12.1
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 1
The Black-Scholes
Model
Chapter 12
12.2
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 2
The Stock Price Assumption
? Consider a stock whose price is S
? In a short period of time of length dt,the
return on the stock is normally distributed:
where m is expected return and s is
volatility
? ?ttSS dsmd??d,
12.3
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 3
The Lognormal Property
(Equations 12.2 and 12.3,page 235)?
It follows from this assumption that
? Since the logarithm of ST is normal,ST is
lognormally distributed
ln ln,
ln ln,
S S T T
S S T T
T
T
? ? ?
?
?
?
?
?
?
?
?
?
?
?
?
? ? ?
?
?
?
?
?
?
?
?
?
?
?
?
0
2
0
2
2
2
? m
s
s
? m
s
s
or
12.4
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull
The Lognormal Distribution
E S S e
S S e e
T
T
T
T T
( )
( ) ( )
?
? ?
0
0
2 2 2 1
v ar
m
m s
12.5
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 5
Continuously Compounded
Return,h ?Equations 12.6 and 12.7),page 236)
S S e
T
S
S
T
T
T
T
?
? ?
?
?
?
?
?
?
0
0
1
2
or
=
o r
2
h
h
h ? m
s s
ln
,
12.6
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 6
The Expected Return
? The expected value of the stock price is
S0emT
? The expected return on the stock is
m –s2/2
? ?
? ? m?
s?m?
)/(ln
2/)/l n (
0
2
0
SSE
SSE
T
T
12.7
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 7
The Volatility
? The volatility of an asset is the
standard deviation of the continuously
compounded rate of return in 1 year
? As an approximation it is the standard
deviation of the percentage change in
the asset price in 1 year
12.8
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 8
Estimating Volatility from
Historical Data (page 239-41)
1,Take observations S0,S1,.,,,Sn at
intervals of t years2.
Calculate the continuously compounded
return in each interval as:
3,Calculate the standard deviation,s,of
the ui′s
4,The historical volatility estimate is:
u SSi i
i
? ??? ???
?
ln
1
t
?s s?
12.9
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 9
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
12.10
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 10
The Derivation of the Black-Scholes
Differential Equation
sh ar e s,
?
+
d e r i v a t i v e,
of co n si st i n g p o r t f o l i o a up se t e W
??
?
??
?
S
zS
S
tS
St
S
S
zStSS
?
?
?
ds
?
?
?d
?
?
?
?
?
?
?
?
s
?
?
?
?
?
?m
?
?
?d
ds?dm?d
1
22
2
2
12.11
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 11
?
?
by g i v e n is t i m e in v a l u e i t s in ch an g e T h e
?
?
by g i v e n is p o r t f o l i o t h e of v a l u e T h e
S
S
t
S
S
d
?
?
?d???d
d
?
?
????
?
The Derivation of the Black-Scholes
Differential Equation continued
12.12
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 12
The Derivation of the Black-Scholes
Differential Equation continued
?
?
?
??
:e q u a t i o n ald i f f e r e n t i S ch o l e s-B l a ck t h e get to
e q u a t i o n s t h e se in a n d ? f o r su b st i t u t e We
H e n ce r a t e,
f r e e-r i sk t h e be m u st p o r t f o l i o t h e on r e t u r n T h e
r
S
S
S
rS
t
S
tr
?
?
?
s?
?
?
?
?
?
dd
d???d
2
2
22
12.13
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 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
? = S – K when t =T
? The solution to the equation is
? = S – K e–r (T – t )
12.14
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 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
12.15
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 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
12.16
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 16
The Black-Scholes Formulas
(See pages 246-248)
Td
T
TrKS
d
T
TrKS
d
dNSdNeKp
dNeKdNSc
rT
rT
s??
s
s??
?
s
s??
?
????
??
?
?
1
0
2
0
1
102
210
)2/
2
()/l n (
)2/
2
()/l n (
)()(
)()(
w h e r e
12.17
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 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
12.18
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 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
12.19
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 19
Warrants & Dilution (pages 249-50)
? 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
12.20
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 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
12.21
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 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
12.22
Options,Futures,and Other Derivatives,5th Edition ? 2002 by John C,Hull 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 ex-
dividend date