ch07 by Salim Kasap

Finance and Derivatives: Theory and Practice SĂŠ bastien Bossu and Philippe Henrotte

Chapter 7 Lognormal model

1. Lognormal model 



The binomial model envisages only two outcomes for the final price of the underlying. The main advantage of this approach is its simplicity; however it is not very realistic. To make the model more convincing, we must develop it into multiple steps. An alternative approach can be worked out exploiting probabilities.

Chapter 7 Lognormal model

1.1. Fair value 



Suppose that the final underlying price ST follows a given probability distribution (uniform, normal, lognormal, …), The fair value D0 of a European derivative security with payoff DT = f (T, ST) can be represented as the expectation of the discounted payoff:

 f (T , ST )  D0 = E  T ÷  (1 + r )  where r is the annual discount rate. Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

Chapter 7 Lognormal model

1.1. Fair value (2) 

In this framework we need 2 assumptions: 

Assumption on the probability distribution of ST



Assumption on the discount rate r

Chapter 7 Lognormal model

1.2. Probability distribution of ST 

The choice of an appropriate probability distribution of ST depends on the nature of the underlying



For stocks we have the following 3 requirements: 1.

ST > 0: the value of a stock should always be positive  F  ( DT = ST ) ⇒  D0 = 0 T ÷ (1 + r )  : the model should be  forward-neutral i.e. correctly price a forward contract delivering one unit of the underlying (otherwise there would be an arbitrage);

Chapter 7 Lognormal model

1.2. Probability distribution of ST (2) 3.

There exists a unique reference price S* such that for all x > 0: ST ST 1

(

)

(

Pr  S * = x  = Pr  S * = x 

)

i.e. the ‘probability’ of the stock price doubling its reference price should be equal to the probability of the stock price halving, and this should hold generally for any x-fold factor. Note: The notation Pr density function fX(x)

( [ X = x] )

corresponds to a probability

Chapter 7 Lognormal model

1.2. Probability distribution of ST (3) 



These three properties are verified by the lognormal distribution with mean (ln F0 – σ2T/2) and standard deviation σT 0.5 Recall:   

F0 is the forward price of the underlying for maturity T σ is a volatility parameter A random variable Y follows a lognormal distribution with mean m and standard deviation s iff the random variable X = ln(Y) follows a normal distribution with mean m (NOT ln m) and standard deviation s (NOT ln s)

Chapter 7 Lognormal model

Figure p.104: Normal and lognormal distributions (1) S0 = 100, F0 = 125, σ = 40%, T = 1 La loi normale Normal peut distribution prendre can give des negative valeurs values négatives

Loi Normal distribution normale

Loi logLognormal normale distribution -100

100

200

300

400

Chapter 7 Lognormal model

Figure p.104: Normal and lognormal distributions (2) S0 = 100, F0 = 125, σ = 10%, T = 1 Normal Loi distribution normale

Loi logLognormal distribution normale

100

150

200

Chapter 7 Lognormal model

1.3. Discount rate (1)  



Choice of appropriate discount rate r is not obvious If the payoff of the derivative were a fixed cash flow C, arbitrage considerations would lead to the riskfree zero-coupon rate z(T) for maturity T . When the payoff is subject to uncertainty, portfolio theory suggests that this risk should be rewarded by a higher return than the risk-free rate.

Chapter 7 Lognormal model

1.3. Discount rate (2) 



We will see in later chapters that the risk can in theory be entirely eliminated through a dynamic strategy known as delta-hedging developed in Chapter 8. Thus choosing the risk-free rate z(T) for r is in fact appropriate. Note that a corollary of choosing the risk-free rate is that the property of forward-neutrality for modeling ST becomes: under the additional assumption that the underlying asset does not pay any cash flow.

Chapter 7 Lognormal model

1.3. Discount rate 

Choice of appropriate discount rate r is not obvious 



If payoff DT = C (fixed cash flow), arbitrage considerations would impose r = z(T) (zero-coupon rate). When DT is subject to uncertainty, portfolio theory suggests that this risk should be rewarded by a higher return. We will see in later chapters that the risk can be eliminated through a dynamic strategy known as delta-hedging.

Thus choosing r = z(T) is appropriate 

Corollary: if S does not pay any cash-flow the forwardneutrality property becomes:

( DT

= ST ) ⇒ ( D0 = S0 )

Chapter 7 Lognormal model

2. Closed-form formulas 

The lognormal model is enough to find closed-form formulas for the value of European calls and puts as a function of the following parameters:    

Option characteristics: underlying S, strike K and maturity T; Forward price of the underlying at maturity T: F0; Risk-free zero-coupon rate for maturity T: r = z(T); Volatility of the underlying: σ.

Chapter 7 Lognormal model

2. Closed-form formulas (2) 

Classical integration techniques yield: 1 c0 = F N (d1 ) − KN (d 2 ) ] T [ 0 (1 + r ) 1 p0 = KN (−d 2 ) − F0 N (−d1 ) ] T [ (1 + r ) With: F0 1 2 F0 1 2 ln + σ T ln − σ T d1 = K 2 , d2 = K 2 σ T σ T

Chapter 7 Lognormal model

Figure p.106: Value and payoff of a European call c

100

Valeur Value of thedu callcall

75 50 Valeur Time temps value

Payoff S

0 0

100

150

200

Chapter 7 Lognormal model

Figure p.106: Value and payoff of a European put p

100 75

Value of the call Valeur duput put

Valeur Time value temps

25 Payoff

0 0

100

150

200

Chapter 7 Lognormal model

3. Monte-Carlo method 



Finding closed-form formulas is difficult for nonvanilla options Numerical methods are often used to approximate their fair value The Monte-Carlo method consists of simulating a very large number of values for ST according to its probability distribution and calculating the derivative payoff in each simulation. The value of the derivative is then approximately equal to the average simulated payoff.

Chapter 7 Lognormal model

3. Monte-Carlo method (2) 



Let N be the number of simulations and s1, s2, …, sN the simulated values drawn from the distribution of ST. Then: 1 1 N D0 ≈ × ∑ f (T ; si ) T (1 + r ) N i =1 The Central Limit Theorem tells us that for an infinite number of simulations the Monte-Carlo method converges to the fair value of the option. 



Pros: universal method, works for ‘all’ European payoffs, easy to generalize to options on multiple assets Cons: convergence is relatively slow (the error is of order 1/N0.5), does not work for American payoffs

Chapter 7 Lognormal model

Figure p.107: 10 000 simulated call payoffs Payoff

Frequency

300

16% 14% 12%

200

10% 8% 6%

100

4% 2% 0 0

100

200

0% 300

Underlying Price (Strike = 100) Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.