1. The Black-Scholes-Merton Model
Chapter 13
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

2. Random Walk
We begin be considering the discrete-time random walk description:
W(t+1)=W(t) + e(t+1); W(0)=W0 e: i.i.d. N(0,1)
The variable t represent time and is measured in discrete integer increments.
W(t) is the level of the cumulant of e(t); it is called a random walk because it appears that W takes random steps up and down through time. We would like to describe a process that has the same characteristics as the random walk but observed more frequently:
W(t+Δ)=W(t) + e(t+Δ); W(0)=W0 e: i.i.d. N(0,Δ)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

3. W(t+dt)=W(t) + e(t+dt);
Random Walk
Now consider the behavior of the process as
Δ → dt
W(t+dt)=W(t) + e(t+dt);
W(0)=W0 e: i.i.d. N(0,dt)
Define dW(t) = W(t+dt)-W(t). We heuristically define dt as the smallest positive real number such that dta=0, whenever a>1. Either of these processes, dW(t) or e(t+dt), is referred to as a white noise.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

4. Wiener process
A variable W follows a Wiener process if it has the following three properties:
The change dW during a small period of time dt is dW=e dt1/2
where e has a standardized normal distribution N(0,1)
The values of dW for any two short intervals of time are independent.
dW is normally distributed with mean 0 and variance dt.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

5. Random Walk
Recall that dW may be thought of as a normally distributed random variable with mean zero and variance dt. We note three properties that follow by construction:
1.      E[dW(t)]=0, (mean)
2.      E(dW(t)dt]=E[dW(t)]dt=0
3.      E[dW(t)2]=dt, (variance)
These properties give rise to two multiplication rules:
Rule 1: dW(t)2=dt
Rule 2: dW(t)dt=0
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

6. An Ito Process for Stock Prices: Geometric Brownian Motion
where m is the expected return s is the volatility.
The discrete time equivalent is
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

7. Properties of S The process of S has the following properties
1. If S starts with a positive value, it will remain positive.
2.  S has an absorbing barrier at 0: Thus, if S hits 0, then S will remain at zero.
3. The conditional distribution of Su given St is lognormal. The conditional mean of ln(Su) for u>t is ln(St)+ μ(u-t) – (1/2) σ2(u-t) and the conditional standard deviation of ln(Su) is σ(u-t)1/2. ln(Su) is normally distributed. The conditional expected value of Su is St exp[μ(u-t)].
4.   The variance of a forecast Su tends to infinity as u does.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

8. The Lognormal Property (Equations 13.2 and 13.3, page 282)
It follows from this assumption that
Since the logarithm of ST is normal, ST is lognormally distributed
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

9. Example
Consider a stock with an initial price of 40$, an expected return of 16% per annum, and a volatility of 20% per annum. From the above equation, the probability distribution of the stock price, ST, in six months time is given by
lnST ~ Φ[lnS0 + (μ-σ2/2)T, σ(T)1/2],
lnST ~ Φ[ln40 + ( /2)0.5, 0.2(0.5)1/2]
lnST ~ Φ[3.759,0.141]
There is a 95% probability that a normally distributed variable has a value within 1.96 standard deviations of its mean. Hence, with 95% confidence interval,
3.759  *0.141< lnST < *0.141
This can be written
e( *0.141) < ST < e( *0.141) , or
32.55 < ST < 56.56
Thus, there is a 95% probability that the stock price in six months will lie between and
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

10. The Lognormal Distribution
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

11. Monte Carlo Simulation
We can sample random paths for the stock price by sampling values for e
Suppose m= 0.14, s= 0.20, and Dt = 0.01, then
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

12. Monte Carlo Simulation – One Path (See Table 12.1, page 272)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

13. Itô’s Lemma (See pages 273-274)
If we know the stochastic process followed by S, Itô’s lemma tells us the stochastic process followed by some function f (S, t )
Since a derivative security is a function of the price of the underlying and time, Itô’s lemma plays an important part in the analysis of derivative securities
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

14. Taylor Series Expansion
A Taylor’s series expansion of f(S, t) gives
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

15. Application of Ito’s Lemma to a Stock Price Process
And Since
dS = μSdt + σS dW,
We have that
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

16. The Concepts Underlying Black-Scholes
The option price and the stock price depend on the same underlying source of uncertainty
We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty
The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate
This leads to the Black-Scholes differential equation
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

17. The Derivation of the Black-Scholes Differential Equation
It follows that by choosing a portfolio of the stock and the derivative, the Wiener process can be eliminated.
The appropriate portfolio is
-1 : derivative
+ : shares
The holder of this portfolio is short one derivative and long an amount of shares.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

18. The Derivation of the Black-Scholes Differential Equation continued
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

19. The Derivation of the Black-Scholes Differential Equation continued
Substituting the above equations for Δf and ΔS to the last one, we have
Because the equation does not involve ΔW,
the portfolio must be riskless during time Δt.
Thus, the portfolio must instantaneously earn the same rate of return as other short-term risk-free security.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

20. The Derivation of the Black-Scholes Differential Equation continued
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

21. The Differential Equation
Any security whose price is dependent on the stock price satisfies the differential equation
The particular security being valued is determined by the boundary conditions of the differential equation
In the case of call options the boundary condition is ƒ = max(S – K, 0) when t =T
In the case of put options the boundary condition is ƒ = max(K - S, 0) when t =T
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

22. The Black-Scholes Formulas (See pages 295-297)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

23. The N(x) Function
N(x) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x
See tables at the end of the book
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

24. Properties of Black-Scholes Formula
As S0 becomes very large c tends to
S – Ke-rT and p tends to zero
As S0 becomes very small c tends to zero and p tends to Ke-rT – S
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

25. Risk-Neutral Valuation
The variable m does not appear in the Black-Scholes equation
The equation is independent of all variables affected by risk preference
The solution to the differential equation is therefore the same in a risk-free world as it is in the real world
This leads to the principle of risk-neutral valuation
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

26. Example
Consider the situation where the stock price six months from the expiration of an option is 42.
The exercise price of the option is 40.
The risk-free interest rate is 10% per annum.
The volatility is 20% per annum.
This means that S0=42, X=40, r=01, σ=0.2 and T=0.5.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

27. Example (Continued) Then =0.7693-0.2 0.51/2=0.6278
Xe-rt =40 Xe-0.05 =38.049
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

28. Example (Continued)
Hence, if the option is a European call, its value c is given by
c = 42 N(0.7693) N(0.6278)
And if the option is a European put option, its value, p, is given by
p = N( ) – 42 N( )
Using the polynomial approximation,
N(0.7693) = N( ) =
N(0.6278) = N( ) =
So that,
c=4.76 and p=0.81
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005