaf januari. Ronald Huisman in het bijzonder wil ik graag bedanken voor zijn begeleiding en gezelligheid gedurende het project op locatie in India! Tot slot wil ik ook van de gelegenheid gebruik maken om alle bedrijven die ons project hebben gesteund te bedanken voor hun steun en interessante bedrijfsbezoeken en presentaties.

In het bijzonder wil ik de volgende bedrijven bedanken: E&Y, KPMG, Deloitte, ESJ, Steens&Partners en Capgemini. Ik wens mijn opvolger van het dertiende FSR bestuur, Ellis Heijboer, heel veel succes met de volgende editie van dit bijzondere project!

Met vriendelijke groet, Nienke Louwmans Voorzitter IRP FSR Bestuur 2009/2010

Verenigingsnieuws • 61

fsrforum • jaargang 12 • editie #5

Activiteitenverslag Corporate Finance Competition (CFC) Karin Knegt, Fouad Mehadi, Esther Schipper

Dit jaar ging op woensdag 19 mei ging de Corporate Finance Competition (CFC) van start. Tijdens deze 3-daagse business course strijden 4 teams om de eerste plaats door middel van het maken van corporate finance gerelateerde cases. De CFC vindt traditiegetrouw elk jaar plaats in een vijf-sterrenhotel. Dit jaar was dat het landgoed Duin & Kruidberg in Santpoort, bij sommigen wellicht bekend uit het boek ‘De Prooi’ over de perikelen rond ABN Amro.

Na de laatste case is de prijsuitreiking en wordt bekendgemaakt dat team Groen heeft gewonnen.

Nadat alle studenten waren aangekomen verzamelden we in de bar om met iedereen te lunchen. De competitie ging meteen van start toen de teams bekend werden en er werd al hevig gediscussieerd over wie de beste capaciteiten had om uiteindelijke winnaar van deze competitie te worden.

Na deze introductie schoven de medewerkers van ABN Amro CFCM ook aan om de studenten beter te leren kennen. Zij zouden meteen de eerste case verzorgen. Na deze eerste case konden de studenten elkaar nog beter leren kennen tijdens een gezellig diner op het balkon van het landgoed.

Aansluitend was er op woensdagavond een case van Ernst&Young. Tijdens deze case werkten de teams samen op een laptop en werd er een model uitgewerkt. De teams zijn al iets beter op elkaar ingespeeld en de avond eindigt in een gedeelde eerste plaats. De eerste dag werd afgesloten met een borrel in de bar waar de avond eindige met een rondje van dé Bank, aangeboden door Gerrit Zalm, die toevallig die dag ook aanwezig was op het landgoed voor een vergadering.

De volgende dag was er al vroeg ontbijt en begon de derde case van RBS. Op dit moment was de spanning om te snijden want er waren twee teams die samen de leiding namen en de andere twee teams stonden nu ook samen op een gedeelde tweede plaats. Tijdens de case wordt er druk overlegd binnen de teams en worden de taken verdeeld. Na de case was er een gezamenlijke lunch waarbij de medewerkers van RBS ook aanwezig waren. Doordat het erg goed weer was en de zon scheen konden we de lunch houden in de tuin van het landgoed.

Op donderdagmiddag is het tijd voor wat ontspanning en vertrekken we met de fiets naar het strand van IJmuiden. Daar gaan we met een drankje lekker in de zon zitten en de middag vliegt voorbij.

62 • Verenigingsnieuws

Op donderdagavond is het doordraaidiner met alle bedrijven.

Tijdens dit diner zitten de deelnemers elke gang bij een ander bedrijf aan tafel enzo kunnen zij al hun vragen aan de bedrijven stellen. Tijdens de borrel na het diner hebben de studenten de kans om ook de bedrijven te spreken waarbij ze niet aan tafel zaten.

Vrijdagochtend gaan we weer op tijd ontbijten met elkaar, want er staan nog twee cases gepland. In de ochtend komt Rabo Securities en in de middag is de case van BDO Corporate Finance. De laatste dag doen de teams extra hun best om nog punten binnen te halen en op een mooie positie te eindigen. In de middag case gaan de teams bij elkaar om de tafel zitten en wordt er onderhandeld over de voorwaarden van een transactie.

Na de laatste case is de prijsuitreiking en wordt bekendgemaakt dat team Groen heeft gewonnen. Als afsluiting wordt er nog een groepsfoto gemaakt en nemen we in de tuin van het landgoed nog een drankje.

De Corporate Finance Competition was dit jaar weer een geslaagd evenement! Studenten met interesse in corporate finance moeten zich volgend jaar zeker inschrijven. Tijdens het evenement leer je de verschillende bedrijven goed kennen en kom je meer te weten over het werk in corporate finance.

Wij van de Corporate Finance Competition Commissie willen alle deelnemende bedrijven, de studenten en Duin en Kruidberg bedanken voor hun inzet en enthousiasme!

Karin Knegt Fouad Mehadi Esther Schipper

Verenigingsnieuws • 63

fsrforum • jaargang 12 • editie #5

XIIIe FSR Bestuur

Bart Lips

Ik ben Bart Lips, 21 jaar en na een jeugd in het gezellige Brabantse Tilburg al 3 jaar trotse inwoner van de enige echte handelsstad van Nederland, namelijk Rotterdam. Ik zal dit jaar Marc van Erkelens opvolgen als Commissaris Externe Betrekkingen bij de FSR. Mijn naam komt wellicht bekend voor, want buiten het feit dat ik vorig jaar erg actief bent geweest binnen de FSR, als commissielid van de Multinational Battle en van de Traders Trophy, ben ik ook het jongere broertje van Bram Lips, de aftredend Commissaris Activiteiten.

Als Commissaris Externe Betrekkingen ga ik zorgen voor de continuiteit van de vereniging, want zonder medewerking van onze partners, kan de FSR niet verder bestaan en al helemaal niet doorgroeien om een steeds bekender gezicht op de Erasmus Universiteit te worden. Verder hou ik me nu nog bezig met de afronding van mijn Bachelor Business Administration en hockey ik ook nog redelijk fanatiek bij Leonidas in het derde heren team. Buiten hockey zijn mijn hobby’s zeilen en bovenal skiën.

Ik hoop dat ik een erg leuk jaar tegemoet ga, waarin ik veel kan leren en bovenal een onvergetelijke tijd samen met mijn andere bestuurgenoten kan hebben.

Ellis Heijboer

Hi, ik zal mezelf even kort voorstellen: Ik ben Ellis Heijboer en zal het komende jaar de functie van Commissaris Activiteiten op mij nemen. Dit betekent dat ik onder andere het mooie IRP en de Big4-cycle mag organiseren. Naast deze bestuursfunctie bij de FSR ben ik momenteel bezig met het afronden van mijn bachelor economie en bedrijfseconomie. Ook werk ik parttime bij een accountantskantoor en ben ik in mijn vrije tijd bezig met sporten, uitgaan en reizen (vandaar mijn liefde voor het IRP ).

Overigens ben ik geen nieuw gezicht bij de FSR. Afgelopen jaar was ik actief lid en aangezien ik niet kon kiezen, heb ik maar gelijk twee commissies gedaan (commissie IRP en accountancycommissie). Dit beviel me zo goed dat ik dit voorjaar gesolliciteerd heb naar een bestuursfunctie bij de FSR. En kijk nu, als FSR-groentje begonnen vorig jaar en nu schrijf ik hier een stukje in het FSR Forum als volwaardig bestuurslid. Ik heb erg veel zin in het komende jaar en ik geloof er in dat het een mooi en leerzaam jaar zal worden! En hopelijk tot snel bij een van de vele FSR-activiteiten!

Kim de Vries

Ik ben Kim de Vries, volgend jaar zal ik de rol van penningmeester vervullen bij de FSR. Hiernaast zal ik ook het FSR Forum gaan vormgeven dit jaar. Zelf ben ik nu mijn bachelor economie en bedrijfseconomie aan het afronden, hierna wil ik de master Accounting, Auditing & Control gaan volgen.

Het afgelopen jaar heb ik een commissie gedaan bij de FSR en heb ik samen met mijn commissiegenoten het International Research Project georganiseerd. Hiervoor zijn we met een groep studenten twee weken naar India afgereisd om daar ter plaatse onderzoek te doen. Dit was een geweldige tijd en ik heb er veel van geleerd, daarom heb ik besloten om dit jaar te solliciteren naar een bestuursfunctie bij de FSR. Komend jaar hoop ik samen met mijn bestuursgenoten, commissieleden en deelnemers aan onze mooie activiteiten er een gezellig en leerzaam jaar van te maken!

64 • Verenigingsnieuws

Luc Gerretsen

Hallo, mijn naam is Luc Gerretsen en ik ben dit jaar de voorzitter van het XIIIe FSR bestuur. Ik ben 22 jaar geleden geboren in het Gelderse Lichtenvoorde, nabij Berlijn voor de gemiddelde Rotterdammer. Inmiddels ben ik alweer 4 jaar woonachtig in de geweldige Maasstad. Tussentijds ben ik nog een halfjaar op exchange geweest naar het verre Hong Kong. Vorig jaar heb ik mijn bachelor economie afgerond in de richting Finance en dit jaar mijn bachelor rechten.

In mijn 2 jaar dat ik actief ben geweest bij de FSR heb ik deelgenomen aan het International Research Project naar Boston en heb ik afgelopen jaar de International Banking Cycle georganiseerd. Voor mijn lidmaatschap bij de FSR was ik actief bij studenten beleggingsvereniging B&R Beurs. Als voorzitter ben ik dit jaar eindverantwoordelijk voor de bezigheden van de vereniging. Net als mijn bestuursgenoten heb ik erg veel zin om van dit jaar een groot succes te maken.

Eefke van der Meer

Mijn naam is Eefke van der Meer en ik ben 21 jaar en woon inmiddels al 2,5 jaar in Rotterdam. Op dit moment heb ik het derde jaar van de bachelor International Business Administration afgerond en daaropvolgend zal ik de master Finance & Investments aan de RSM gaan doen. Tijdens mijn bachelor en ik vier maanden op uitwisseling geweest naar Ohio State University in Columbus (USA), wat echt een fantastische ervaring was.

Nu, na mijn derde jaar vind ik het een mooi moment voor een nieuwe uitdaging. Ik zal aankomend jaar secretaris zijn van het XIIIe bestuur van de FSR, iets waar ik ongelofelijk veel zin in heb. Zoals het ernaar uitziet wordt het een jaar vol uitdagingen, fantastische evenementen en borrels waar ik heel veel leden van de FSR hoop te ontmoeten.

Sandhya Poeran

Mijn naam is Sandhya Poeran en ik ben 21 jaar. Op dit moment ben ik bezig met het afronden van mijn bachelor Economie en Bedrijfseconomie, waarna ik de master Accounting, Auditing en Control ga volgen. Om ook aankomende studenten te informeren over de studie en de mooie havenstad Rotterdam ben ik in het tweede jaar Ambassador geworden voor de opleiding. Tijdens mijn bacholer ben ik vier maanden op uitwisseling geweest naar de Luigi Bocconi Universiteit in Milaan, wat een geweldige ervaring was.

Dit jaar ga ik een nieuwe uitdaging aan in de functie van Commisaris Activiteiten in het XIIIe bestuur. Dit betekent dat ik onder andere de Corporate Finance Competition, Financial Business Cycle en de Bachelor Accountancy Day mag organiseren. Komend jaar hoop ik samen met mijn bestuursgenoten, commissieleden en de deelnemers aan onze mooie activiteiten er een mooi, leerzaam en onvergetelijk jaar van te maken. Ik heb er heel veel zin in en hoop jullie bij een van de vele FSR-activiteiten te zien.

Verenigingsnieuws â&#x20AC;˘ 65

Je derde week bij Financiën...

...beheer jij staatsbedrijven.

Bij het ministerie van Financiën werk je aan grote omvangrijke projecten waarmee veel geld is gemoeid. Het Rijk is mede-eigenaar van een aantal grote ondernemingen zoals Schiphol, Havenbedrijf Rotterdam en Gasunie.

Het ministerie van Financiën is hier namens de staat aandeelhouder.

Een grote verantwoordelijkheid die je vanaf het begin moet dragen.

Bij Financiën tel je meteen mee.

Werken bij het Rijk. Als je verder denkt www.werkenbijhetrijk.nl

Financiën zoekt startende bedrijfseconomen Ben je op zoek naar een baan waarin je direct veel verantwoordelijkheid krijgt en mee kunt denken over uitdagende projecten met maatschappelijke gevolgen? Dan is werken bij het ministerie van Financiën iets voor jou. We zijn op zoek naar talentvolle bedrijfseconomen die zich willen inzetten voor een financieel gezond en welvarend Nederland.

Kijk voor meer informatie op www.minfin.nl of bel 070 - 342 89 69.

Je sollicitatie mail je naar recruitment@minfin.nl.

fsrforum • jaargang 12 • editie #5

FSR Oud-bestuurder Geert van Roon

Wat is het mooiste moment in je FSR tijd?

Heel lastig om één moment op te noemen. Een paar absolute hoogtepunten waren toch wel het (uit)lopen van de halve marathon in San Sebastian (zonder training uiteindelijk toch uitgelopen en met een marge van een paar minuten nog binnen te maximale tijd gebleven ook), de vele borrels & etentjes, de weekendjes weg met mijn bestuursgenootjes en de wisselings ALV waarbij je na een jaar keihard werken uiteindelijk het stokje overdraagt aan je opvolger.

Wat is het meest genante moment in je FSR tijd?

Sommige genante momenten zijn nu niet bepaald verstandig om in dit ‘wetenschappelijke magazine’ te publiceren, maar één legendarisch verhaal speelde zich af in Parijs: Ons jaarlijkse actieven-weekend vond plaats in Parijs. De eerste avond was een diner georganiseerd in een echt Bourgondisch, Frans restaurant. Het was zelfs zo Bourgondisch dat de vaten wijn op tafel stonden samen met de paté’s, worsten, rauwe groenten, broden enz. Niet bepaald een verfijnde keuken, maar veel wordt er toch al nooit gegeten tijdens FSR etentjes. Na een aantal uren het een na het andere wijnvat te hebben leeggedronken en de tent nogal op stelten te hebben gezet (o.a. een aantal leraren kansloos zien te overtuigen dat de tafel met minderjarige Noorse meisjes heus wel veilig met ons op pad kon gaan en onder het mom van Hans en Grietje een zoutspoor vanaf onze tafel naar de toiletten te hebben gemaakt zodat we altijd nog kruipend onze plek terug zouden kunnen vinden), vloog er tijdens een van mijn bezoeken aan de toiletten een deur tegen mijn voorhoofd aan. Gevolg: een snee van 3 cm. Het lullige was dat een bestuursgenootje van mij deze deur tegen mij aangooide en ik mijzelf al met een enorme snee boven mij al bestaande litteken door het leven zag gaan. Het vervelendste van alles was nog wel dat gezien ons gedrag, de mensen van het restaurant niet bereid waren mij pleisters of een doekje voor het bloeden te geven. Uiteindelijk heb ik mijn eigen hoofdwond gedicht en geplakt met pleisters die ik kocht in een lugubere nachtwinkel en de rest van het weekend liep ik met een tulband op… Het mooiste achteraf is dat dit litteken totaal niet meer te zien is (dit i.t.t. mijn litteken dat destijds gehecht is door een professionele arts)!

Naam: Geert van Roon Leeftijd: 26 jaar Woonplaats: Amsterdam Werkzaam bij: KPMG Corporate Finance Huidige functie: Executive Welk bestuur: het mooiste en na 5 jaar nog steeds het meest hechte bestuur (VIIIste dus) Welke functie: Secretaris / Vice voorzitter Studie: Financial Economics Jaar van afstuderen: 2009 In welke auto rijd je: Alfa Romeo 147 Wat drink je op de vrijdagavond: Afhankelijk van het gezelschap bier, wijn, bubbels, sterk Levensmotto: Work Hard, Play Hard

terwijl ik er pas na een paar minuten achter kwam dat hij een ober was en hij zijn hand alleen maar uit stak om mijn lege bord aan te nemen! Ook liet ik tijdens het praten met deze dame een stuk vis vallen op het tapijt van Juliana (in een ruimte waar je toch al niet mocht eten, maar goed omdat wij volwassen genoeg waren maakten men hier een uitzondering voor). Gelukkig belandde een groot stuk ervan in de omslag van mijn broekspijp en kon ik dit deel heel onopvallend doen ‘verdwijnen’. Tot slot ben ik die avond ook nog een keer met mijn hak vol op haar teen gaan staan… Al met al een onvergetelijke dag!

Wat heb je aan je FSRtijd gehad?

Heel erg veel. Een bestuursjaar bij de FSR kan ik iedereen aanraden! In meerdere opzichten was dit voor mij een gouden greep. Naast een duidelijke groei in mijn fysieke (+10kg) en persoonlijke ontwikkeling (waaronder het heel intensief samenwerken met je bestuursgenootjes en commissieleden) heb ik er ook veel vrienden voor het leven aan over gehouden – waaronder mijn bestuursgenootjes. Na een aantal maanden besloten wij als bestuur om ons studiejaar als “verloren” te beschouwen en op alle mogelijke manieren het maximale uit dit bestuursjaar te halen... Ik weet nog goed dat we als bestuur tijdens de tentamens alleen nog maar ons gezicht lieten zien om flyers of potloden uit te delen en dan snel een tentamen mee te nemen zodat we het jaar erop weer een extra oefententamen achter de hand hadden.

Waar werk je?

Wat is je beste bestuursprestatie?

Buiten het overall succesvol hebben bestuurd en geleid van de FSR mogen wij als VIIIste bestuur denk ik vooral ook met trots terugkijken op de voor het eerst behaalde landelijke promotie door middel van een artikel in de Spits en de billboards die overal op de Universiteit hingen. Normaal waren deze billboards alleen maar beplakt met bedrijfsadvertenties maar toen ineens maanden achter elkaar alleen maar FSR activiteiten en algemene promotie. Ook weet ik nog dat wij de magische grens van 1.000 leden op een schandalige manier gevierd hebben in een van de slechtere horeca-gelegenheden van Rotterdam.

Nog een ander mooi verhaal?

Van alle bestuurs-kennismakings-dagen met bedrijven kan ik mij nog heel goed de kennismaking met Ernst & Young herinneren aan het begin van ons bestuursjaar…. Het was een zonnige dag en na een avond weer eens volledig doorgehaald te hebben stond de bestuursdag met E&Y op het programma.

We gingen een dag varen op de oude motorboot van Bernard en Juliana. Laat ik mijn verhaal kort proberen te houden: het was een fantastische dag maar vanaf het moment dat ik aan boord stapte was ik compleet van de kaart door de aanwezigheid van één van de E&Y medewerksters. Ik zal geen naam noemen aangezien zij er nog steeds werkt, maar die dag verliepen veel dingen anders dan gepland…

Wat ik mij nog kan herinneren heb ik mij – in het bijzijn van haar – uitvoerig voorgesteld aan een meneer en met hem ook uitgebreid over E&Y heb gepraat,

Na mijn afstuderen in december 2009 aan de slag gegaan bij KPMG Corporate Finance. Ik werd binnen Corporate Finance op de afdeling Mergers & Acquisitions (M&A), waarbij wij fusies en overnames begeleiden.

Wat zijn je werkzaamheden?

Als Executive bij KPMG Corporate Finance ben je vanaf dag 1 volledig betrokken bij lopende projecten en krijg je al heel snel verantwoordelijkheden. Als voorbeeld zat ik bijvoorbeeld na anderhalve week al een informatie-memorandum te schrijven voor een klant en zat ik een week later samen met een collega bij deze klant op kantoor om dit aan hem te presenteren. Het voordeel van KPMG ten opzichte van andere partijen binnen onze sector (zonder deze te willen afkraken) is dat wij marktleider zijn binnen mid-market transacties en hierdoor een groot aantal transacties per jaar doen. Doordat we veel deals doen ben je als starter bij ons al in je eerste jaren betrokken bij veel verschillende transacties van verschillende grootte en binnen verschillende sectoren. Hierdoor sta je soms verbaasd over hoe stijl je leercurve is.

Verenigingsnieuws • 67

WE ARE SCOUTING FOR BRILLIANT MINDS ONLY START YOUR CAREER IN TRADING

APPLY AT WWW.OPTIVER.COM

fsrforum • jaargang 12 • editie#4

Do you have what it takes to become a Trader at Optiver?

Bedrijfspresentatie Schrijver

Optiver is een internationaal handelshuis. Onze core business is elektronische Market Making en arbitrage activiteiten in financiële producten, zoals derivaten, aandelen en obligaties. We handelen voor eigen rekening en risico en hebben geen klanten. Met kantoren in Amsterdam, Sydney en Chicago, 600 medewerkers en 30 nationaliteiten wordt er wereldwijd 24 uur per dag gehandeld.

Jelle Jans is sinds 2008 werkzaam als Trader bij Optiver. Dit interview is voor hem een ideale manier om aan FSR te laten weten wat hij doet bij Optiver.

Waarom heb je gekozen voor Optiver?

Tijdens mijn studie was ik altijd erg geïnteresseerd in de financiële wereld. Als croupier kwam ik in aanraking met kansen, handelen en pokeren. Ik vond het werken met geld en cijfers erg dynamisch en fascinerend. Via via ben ik bekend geraakt met het fenomeen Market Making. Ik ben mij gaan verdiepen in bedrijven die actief zijn in deze branche en kwam uit bij de grootste van Nederland: Optiver.

Wat is je functie binnen Optiver?

Als handelaar bij Optiver geef ik prijzen af aan financiële (afgeleide) producten. Op basis van informatie over de huidige markt bepaal ik de prijs waarvoor ik een optie wil kopen en de prijs waarvoor ik dit product wil verkopen. Ik zit achter de computer met informatie van over de hele wereld op mijn schermen. Op basis hiervan moet ik continu mijn prijzen bepalen en transacties doen die binnen fracties van een seconde zijn uitgevoerd door de computer.

De grootste uitdaging is om met deze schaarste van tijd en informatie elke keer met een strategie te komen die een zo hoog mogelijke winst genereert.

Tegelijkertijd moet ik elk risico voor Optiver minimaliseren ofwel hedgen. Doordat we in teamverband handelen kunnen we op ieder moment van elkaar leren en informatie overdragen. Geen dag is namelijk hetzelfde in de financiële markten.

Waarom zouden studenten voor Optiver moeten kiezen?

Het handelen is elke dag anders en ik ben mij nog dagelijks aan het ontwikkelen. Door de directe verantwoordelijkheid die je krijgt, is deze functie zeer uitdagend. Optiver heeft een platte organisatiestructuur met directe lijnen. Het motto van Optiver is ‘Look casual, Work smart’. Bovendien werk ik elke dag in een jong en internationaal gezelschap.

Het motto van Optiver is

‘Look casual, Work smart’.

Bovendien werk ik elke dag in een jong en internationaal gezelschap.

Interesse?

Voor afgestudeerden die gedreven en competitief zijn, hebben we verrassende carrièremogelijkheden in een dynamische werkomgeving. Heb jij affiniteit met de financiële markten, zeer goede rekenkundige en analytische vaardigheden, een winnaars mentaliteit en het vermogen om onder druk beslissingen te nemen? Dan is wellicht een carrière in Trading of Wholesale Trading iets voor jou.

Kijk op www.optiver.com voor meer informatie of bel Femke Huijbers, Recruiter Trading op 020 70 87 000.

Bedrijfspresentatie Optiver • 69

ONDERSCHEIDEN EN KANSEN GRIJPEN! COMMISSIELEDEN GEZOCHT Grijp jij deze kans?

Heb je interesse of wil je meer weten?

Kom dan langs op onze kamer (H14-06), bel ons op 010- 408 18 30 of mail naar hr@fsr.nu

fsrforum • jaargang 12 • editie #5

FSR Activities 2010/2011

MASTER KICK-OFF DAY September 2010

INTERNATIONAL BANKING CYCLE

BIG 4 CYCLE

October/November 2010

October 2010

MULTINATIONAL DINNER November 2010

TRADERS TROPHY November 2010

ACTIVE MEMBERS DAY November 2010

FINANCIAL BUSINESS CYCLE January/February 2011

ACCOUNTANCY DAY

NATIONAL INVESTMENT COMPETITION

December 2010*

February until May 2011

YOUNG FINANCIALS DINNER March 2011

EUROPEAN FINANCE TOUR April 2011

CORPORATE FINANCE COMPETITION June 2011

INVESTMENT BANKING MASTERCLASS May 2011

INTERNATIONAL RESEARCH PROJECT MAY 2011

BACHELOR ACCOUNTANCY DAY May 2010*

MULTINATIONAL BATTLE March 2011

ACTIVE MEMBERS WEEKEND April 2011

*The exact date and time of this event will be determined at a later point in time. Please be sure to regularly check our website (www.fsr.nu) for more information.

Verenigingsnieuws • 71

FSR Alumnivereniging Options

Waarde lezer, Het onderwerp van deze editie van het FSR Forum is opties. De meeste lezers van dit magazine zullen bij dit onderwerp direct denken aan een afgeleid financieel product en allerhande hieraan verwante onderwerpen zoals de waardering hiervan.

Graag zou ik het onderwerp wat breder willen belichten. Het afgelopen weekend heb ik kennisgemaakt met het f.t. bestuur van de FSR voor het aankomend jaar, tijdens een brainstormsessie met de Raad van Commissarissen. Deze jonge enthousiaste studenten hebben hun opties nadrukkelijk afgewogen en besloten voor een hele mooie optie te kiezen, namelijk een FSR bestuursjaar. Ik wil de komende FSR bestuurders feliciteren en complimenteren met deze briljante optiestrategie. Een FSR bestuursjaar is namelijk een optie die altijd “in the money” is en zal blijven!

Graag wil ik ook van deze gelegenheid gebruikmaken om alle FSR actieven die dit jaar hebben bijgedragen deze vereniging naar een hoger plan te tillen uit te nodigen lid te worden van de FSR Alumnivereniging, de ultieme beloning voor een jaar noeste arbeid. De belangrijkste voordelen van het lidmaatschap zijn het sterke netwerk en de gezellige activiteiten waarbij de FSR sfeer nadrukkelijk aanwezig is. Zo hebben we onlangs met een grote groep alumni de WK wedstrijd tegen Japan bekeken onder het genot van een hapje en een drankje. Om vervolgens met een rib boot met een snelheid van rond de 90 kilometer per uur over de Maas te knallen en de dag af te sluiten met een BBQ in het Euromast park. Alwaar het, met een beetje hulp van “partytent Jaap”, ondanks een plaatselijke regenbui goed toeven was.

Lid worden kan door een mail te sturen naar alumni@fsr.nu of door een lidmaatschapsformulier in te vullen op de FSR kamer, ouderwets weer in H14-6, heerlijk!

Namens het VIe Alumnibestuur Met vriendelijke groet, Rex Neijtzell de Wilde Voorzitter FSR Alumnivereniging

72 • Verenigingsnieuws

of heb jij* een beter idee om alle facetten van de financiële wereld te ontdekken? &INANCIAL 4RAINEESHIP www.werkenbijpwc.nl !SSURANCE s 4AX s !DVISORY

*connectedthinking © 2010 PricewaterhouseCoopers B.V. Alle rechten voorbehouden.

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