11.1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Black-Scholes
Model
Chapter 11
11.2
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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 S is assumed to be
normal with mean mSdt and standard
deviation,that is,S follows geometric
Brownian motion ds=m Sdt+?Sdz,Then
m is expected return and? is volatility
S tD
11.3
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Lognormal Property
It follows from this assumption that
Since the logarithm of ST is normal,ST is
lognormally distributed
2
0
2
0
l n l n,
2
or
l n l n,
2
T
T
S S T T
S S T T
m?
m?
11.4
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Modeling Stock Prices in Finance
In finance,frequently we model the evolution
of stock prices as a generalized Wiener
Process
Also,assume prices are distributed lognormal
and returns are distributed normal
dzSdtSdS?m
11.5
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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?
11.6
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Continuously Compounded Rate of
Return,? (Equation (11.7))
S S e
T
S
S
T
T
T
T
0
0
1
2
or
=
o r
2
m
ln
,
11.7
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Expected Return
The expected value of the stock price is
E(ST)=S0emT
The expected continuously compounded return
on the stock is
E(?)=m –?2/2 (the geometric average)
m is the the arithmetic average of the returns
Note that E[ln(ST)] is not equal to ln[E(ST)]
ln[E(ST)]=ln S0+m T,
E[ln(ST)] =ln S0+(m-?2/2)T
11.8
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Expected Return Example
Take the following 5 annual returns,10%,12%,8%,9%,and 11%
The arithmetic average is
However,the geometric average is
Thus,the arithmetic average overstates the geometric average.
The geometric is the actual return that one would have earned.
The approximation for the geometric return is
This differs from g as the returns are not normally distributed.
10.050.0*)11.009.008.012.010.0( 51
1
511
_
n
i
in xx
(? 0 9 9 9 1.0111.1*09.1*08.1*12.1*10.11)1( 511
1
nn
i i
xg
(? 0 9 9 8 8.02/0 1 5 8 1 1.010.02/ 22m
11.9
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Is Normality Realistic?
If returns are normal and thus prices are
lognormal and assuming that volatility is at
20% (about the historical average)
– On 10/19/87,the 2 month S&P 500 Futures
dropped 29%
This was a -27 sigma event with a probability of
occurring of once in every 10160 days
– On 10/13/89,the S&P 500 index lost about 6%
This was a -5 sigma event with a probability of
0.00000027 or once every 14,756 years
11.10
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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
11.11
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Assumptions Underlying
Black-Scholes
1,The stock follows a Brownian motion with constant m and?
2,Short selling of securities with full use of proceeds is permitted
3,No transaction cost or taxes
4,Securities are perfectly divisible
5,No dividends paid during the life of the option
6,There are no arbitrage opportunities
7,Security trading is continuous
8,The risk-free rate of interest,r,is constant and is the same for
all maturities
11.12
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
1 of 3,The Derivation of the
Black-Scholes Differential Equation
D D D
D D D
S S t S z
S
S
t S
S t
S
S z
S
m?
m
W e set up a por tf ol i o consi sti n g of
,der i v ati ve
+
,shar es
2
2
2 2
1
11.13
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
T he v al ue of the por tf ol i o i s g i v en by
T he c hang e i n i ts v a l ue i n ti m e i s g i v en by
D
D? D D
S
S
t
S
S
2 of 3,The Derivation of the
Black-Scholes Differential Equation
11.14
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
3 of 3,The Derivation of the
Black-Scholes Differential Equation
2
22
T h e r e tur n o n the p o r tf o l io m u st b e the r isk- f r e e r a te,H e n c e
W e su b stit u te f o r a n d in the se e q u a ti o n s to g e t the
B l a c k - Sc h o l e s d if f e r e n ti a l e q u a ti o n,
1
2
f
fS
ff
rS S
tS
rt
S
D
D
D
D
2
rf?
11.15
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Algebra of the Differential
Equation I (?=D)
notice that all of the m’s cancel out
tS
S
f
frtS
S
f
t
f
tS
S
f
frtS
S
f
t
f
tS
S
f
frtS
S
f
tS
S
f
S
S
f
t
f
tS
S
f
frwStS
S
f
wS
S
f
tS
S
f
S
S
f
t
f
tS
S
f
frS
S
f
f
tr
S
f
f
mm
mm
22
2
2
22
2
2
22
2
2
22
2
2
2
1
0
2
1
0
00
2
1
2
1
11.16
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Algebra of the Differential
Equation II
Again,this is the Black-Scholes partial
differential equation
rfSS fSSfrtf
SSfrrfSS ftf
SSffrSS ftf
tSSffrtSS ftf
22
2
2
22
2
2
22
2
2
22
2
2
2
1
2
1
2
1
2
1
11.17
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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 )
11.18
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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
11.19
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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
11.20
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Application to Forward contracts
on a Stock
Long forward contract with maturity time T and
delivery price K
The value of the contract at T=ST-K
In the risk-neutral world,f,the value of the
forward contract at t<T is
f=e-r(T-t) (ST-K)
where K is constand and (ST)=Sem(T-t)
Take m=r
f=S-K e-r(T-t) (same as eq.3.9)
E
E
11.21
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Black-Scholes Formulas 0
0 1 2
2 0 1
1
0
21
2
l n( / ) ( / 2)
wh
( ) ( )
e r e
2
l n( / ) ( / 2)
( )
(
)
rT
rT
c S N d X e
S X r T
d
T
S
Nd
p X e N
X r T
d d T
T
d S N d
and N(x) is the CDF for standard normal distribution,Section 11.8 gives a
polynomial approximation with 6-decial-place of accuracy
11.22
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Example using Black-Scholes
Consider a European call with S0=105,K=100,
r =10%,D = 0,T = 0.25,and? = 30%,
Calculate c.
First,calculate d1 and d2
4174.0
5673.0
25.030.0
)25.0)(2/30.010.0()100/105l n (
)2/()/l n (
12
2
2
0
1
Tdd
T
TrKS
d
11.23
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Example using Black-Scholes
(continued)
Hence,
N(d1) = 0.7123+0.73*(0.7157-0.7123) = 0.7148
N(d2) = 0.6618
So,
0 1 2
0,1 0 * 0,2 5
( ) e ( )
1 0 5 * 0,7 1 4 8 1 0 0 e * 0,6 6 1 8
7 5,0 5 4 0 6 4,5 4 6 0
- r Tc S N d K N d
10,50 50
11.24
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Intuition Resulting from the
Black-Scholes Formulas
NPV
Expected value of a variable that
is ST if ST >K and 0 otherwise
Strike Price * Pr of exercise
)]()(e[e
)(e)(
210
210
dKNdNSc
dNKdNSc
rT- r T
- r T
11.25
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Implied Volatility
Implied Volatility is simply the volatility implied
by the market price of the option
In other words,given the market price of the
option and the B-S inputs for S0,K,r,D,T,
what? is necessary to equate the B-S price
with the market price
The is a one-to-one correspondence between
prices and implied volatilities
Note,the B-S formula CAN NOT be inverted to
solve for?,hence one must use an iterative
technique
11.26
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Example with Implied Volatility
This example uses trial and error in EXCEL,
However,I suggest that you use EXCEL’s
solver if at all possible
You have a European call with c=2.49
S0=90,K=100,r=0.08,D=0,and T=0.25
c?
1.03 20%
1.72 25%
2.49 30%
3.31 35%
4.15 40%
11.27
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Causes of Volatility
Random arrival of new information
– However,volatility is larger when the exchanges
are open than when they are closed
– Most information arrives while the exchanges are
closed
Trading
– Hence,for annualizing daily volatility use 252
trading days per year
11.28
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Volatility
The volatility is the STD of the continuously
compounded rate of return? in 1 year
(eq,11.7)
As an approximation it is the STD of the
proportional change DS/S in the stock price
in 1 year (eq,10.9)
It is also the STD of ln S at the end of 1
year (eq,11.2)
11.29
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Estimating Volatility from
Historical Data
1.Take observations S0,S1,.,,,Sn at intervals of t years
2.Define the continuously compounded return 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 *
*
() 2w i t h S T D e r r o r n
11.30
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Estimating Volatility from Historical
Data Example
W e e k
C l o s i n g
S t o c k
P r i c e S
i
/S
i-1
R e t u r n
u
i
=
l n ( S
i
/S
i-1
)
l n ( S
i
/S
i-1
)
S q u a r e d
0 1 0 1 / 4
1 1 0 1 / 8 0,9 8 7 8 0 - 0,0 1 2 2 7 0,0 0 0 1 5
2 1 0 1 / 4 1,0 1 2 3 5 0,0 1 2 2 7 0,0 0 0 1 5
3 1 0 1 / 2 1,0 2 4 3 9 0,0 2 4 1 0 0,0 0 0 5 8
4 1 0 1 / 8 0,9 6 4 2 9 - 0,0 3 6 3 7 0,0 0 1 3 2
5 1 0 5 / 8 1,0 4 9 3 8 0,0 4 8 2 0 0,0 0 2 3 2
6 1 0 1 / 2 0,9 8 8 2 4 - 0,0 1 1 8 3 0,0 0 0 1 4
7 1 0 3 / 4 1,0 2 3 8 1 0,0 2 3 5 3 0,0 0 0 5 5
8 1 0 7 / 8 1,0 1 1 6 3 0,0 1 1 5 6 0,0 0 0 1 3
9 1 0 3 / 4 0,9 8 8 5 1 - 0,0 1 1 5 6 0,0 0 0 1 3
10 1 0 1 / 2 0,9 7 6 7 4 - 0,0 2 3 5 3 0,0 0 0 5 5
S u m 0,0 2 4 1 0 0,0 0 6 0 4
11.31
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Estimating Volatility from
Historical Data Example
(continued)
02 578.0
)110(10
)02 410.0(
110
00 604.0
)1(
1
1
1
or)(
1
1
2
1
2
11
2
1
2
n
i
n
ii
n
i
i unnunuuns? The standard deviation of the ui's is
Now,this is the weekly volatility,To annualize,simply
multiply by the square root of the periodicity.
%59.181859.0
52*02578.0*
%16.40416.010*2/1859.0
2/**
n
11.32
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Warrants & Dilution (稀释 )
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
11.33
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Valuation of European Warrant
(Page 254)
A Company has N outstanding shares and issues M
European warrants NOW.
The holder has the right to buy? shares at T at a price
of X per share.
If the warrant is exercised at T,the Company’s equity
increased from VT to VT+M?X
Share price after exercise=(VT+M?X)/(N+M?)
The payoff per warrant to the holder is
TVm a x [ ( ),0 ] m a x ( )TM X VNXX
N M N M N
11.34
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Valuation of European Warrant
(continued)
The value of the warrant is
where V is the value of the company’s equity,
At time 0,V0=NS0+MW (W=?)
v a l u e o f r e g u l a r c a l l o p t i o n s o n V / N,NNM
0
0
V MSW
NN
11.35
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Valuation of European Warrant
(continued)
The procedure to give the warrant price W by
modifying the B-S formula
1,The stock price S0 us replaced by S0+(M/N)W
2,?is the volatility of the company’s equity (including
both the shares and the warrants)
3,The B-S formula is multiplied by the coefficient
N
NM
11.36
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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
Example 11.7 (page 258)
11.37
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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
11.38
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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
Example 11.8 (page 260)
11.39
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Early Exercise on American Calls for
Dividend Paying Stocks
For early exercise to be optimal,the dividend must
outweigh the interest savings on the strike price
Only need to look at days that dividends are paid
Looking at the last dividend if
then no early exercise at that time
Similarly,for any i < n if
then no early exercise at those times
)e1( )( ntTrn KD
)e1( )( 1 ii ttri KD
11.40
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Assignment
11.4 +11.5,11.8,11.11,11.13,11.15,
11.16,11.17,11.18,11.19,11.23
Assignment Questions
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Black-Scholes
Model
Chapter 11
11.2
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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 S is assumed to be
normal with mean mSdt and standard
deviation,that is,S follows geometric
Brownian motion ds=m Sdt+?Sdz,Then
m is expected return and? is volatility
S tD
11.3
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Lognormal Property
It follows from this assumption that
Since the logarithm of ST is normal,ST is
lognormally distributed
2
0
2
0
l n l n,
2
or
l n l n,
2
T
T
S S T T
S S T T
m?
m?
11.4
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Modeling Stock Prices in Finance
In finance,frequently we model the evolution
of stock prices as a generalized Wiener
Process
Also,assume prices are distributed lognormal
and returns are distributed normal
dzSdtSdS?m
11.5
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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?
11.6
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Continuously Compounded Rate of
Return,? (Equation (11.7))
S S e
T
S
S
T
T
T
T
0
0
1
2
or
=
o r
2
m
ln
,
11.7
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Expected Return
The expected value of the stock price is
E(ST)=S0emT
The expected continuously compounded return
on the stock is
E(?)=m –?2/2 (the geometric average)
m is the the arithmetic average of the returns
Note that E[ln(ST)] is not equal to ln[E(ST)]
ln[E(ST)]=ln S0+m T,
E[ln(ST)] =ln S0+(m-?2/2)T
11.8
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Expected Return Example
Take the following 5 annual returns,10%,12%,8%,9%,and 11%
The arithmetic average is
However,the geometric average is
Thus,the arithmetic average overstates the geometric average.
The geometric is the actual return that one would have earned.
The approximation for the geometric return is
This differs from g as the returns are not normally distributed.
10.050.0*)11.009.008.012.010.0( 51
1
511
_
n
i
in xx
(? 0 9 9 9 1.0111.1*09.1*08.1*12.1*10.11)1( 511
1
nn
i i
xg
(? 0 9 9 8 8.02/0 1 5 8 1 1.010.02/ 22m
11.9
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Is Normality Realistic?
If returns are normal and thus prices are
lognormal and assuming that volatility is at
20% (about the historical average)
– On 10/19/87,the 2 month S&P 500 Futures
dropped 29%
This was a -27 sigma event with a probability of
occurring of once in every 10160 days
– On 10/13/89,the S&P 500 index lost about 6%
This was a -5 sigma event with a probability of
0.00000027 or once every 14,756 years
11.10
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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
11.11
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Assumptions Underlying
Black-Scholes
1,The stock follows a Brownian motion with constant m and?
2,Short selling of securities with full use of proceeds is permitted
3,No transaction cost or taxes
4,Securities are perfectly divisible
5,No dividends paid during the life of the option
6,There are no arbitrage opportunities
7,Security trading is continuous
8,The risk-free rate of interest,r,is constant and is the same for
all maturities
11.12
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
1 of 3,The Derivation of the
Black-Scholes Differential Equation
D D D
D D D
S S t S z
S
S
t S
S t
S
S z
S
m?
m
W e set up a por tf ol i o consi sti n g of
,der i v ati ve
+
,shar es
2
2
2 2
1
11.13
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
T he v al ue of the por tf ol i o i s g i v en by
T he c hang e i n i ts v a l ue i n ti m e i s g i v en by
D
D? D D
S
S
t
S
S
2 of 3,The Derivation of the
Black-Scholes Differential Equation
11.14
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
3 of 3,The Derivation of the
Black-Scholes Differential Equation
2
22
T h e r e tur n o n the p o r tf o l io m u st b e the r isk- f r e e r a te,H e n c e
W e su b stit u te f o r a n d in the se e q u a ti o n s to g e t the
B l a c k - Sc h o l e s d if f e r e n ti a l e q u a ti o n,
1
2
f
fS
ff
rS S
tS
rt
S
D
D
D
D
2
rf?
11.15
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Algebra of the Differential
Equation I (?=D)
notice that all of the m’s cancel out
tS
S
f
frtS
S
f
t
f
tS
S
f
frtS
S
f
t
f
tS
S
f
frtS
S
f
tS
S
f
S
S
f
t
f
tS
S
f
frwStS
S
f
wS
S
f
tS
S
f
S
S
f
t
f
tS
S
f
frS
S
f
f
tr
S
f
f
mm
mm
22
2
2
22
2
2
22
2
2
22
2
2
2
1
0
2
1
0
00
2
1
2
1
11.16
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Algebra of the Differential
Equation II
Again,this is the Black-Scholes partial
differential equation
rfSS fSSfrtf
SSfrrfSS ftf
SSffrSS ftf
tSSffrtSS ftf
22
2
2
22
2
2
22
2
2
22
2
2
2
1
2
1
2
1
2
1
11.17
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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 )
11.18
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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
11.19
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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
11.20
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Application to Forward contracts
on a Stock
Long forward contract with maturity time T and
delivery price K
The value of the contract at T=ST-K
In the risk-neutral world,f,the value of the
forward contract at t<T is
f=e-r(T-t) (ST-K)
where K is constand and (ST)=Sem(T-t)
Take m=r
f=S-K e-r(T-t) (same as eq.3.9)
E
E
11.21
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Black-Scholes Formulas 0
0 1 2
2 0 1
1
0
21
2
l n( / ) ( / 2)
wh
( ) ( )
e r e
2
l n( / ) ( / 2)
( )
(
)
rT
rT
c S N d X e
S X r T
d
T
S
Nd
p X e N
X r T
d d T
T
d S N d
and N(x) is the CDF for standard normal distribution,Section 11.8 gives a
polynomial approximation with 6-decial-place of accuracy
11.22
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Example using Black-Scholes
Consider a European call with S0=105,K=100,
r =10%,D = 0,T = 0.25,and? = 30%,
Calculate c.
First,calculate d1 and d2
4174.0
5673.0
25.030.0
)25.0)(2/30.010.0()100/105l n (
)2/()/l n (
12
2
2
0
1
Tdd
T
TrKS
d
11.23
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Example using Black-Scholes
(continued)
Hence,
N(d1) = 0.7123+0.73*(0.7157-0.7123) = 0.7148
N(d2) = 0.6618
So,
0 1 2
0,1 0 * 0,2 5
( ) e ( )
1 0 5 * 0,7 1 4 8 1 0 0 e * 0,6 6 1 8
7 5,0 5 4 0 6 4,5 4 6 0
- r Tc S N d K N d
10,50 50
11.24
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Intuition Resulting from the
Black-Scholes Formulas
NPV
Expected value of a variable that
is ST if ST >K and 0 otherwise
Strike Price * Pr of exercise
)]()(e[e
)(e)(
210
210
dKNdNSc
dNKdNSc
rT- r T
- r T
11.25
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Implied Volatility
Implied Volatility is simply the volatility implied
by the market price of the option
In other words,given the market price of the
option and the B-S inputs for S0,K,r,D,T,
what? is necessary to equate the B-S price
with the market price
The is a one-to-one correspondence between
prices and implied volatilities
Note,the B-S formula CAN NOT be inverted to
solve for?,hence one must use an iterative
technique
11.26
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Example with Implied Volatility
This example uses trial and error in EXCEL,
However,I suggest that you use EXCEL’s
solver if at all possible
You have a European call with c=2.49
S0=90,K=100,r=0.08,D=0,and T=0.25
c?
1.03 20%
1.72 25%
2.49 30%
3.31 35%
4.15 40%
11.27
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Causes of Volatility
Random arrival of new information
– However,volatility is larger when the exchanges
are open than when they are closed
– Most information arrives while the exchanges are
closed
Trading
– Hence,for annualizing daily volatility use 252
trading days per year
11.28
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Volatility
The volatility is the STD of the continuously
compounded rate of return? in 1 year
(eq,11.7)
As an approximation it is the STD of the
proportional change DS/S in the stock price
in 1 year (eq,10.9)
It is also the STD of ln S at the end of 1
year (eq,11.2)
11.29
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Estimating Volatility from
Historical Data
1.Take observations S0,S1,.,,,Sn at intervals of t years
2.Define the continuously compounded return 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 *
*
() 2w i t h S T D e r r o r n
11.30
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Estimating Volatility from Historical
Data Example
W e e k
C l o s i n g
S t o c k
P r i c e S
i
/S
i-1
R e t u r n
u
i
=
l n ( S
i
/S
i-1
)
l n ( S
i
/S
i-1
)
S q u a r e d
0 1 0 1 / 4
1 1 0 1 / 8 0,9 8 7 8 0 - 0,0 1 2 2 7 0,0 0 0 1 5
2 1 0 1 / 4 1,0 1 2 3 5 0,0 1 2 2 7 0,0 0 0 1 5
3 1 0 1 / 2 1,0 2 4 3 9 0,0 2 4 1 0 0,0 0 0 5 8
4 1 0 1 / 8 0,9 6 4 2 9 - 0,0 3 6 3 7 0,0 0 1 3 2
5 1 0 5 / 8 1,0 4 9 3 8 0,0 4 8 2 0 0,0 0 2 3 2
6 1 0 1 / 2 0,9 8 8 2 4 - 0,0 1 1 8 3 0,0 0 0 1 4
7 1 0 3 / 4 1,0 2 3 8 1 0,0 2 3 5 3 0,0 0 0 5 5
8 1 0 7 / 8 1,0 1 1 6 3 0,0 1 1 5 6 0,0 0 0 1 3
9 1 0 3 / 4 0,9 8 8 5 1 - 0,0 1 1 5 6 0,0 0 0 1 3
10 1 0 1 / 2 0,9 7 6 7 4 - 0,0 2 3 5 3 0,0 0 0 5 5
S u m 0,0 2 4 1 0 0,0 0 6 0 4
11.31
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Estimating Volatility from
Historical Data Example
(continued)
02 578.0
)110(10
)02 410.0(
110
00 604.0
)1(
1
1
1
or)(
1
1
2
1
2
11
2
1
2
n
i
n
ii
n
i
i unnunuuns? The standard deviation of the ui's is
Now,this is the weekly volatility,To annualize,simply
multiply by the square root of the periodicity.
%59.181859.0
52*02578.0*
%16.40416.010*2/1859.0
2/**
n
11.32
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Warrants & Dilution (稀释 )
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
11.33
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Valuation of European Warrant
(Page 254)
A Company has N outstanding shares and issues M
European warrants NOW.
The holder has the right to buy? shares at T at a price
of X per share.
If the warrant is exercised at T,the Company’s equity
increased from VT to VT+M?X
Share price after exercise=(VT+M?X)/(N+M?)
The payoff per warrant to the holder is
TVm a x [ ( ),0 ] m a x ( )TM X VNXX
N M N M N
11.34
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Valuation of European Warrant
(continued)
The value of the warrant is
where V is the value of the company’s equity,
At time 0,V0=NS0+MW (W=?)
v a l u e o f r e g u l a r c a l l o p t i o n s o n V / N,NNM
0
0
V MSW
NN
11.35
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Valuation of European Warrant
(continued)
The procedure to give the warrant price W by
modifying the B-S formula
1,The stock price S0 us replaced by S0+(M/N)W
2,?is the volatility of the company’s equity (including
both the shares and the warrants)
3,The B-S formula is multiplied by the coefficient
N
NM
11.36
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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
Example 11.7 (page 258)
11.37
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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
11.38
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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
Example 11.8 (page 260)
11.39
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Early Exercise on American Calls for
Dividend Paying Stocks
For early exercise to be optimal,the dividend must
outweigh the interest savings on the strike price
Only need to look at days that dividends are paid
Looking at the last dividend if
then no early exercise at that time
Similarly,for any i < n if
then no early exercise at those times
)e1( )( ntTrn KD
)e1( )( 1 ii ttri KD
11.40
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Assignment
11.4 +11.5,11.8,11.11,11.13,11.15,
11.16,11.17,11.18,11.19,11.23
Assignment Questions
