Black Scholes Model

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Valuing Stock Options: The Black-Scholes-Merton Model
Chapter 13

Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

1

The Black-Scholes-Merton Random Walk Assumption


Consider a stock whose price is S



In a short period of time of length Dt, the return on the stock (DS/S) is assumed to be normal with:  mean m Dt  standard deviation s Dt

 m is the annualized expected return and s is the annualized volatility.
Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

2

Why can we say that?


Assume that the Normal(m,s2) annual return is made up of the sum of n returns of shorter horizons (eg. monthly, weekly): m  E ( ri )  E (ri ) nE (ri ) i 1 i 1 n n 2 n n

thus

E (ri )  m / n thus Var (ri )  s 2 / n

s  Var ( ri )  Var (ri ) nVar (ri ) i 1 i 1



We have n=1/Dt intervals of length Dt in a year (eg. for monthly n=1/(1/12) = 12 intervals of length 1/12 of a year), therefore:
E (ri )  m / n  m / (1/ Dt )  mDt Var (ri )  s 2 / n  s 2 / (1/ Dt )  s 2 Dt Sigma(ri )  s Dt
Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

3

The Lognormal Property


These assumptions imply that ln ST is normally (Gaussian) distributed with mean:

ln S 0  (m  s 2 / 2)T and standard deviation:

s T



Because the logarithm of ST is normal, the future value or price (at time T) of the stock ST is said to be lognormally distributed.
4

Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013

Proof, if you like details… 
        

Let a security price St evolve over time according to: dSt /St = m dt + s dZt where dZt~N(0,Dt) (Dt=variance) Note that this also means that dSt /St = m dt + s Dt N(0,1) Multiplying by St on both sides…...

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