Derivatives markets 3rd edition mcdonald solutions manual by Farah028

Chapt er 2 An Int roduc tion to Forward s a nd Options Ques tio n 2.1 The pa yoff di a gram of th e stock is j ust a gr aph of the stock price as a fun cti on of the stock price:

In orde r to obtain the pr ofit diagram at ex pirati on, we h ave to fin ance the ini ti al investment.We dosobysellingabondfor$50. After on e yea r , w e have to pa y b ack: $50 Ă&#x2014; (1 + 0.1)=$55.The secondfigure(onthenext page) shows the gr ap h of the stock, o f the bo nd toberepaid,andof thesumofthetwoposi tions, which is the profit grap h. The arro ws show that at a stock price of $55, the profit at ex pirati on is indeed z ero.

Ch ap t er 2 /An In trod u ctio n to F o rward s an d Op ti on s 9

Ques tio n 2.2 S ince we sol d the stock i nit iall y, ou r pa yof f at ex pirati on from bein g short the stock is ne gati ve.

10 P art On e/In su ran ce, Hed ging, an d S imp le S trategies

In ord er to obtain the profit diagram at ex pirati on, we have to lend out the mone y we r eceived fromtheshortsaleofthestock.Wedo so b y bu yi n g a bond for $50. Aft er one ye ar , w e receive fromtheinvestmentinthebond:$50Ă&#x2014; (1 + 0.1 ) = $55. The se cond fi gur e s hows the graphofthe soldstock,ofthemoneywe receive f rom the investm ent in the bond, and of thesumofthetwo positions,whichisthe profit graph. Th e a rrows show that at a stock priceof$55,theprofitat expirationis indeed z ero.

Ques tio n 2.3 The posi ti on that is the opposi te of a purch ased c all is a writ ten call . A se ll er of a c all optionis saidtobetheoptionwriterorto have a short posi ti on. The call opti on writ er isthecounterparty totheoptionbuyer,and his pa yo ffs and profit s a re just the opposi te of those of the call opti on bu ye r. S im il arl y, the posi ti on t hat is the opposi te o f a purchas ed put opti on is a w ritten put opti on. Again,thepayoffandprofitforawritten put a re j ust t he opposi te of those of the purch ased put. It is im portant to note that the opposi te of a purchased call is NOT the purchased put. If you do notseewhy,pleasedrawapayoffdiagram wit h a purchased call and a purc hased put.

Ch ap t er 2 /An In trod u ctio n to F o rward s an d Op ti on s 11

Ques tio n 2.4 a) The pa yoff to a long for ward at ex pirati on is equ al t o: P a yof f to long forwa rd = S pot price at ex pirat ion – forwa rd price Therefo re, w e c an const r uct t he following tabl e: P rice of asset in six months 40 45 50 55 60

Agreed forw ard pric e 50 50 50 50 50

P a yof f to the long forw ar d −10 −5 0 5 10

b) The pa yoff to a purch ase d call opti on at ex pirati on is: P a yof f to call opti on = m ax [ 0, spot price at ex pirati on – strike price] The strike is giv en: It i s $ 50. Therefo re, w e c an co nstruct t he following tabl e: P rice of asset in six months 40 45 50 55 60

S trike price 50 50 50 50 50

P a yof f to the call opti on 0 0 0 5 10

c) If we compar e the two c ontracts, we im mediatel y see that the c all opti on has a protectionfor adversemovementsinthe price o f the ass et: If th e spot price is below $50,thebuyerofthe calloptioncanwalkaw a y a nd ne ed not incur a lo s s. The bu yer of the longforwardincursa loss,buthehasthe same pa yo ff as the bu ye r of the call opti on if thespotpriceisabove$50. Therefore,the call opti on shoul d be more ex pensive. It is thisattractiveoptiontowalkaway thatwe have to p a y fo r.

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Ques tio n 2.5 a) The pa yoff to a short for ward at ex pirati on is equ al t o: P a yof f to short forw ard = forwa rd price – spot pri c e at ex pirati on Therefo re, w e c an const r uct t he following tabl e: P rice of asset in s ix months Agreed forw ard pric e P a yof f to th e short forw a rd 40 45 50 55 60

50 50 50 50 50

10 5 0 −5 −10

b) The pa yoff to a purch ase d put opt ion at ex pirati on is: P a yof f to put opt ion = max [ 0, strike price – spot price at ex pirati on] The strike is giv en: It i s $ 50. Therefo re, w e c an co nstruct the foll owing tabl e: P rice of asset in six months 40 45 50 55 60

S trike price 50 50 50 50 50

P a yof f to the call opti on 10 5 0 0 0

c) The same lo gic as in qu esti on 2.4 ( c) appli es. If we compa re th e two con tracts, we see that the put opti on has a prot ecti on for increas es in the price of the asset: If the s pot price is above $50, the bu yer of the put opti on can w alk aw a y a nd need not incu r a loss . The bu ye r of th e short forw ard in curs a lo ss and must me et he r obl igati ons. Ho weve r, she h as the s ame p a yoff asthebuyeroftheputoptionifthe spot price is below $50. There fore, th e put optionshould bemoreexpensive.Itisthis att racti ve opti on to walk aw a y if thi n gs ar e not as we w ant that we have to p a y for.

Ques tio n 2.6

W e need to solve the foll owing equati on to determi ne the effecti ve annu al int erest rate:$91× (1+r)=$100.Weobtainr=0.0989, which means that the e ffe cti ve annual int erest rate is approx im atel y 9.9 per cen t .

Ch ap t er 2 /An In trod u ctio n to F o rward s an d Op ti on s 13

R emember that when we drew profit diagrams for the forward or call opti on, we drew the pa yoff on the vertical ax is, and the index price at the ex pirati on of the contract on the horiz ontal axis.In thiscase,theparticularityisthat the de fault -fr ee , z ero-coupon bond will pa y ex actl y $10 0, no matter what the stock pri ce is. The refo re, the pa yo ff dia gr am is just a horizontalline,intersecting they-axisat $100. The tex tbook provides the answ er to the qu esti on conce rning the pro fit diagr am in the se cti on “Zero-CouponBondsinPayoffandProfit Dia gra ms.” W hen we w ere calc ulat ing pro fits, wesaw thatwehadtofindthefuturevalueof the ini ti al investm ent. In thi s c ase, ou r ini ti al investm ent is $91. How do we find the future value? W e use the cur rent risk -fr ee int e res t rate and mul ti pl y the initialinvestmentbyit.However,asour bond is default -fr ee and do es not bear coupons,the effectiveannualinterestrate is ex actl y the 9.9 pe rcent w e h ave calculat ed before.Therefore,the futurevalueof$91is $9 1 × (1 + 0.0989) = $10 0, and our profit in si x months i s z ero!

Ques tio n 2 .7 a) It does not cost an ythi ng to enter int o a forw ard contra ct —we do no t pa y a pr emi um. Therefo re, th e pa yo ff di agram of a forw ard cont ract coincides with the p rofit dia gram. The gr aphs have th e following shap e:

b) W e have seen in questi on 2.1 that i n order to obtain the profit diagr a m at ex pirati on of a purchase o f XY Z stock, we have to finan ce the ini ti al investm ent. W e did so b y s ell ing a bond for $50. Afte r one ye a r , w e had to pa y b ack : $50 Ă&#x2014; (1 + 0.1) = $55. Therefo re, ou r tot al profitatexpirationfromthepurchase of a stock t hat was finan ced b y a lo an was: $ ST â&#x2C6;&#x2019; $55, whereSTisthevalueofoneshareofXYZ at e x pirati on. But thi s profit from bu yin g the stock,andfinancingitisthesameas the profit from our long forw ard contract , and both

14 P art On e/In su ran ce, Hed ging, an d S imp le S trategies

posi ti ons do not require a n y ini ti al cash â&#x20AC;&#x201D;but t hen, there is no adv anta ge in investi ng in either inst rument. c) The divi dend is onl y pai d to the owner of the sto ck. The owne r of the lon g fo rward contra ct isnotentitledtoreceivethedividend because sh e onl y has a claim to buy the stock inthe futureforagivenprice,butshedoes not own it ye t. Ther efor e, it does m att er now whether weownthestockorthelongforward contr act. B e cause ev er ythi ng else is the sameasinpart a)andb),itisnowbenefi cial to own the share: We can re ceive an addit ionalpaymentinthe formofthedividendif we own th e stock at th e ex -divi dend date. Thisquestionhintsatthe veryimportant fa ct t hat we have to be c are ful to t ake int o ac countallthebenefitsandcostsof anasset when we tr y to c ompare pric es. W e will encounter sim il ar problem s in l ater chapte rs.

Ques tio n 2.8

W e saw in questi on 2.7 (b) that there is no advantage in bu yin g eit he r the stock or the forward contractifwecanborrowtobuya stock toda y (s o both strate gies do not r equire anyinitialcash) andiftheprofitfromthis strateg y is the same as the profit of a long forw ardcontract.Theprofit ofalongforward contra ct with a price for deli ver y of $53 is equalto:$ST−$53,whereSTisthe (unknown) value o f one share of X Y Z at ex pirati on of the forwardcontractinoneyear.Ifwe borrow$50 tod a y to bu y one sha re o f XY Z stock (that costs $50), we hav e to rep a y in on e ye ar: $50 × (1 + r ). Our tot al profit in one yea r from b orrowin g to bu y one sha r e of XY Z is ther efor e: $ST − $50 × (1 + r ). Now we can equate th e two pr ofit equati ons and sol ve f or the int erest r ate r : $ST − $53 = $ ST − $50 × (1 + r ) ⇔$53 = $50 × (1 + r ) $53 ⇔ 1=r $50 ⇔r = 0.06 Therefo re, the on e- ye ar effe cti ve int erest r ate that is c onsi stent with no advanta ge to eit he r bu yin g the stock or forw ard contr act i s 6 per cent.

Ques tio n 2.9 a) If the for ward pric e is $1,100, then the bu ye r of the one - ye ar forw ard c ontract rec eives at expirationafteroneyearaprofitof:$ST − $ 1,10 0, where ST is the (unkn own) value of the S&Rindexatexpirationoftheforward contra ct in one yea r. R emembe r t hat it costs nothi ng to enter the for ward contr act. Let us a gain follow our strate g y o f borrowin g m one y to finan ce the purc hase of the index toda y, so that we do not need an y ini ti al c ash. If we borro w $ 1,000 toda y to bu y the S &R index (that costs $ 1,000), we have to r epa y in on e ye ar: $ 1,000 × (1 + 0. 10) = $ 1,100. Our tot al profit in one year f r om borrowing to bu y th e S &R index is therefore: $ ST − $ 1,100. The profit s from t he two st rat egi es ar e identical.

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b) The fo rward pric e o f $ 1,200 is wors e fo r us i f we want to bu y a for ward contra ct. To understand thi s, suppose the index after on e ye a r is $ 1,150. W hil e we h ave alr ead y m ade mone y in pa rt a) wi th a f orward pri ce of $ 1,100, we ar e sti ll losi ng $50 with the new price of

$1,200. As there was no advanta ge in bu yin g eit her the stock or fo rwa rd at a pric e of $ 1,100, wenowneedtobe“bribed”toenter int o the for ward contr act. W e some how need tofindan equationthatmakesthetwo strate gies compar abl e a gain. S uppose that welendsomemoney initiallytogetherwith en tering int o the forw ard c ontract so that w e will r eceive $100 aft er one ye a r. Then, the p a yo ff from our modi fied for ward strat e g y is: $ ST − $ 1, 200 + $100 = $ ST − $ 1,100, which equ als the pa yof f of the “borrow to bu y index ” strate g y. W e have foundthe futurevalueofthepremium somebod y n eeds us to pa y. W e sti ll need to findoutwhatthe premiumwewillreceivein one ye ar is wo rth today. W e need to di scount i t: $100/ (1 + 0.10) = $90.91. c) S im il arl y, the forw ard pr ice of $ 1,000 is advant a geous for us. As ther e w as no advanta gein buyingeitherstockorforwardatapri ce o f $ 1,10 0, we now n eed to “b rib e” someone tosell thisadvantageousforward contract to us. W e so mehow need to find an e quationthatmakes thetwostrategies compa rable a gain. S uppose th a t we bor row somemoneyinitiallytogether with entering int o the for ward contr act so that we will have to pa y back $1 00 after on e yea r. Then, the pa yo ff from ou r modi fied for ward st rate g y is: $ ST − $ 1,000 − $1 00 = $ ST − $ 1,100, which equals the p a yoff of the “bor row to bu y i ndex ” strateg y. W e h ave found the future value of the premi um we need to pa y. W e sti ll need to find out what thi s premi um we have to pa y in one ye ar is worth toda y. W e sim pl y need to discount it : $100/(1 + 0.10) = $90.91. W e shoul d be will ing to pa y $90.91 to enter int o the one - ye ar fo rward contra ct with a forw ard price of $1,000.

Ques tio n 2.1 0 a) Figu re 2. 6 depicts the pr ofit from a long call opt ion on the S &R index with six mont hs to expirationandastrikepriceof$1,000 if the fut ure pric e of the opti on p remi um is $95.68. Theprofitofthelongcalloptioni s: max[ 0, ST − $1,000] − $95.68 ⇔max[ −$95.6 8, ST − $1,095.68] where ST is the (unkno wn) value of the S &R index at ex pirati on of th e c all opti on in six mont hs. In orde r to find the S &R index pric e at which the call opti on dia gr am int erse cts the x-ax is, we have to set the above equati on equal t o z ero. W e get: ST − $ 1,095.68 = 0 ⇔ST = $1,095.68. This is the onl y solut ion, as the other part of the max im um functi on, −$95.6 8, is alwa ys less t han z ero. b) The profit of the six month forward contr act with a forwa rd price of $1,02 0 is: $ ST − $ 1,020. In ord er to find the S &R index price at which the call opti on and the forward contra ct have thesameprofit,weneedtosetboth parts of the max im um functi on of the profit ofthecall optionequaltotheprofitofthe fo rward cont r act and se e which pa rt pe rmits asolution.First, weseeimmediatelythat $ ST − $ 1,020 = $ ST − $ 1,095.68 does not h ave a solut ion. But we

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can solve the other leg: $ST − $ 1,020 = −$95.68 ⇔ST = $924.3 2, which is the value given in the ex ercise.

Ques tio n 2.11 a) Figu re 2. 8 depicts the pr ofit from a long put optionontheS&Rindexwithsixmonthsto ex pirati on and a strike price of $ 1,000 if the future value of the put premi um is $75.68. The profit of the long put opt i on is: max[ 0, $1, 000 − ST ] − $75.68 ⇔max[ −$75.6 8, $924.32 − ST ] where ST is the (unkno wn) value of th e S &R i ndex at ex pirati on of the put opti on in six mont hs. In orde r to find the S &R index pric e at which the put opti on dia gr am int ers ects the x-ax is, we h ave to set th e abov e equ ati on equ al t o z ero. W e get: $924.32 − ST = 0 ⇔ST = $924.32. This is the only solut ion, as the other part of the max im um functi on, −$75.6 8, is alwa ys less t han z ero. b) The profit of the short si x -mont h forward contra c t wit h a forwa rd price o f $ 1, 020 is : $ 1,020 − $ST. In o rder to find the S &R ind ex price at which the put opti on an d the sold forwa rd contract h ave the s ame pr ofit, we need to s et bot h parts of the max im um functi on of the profit oftheputoptionequaltotheprofit of the forw ard contr act and se e wh ich part permitsa solution.First,weseeimm ediatel y that $ 1,020 − $ ST = $924.32 − $ ST does not have a solut ion. But we can solve the othe r le g: $ 1,020 − ST = −$75.68 ⇔ST = $ 1, 095.68, which is the value given in the ex ercise.

Ques tio n 2.1 2 a) Lon g Forw ard The max im um loss occu rs if the stock p rice at e x pirati on is z ero (the sto ck pric e c annot be lessthanzerobecausecompanieshaveli mi ted li abil it y). Th e forw ard then pa ys 0 – Forward price.Themaximumgainisunlimit ed. The stock price at ex pirati on could theoreticallygrow toinfinity;thereis no bo und. W e mak e a lot o f m one y if the stock p ricegrowstoinfinity(or toaverylarge amount ). b) S hort Forwa rd The profit for a short f orward contr act is forw ard price – stock price at ex pirati on. The maximumlossoccursifthestockpricer ises sharp l y; there is no bound to it, so it couldgrow toinfinity.Themaximumgainoc curs if the stoc k price is z ero. c) Lon g C all

W e will not ex ercise the call opti on if the stock price at ex pirati on is less than the strike price.Consequently,theonlythi ng w e lose is the f uture value of the premi um wepaid initiallytobuytheoption.Asthe stock price can grow ve r y la r ge (and wi thout bound), and

Ch ap t er 2 /An In trod u ctio n to F o rward s an d Op ti on s 17

our pa yo ff grows li nearl y in the termi nal stock pr ice once it is higher than the strike, there is no li mi t t o our gain. d) S hort C all W e have no control ove r the ex ercise de cisi on when we w rite a call . T he bu ye r of the call opti on decides whether t o ex ercise it or not, and he will onl y ex ercise the call if he makesa profit.Aswehavetheopposite side, we will never make an y mon e y at th e ex pirationofthe calloption.Ourprofitis restricted to the futur e value of the pr emi um,andwemakethis maximumprofit whenever the stock price at e x pirati on is smallerthanthestrikeprice. However, the stock pri c e at ex pirati on can b e v e r y l ar geandhasnobound,andasourloss grows linearl y in t he ter mi nal st ock price, ther e is no li mi t t o our loss . e) Lon g P ut W e will not ex ercise the put opti on if the stock price at ex pirati on is large r than the strike price.Consequently,theonlythin g w e lose when ever the te rminal stock p rice islargerthan thestrikeisthefuturev alue of the p remi um we paid ini ti all y to bu y theoption.Wewill profitfromadeclinein the stock prices. Howeve r, stock prices cannotbesmallerthanzero, soourmaximum gain is restricted to strike pric e l ess the futurevalueofthepremium,andit occurs at a termi nal st ock price of z ero. f) S hort P ut W e have no control ove r the ex ercise decisi on when we writ e a put. The bu ye r of the put opti on decides whether t o ex ercise o r not, and h e will onl y ex ercise if h e makes a profit .As wehavetheoppositeside,wewillnever mak e an y mone y at the ex pirati on of the put option. Ourprofitisrestrictedtothe futur e value of th e p remi um, and w e mak e th is maximumprofit wheneverthestockpriceat ex pirati on is greate r than the strike price.However,welose moneywheneverthe st ock price is smaller tha n the strike ; hence,thelargestlossoccurs whenthe stock pric e att a ins it s smallest possi ble value, z ero. W e lose the st rike price b ec ause somebodysellsusanassetforthe strike th at is w orth nothi ng. W e ar e onl y compensatedby thefuturevalueofthepr emi um we rec eived.

Ques tio n 2.1 3 a) In ord er to be able to draw profit dia grams, we need to find the future values of t he call premi a. The y are:

i) 35-strike call : $9.12 × (1 + 0.08) = $9.8496 ii) 40-strike call : $6.22 × (1 + 0.08) = $6.7176 iii) 45-strike call : $4.08 × (1 + 0.08) = $4.4064

18 P art On e/In su ran ce, Hed ging, an d S imp le S trategies

W e can now graph th e p a yo ff and p rofit dia gram s for the call opti ons. Th e pa yoff di a gram looks as follows:

W e get the p rofit dia gra m b y dedu cti ng the opti on premi a from the p a yoff gr aphs. The p rofit diagr am l ooks as follows :

Ch ap t er 2 /An In trod u ctio n to F o rward s an d Op ti on s 19

b) Intui ti vel y, whenev er the 45 -strike opti on pa ys of f (i.e., has a pa yof f bigger than z ero), the 40-strike and the 35 -strike opti ons pa y of f. How ever, ther e are some inst ances in which the 40-strike opti on pa ys o ff and the 45 -strike op ti on does not. S im il arly, th ere a re some inst ances in which the 35 -strike opti on pa ys o ff bu t neit her the 40 -strike nor the 45 -s trikepay off.Therefore,the35-strike offe rs more potential than the 40 - and 45 -strike,andthe40strikeoffersmore potential than the 45-strike. W e pa y fo r theseadditionalpayoff possibilitiesby ini ti all y p a yin g a high er pr emi um.

Ques tio n 2.1 4 In orde r to be able to dra w profit di a grams, we ne ed to find the future valu es of the put p remi a. The y ar e: a) 35-strike put: $1.53 Ă&#x2014; (1 + 0.08) = $1.6524 b) 40-strike put: $3.26 Ă&#x2014; (1 + 0.08) = $3.5208 c) 45-strike put: $5.75 Ă&#x2014; (1 + 0.08) = $6.21 W e get t he foll owin g pa yoff dia grams:

20 P art On e/In su ran ce, Hed ging, an d S imp le S trategies

W e get the p rofit dia gra m b y dedu cti ng the opti on premi a from the p a yoff graphs. Th e profit diagr am l ooks as follows :

Intui ti vel y, when ever the 35 -strike put opti on pa ys off (i.e., has a pa yof f bigger th an z ero), the 40-strike and the 35 -strike opti ons also pa y o ff. H owever, ther e ar e some inst ances in which the 40-strike opti on pa ys off and the 35 -strike opti on does not. S im ilarl y, there are some instancesin whichthe45-strikeoptionpays off but neit her the 40 -strike nor th e 35 -stri kepayoff.Therefore, the45-strikeoffers mor e potential than th e 40 - and 35 -strik e, and the40-strikeoffersmore potentialthanthe 35 -strike. W e pa y fo r these add it ional pa yoff possibilitiesbyinitiallypayinga higherp remi um. It m ake s sens e that the premi um is i ncreasin g in t he strike price.

Ques tio n 2.1 5 The nic e thi n g that l ead us to the noti on o f indi ff erenc e b etwe en a for war d contra ct and a loan - financedstockindexpurchasewheneverthe fo r ward pric e equ aled the f uture price o f the loan wasthatwecouldalreadytelltodaywhat w e had to pa y b ack in the future. In other wo rds, the returnontheloan,therisk-freeint erest rate r , w as known toda y, and we removed uncertainty aboutthepaymenttobemade.Ifw e we re to fina nce the pur cha se o f the i ndex b y shortselling IBMstock,wewouldintroduce addit ional unce rta int y be cause the futur e v alueoftheIBMstock isunknown.Therefore, we could not calcul ate t oda y th e amount to be r epaid,anditwouldbe impossibletoestablis h an equivale nce betwe en the forward and loan - financedindexpurchase today.Thec alculati on of a profit dia gr am woul d onl y be possibleifweassumedanarbitrary valuefor IBM at ex pirati on of the futures, and w e would have to dra w man y pro fit di a grams wi th different v alues for IBM to get an id ea of the m an y possi ble profits w e cou ld m ake.

Ch ap t er 2 /An In trod u ctio n to F o rward s an d Op ti on s 21

Ques tio n 2.1 6 The following is a cop y of a spre adsheet t hat sol v es the problem:

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