The Black-ScholesMerton Model
Chapter 13
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
13.1
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
13.2
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
13.3
Wiener process
A variable W follows a Wiener process if it has
the following three properties:
1.
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)
1.
The values of dW for any two short intervals of
time are independent.
2.
dW is normally distributed with mean 0 and
variance dt.

Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
13.4
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
13.5
An Ito Process for Stock Prices:
Geometric Brownian Motion
dS mS dt sS dW
where m is the expected return s is
the volatility.
The discrete time equivalent is
S  mSt  sS t
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
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, 6 Edition, Copyright © John C. Hull 2005
th
13.7
The Lognormal Property
(Equations 13.2 and 13.3, page 282)

It follows from this assumption that


s2 
ln ST  ln S0    m 
 T, s T 
2


or



s2 
ln ST   ln S0   m 
 T, s T 
2




Since the logarithm of ST is normal, ST is
lognormally distributed
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
13.8
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 + (0.16-0.22/2)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  1.96*0.141< lnST <3.759 + 1.96*0.141
This can be written
e(3.759-1.96*0.141) < ST < e(3.759+1.96*0.141) , or
32.55 < ST < 56.56
Thus, there is a 95% probability that the stock price in six months will
lie between 32.55 and 56.56.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
The Lognormal Distribution
E ( ST )  S0 e mT
2 2 mT
var ( ST )  S0 e
(e
s2T
 1)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Monte Carlo Simulation


We can sample random paths for the
stock price by sampling values for 
Suppose m= 0.14, s= 0.20, and t = 0.01,
then
S  mSt  sS t
S  0.0014 S  0.02 S
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Monte Carlo Simulation – One Path (See Table
12.1, page 272)
Period
Stock Price at
Random
Start of Period Sample for 
Change in Stock
Price, S
0
20.000
0.52
0.236
1
20.236
1.44
0.611
2
20.847
-0.86
-0.329
3
20.518
1.46
0.628
4
21.146
-0.69
-0.262
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
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
Taylor Series Expansion

A Taylor’s series expansion of f(S, t)
gives
2f

f
f
df  dt  dS  1 2 dS 2...
t
S
2 S
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
Application of Ito’s Lemma
to a Stock Price Process
And Since
dS = μSdt + σS dW,
We have that
f
f 1  2 f 2 2
f
df ( mS  
s S )dt  sSdW
2
S
t 2 S
S
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
The Concepts Underlying BlackScholes




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
13.16
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
f
+S : shares
The holder of this portfolio is short one derivative and
f
long an amount S of shares.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
13.17
The Derivation of the Black-Scholes
Differential Equation continued
The value of the portfolio  is given by
ƒ
  ƒ 
S
S
The change in its value in time t is given by
ƒ
    ƒ 
S
S
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
13.18
The Derivation of the Black-Scholes
Differential Equation continued
Substituting the above equations for Δf and ΔS to
the last one, we have
f 1  2 f 2 2
 ( 
s S )t
2
t 2 S
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
13.19
The Derivation of the Black-Scholes
Differential Equation continued
The return on the portfolio must be the risk - free
rate. Hence
  r t
We substitute for  ƒ and S in these equations
to get the Black - Scholes differenti al equation :
2
ƒ
ƒ

ƒ
2 2
 rS
½ s S
 rƒ
2
t
S
S
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
13.20
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
13.21
The Black-Scholes Formulas
(See pages 295-297)
c  S 0 N (d1 )  K e
 rT
N (d 2 )
p  K e  rT N (d 2 )  S 0 N (d1 )
2
ln( S 0 / K )  (r  s / 2)T
where d1 
s T
ln( S 0 / K )  (r  s 2 / 2)T
d2 
 d1  s T
s T
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
13.22
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
13.23
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
13.24
Risk-Neutral Valuation




The variable m does not appear in the BlackScholes 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
13.25
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
13.26
Example (Continued)
Then
ln( S0 / X)(r s 2 / 2)T ln( 42/ 40)(0.10.22 / 2)0.5
d1 

0.7693
s T
0.2 0.5
d2 d1 s T =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
13.27
Example (Continued)
Hence, if the option is a European call, its value c is
given by
c = 42 N(0.7693) - 38.049 N(0.6278)
And if the option is a European put option, its value, p, is
given by
p = 38.049 N(-0.6278) – 42 N(-0.7693)
Using the polynomial approximation,
N(0.7693) = 0.7791
N(-0.7693) = 0.2209
N(0.6278) = 0.7349
N(-0.6278) = 0.2651
So that,
c=4.76 and
p=0.81
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005
13.28