11.1
The Black-Scholes
Model
Chapter 11
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.2
The Stock Price Assumption
• Consider a stock whose price is S
• In a short period of time of length Dt the
change in then stock price is assumed
to be normal with mean mSdt and
standard deviation
sS Dt
 m is expected return and s is volatility
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
The Lognormal Property
(Equation 11.2, page 238)
• 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, 4th Edition © 1999 by John C. Hull
11.3
11.4
The Lognormal Distribution
E ( ST )  S0 e mT
2 2 mT
var ( ST )  S0 e
(e
s2T
 1)
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.5
Continuously Compounded
Return, h (Equation (11.7), page 240)
ST  S0 e hT
or
1
ST
h = ln
T
S0
or

s2
h   m 
,
2

s 

T
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.6
The Expected Return
• The expected value of the stock price is
•
SemT
The expected return on the stock is
m – s2/2
Note that E[ln(ST)] is not equal to ln[E(ST)]
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.7
The Volatility
• The volatility is the standard deviation
•
of the continuously compounded rate
of return in 1 year
As an approximation it is the standard
deviation of the proportional change
in 1 year
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
Estimating Volatility from
Historical Data (page 242-4)
1. Take observations S0, S1, . . . , Sn at
intervals of t years
2. Define the continuously compounded
return as:
 Si 

ui  ln
 Si 1 
3. Calculate the standard deviation, s , of
the ui ´s
s
4. The historical volatility estimate is: s* 
t
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.8
11.9
The Concepts Underlying
Black-Scholes
•
•
•
•
The option price & 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, 4th Edition © 1999 by John C. Hull
11.10
The Derivation of the
Black-Scholes Differential Equation
1 of 3:
DS  mS Dt  sS Dz
2
 ƒ
ƒ
 ƒ 2 2
ƒ
D ƒ   mS 
 ½ 2 s S  Dt  sS Dz
t
S
S
 S

We set up a portfolio consisting of
 1: derivative
ƒ
+ : shares
S
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.11
The Derivation of the
Black-Scholes Differential Equation
2 of 3:
The value of the portfolio  is given by
ƒ
  ƒ 
S
S
The change in its value in time Dt is given by
ƒ
D   D ƒ 
DS
S
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.12
The Derivation of the
Black-Scholes Differential Equation
3 of 3:
The return on the portfolio must be the risk - free rate. Hence
D  r Dt
We substitute for D ƒ and DS in these equations to get the
Black - Scholes differential equation:
2
ƒ
ƒ

ƒ
2 2
 rS
½ s S
 rƒ
2
t
S
S
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.13
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 a forward contract the boundary condition is
ƒ = S – K when t =T
The solution to the equation is
ƒ = S – K e–r (T
–t)
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.14
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, 4th Edition © 1999 by John C. Hull
11.15
Applying Risk-Neutral
Valuation
1. Assume that the expected
return from the stock price is
the risk-free rate
2. Calculate the expected
payoff from the option
3. Discount at the risk-free
rate
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.16
The Black-Scholes Formulas
(See pages 250-252)
c  S 0 N ( d1 )  X e
p Xe
 rT
 rT
N (d 2 )
N (  d 2 )  S 0 N (  d1 )
2
ln( S0 / X )  (r  s / 2)T
where d1 
s T
2
ln( S0 / X )  (r  s / 2)T
d2 
 d1  s T
s T
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.17
Implied Volatility
• The implied volatility of an option is the
•
•
volatility for which the Black-Scholes
price equals the market price
The is a one-to-one correspondence
between prices and implied volatilities
Traders and brokers often quote implied
volatilities rather than dollar prices
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.18
Causes of Volatility
• Volatility is usually much greater when
•
the market is open (i.e. the asset is
trading) than when it is closed
For this reason time is usually
measured in “trading days” not calendar
days when options are valued
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.19
Warrants & Dilution (pages 253-4)
• When a regular call option is exercised
•
•
•
the
stock that is delivered must be purchased in
the open market
When a warrant is exercised new Treasury stock
is issued by the company
This will dilute the value of the existing stock
One valuation approach is to assume that all
equity (warrants + stock) follows geometric
Brownian motion
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.20
Dividends
• European options on dividend-paying
•
•
stocks are valued by substituting the
stock price less the present value of
dividends into Black-Scholes
Only dividends with ex-dividend dates
during life of option should be included
The “dividend” should be the expected
reduction in the stock price expected
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.21
American Calls
• An American call on a non-dividend-paying
•
stock should never be exercised early
An American call on a dividend-paying stock
should only ever be exercised immediately
prior to an ex-dividend date
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull
11.22
Black’s Approach to Dealing with
Dividends in American Call Options
Set the American price equal to the
maximum of two European prices:
1. The 1st European price is for an
option maturing at the same time as the
American option
2. The 2nd European price is for an
option maturing just before the final exdividend date
Options, Futures, and Other Derivatives, 4th Edition © 1999 by John C. Hull