12-0
Finance 457
The Black–Scholes Model
12
Chapter Twelve
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12-1
Finance 457
Chapter Outline
12.1 Log-Normal Property of Stock Prices
12.2 The distribution of the Rate of Return
12.3 The Expected Return
12.4 Volatility
12.5 Concepts Underlying the Black-Scholes-Merton
Differential Equation
12.6 Derivation of the Black-Scholes-Merton
Differential Equation
12.7 Risk Neutral Valuation
12.8 Black-Scholes Pricing Formulae
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12-2
Finance 457
Chapter Outline (continued)
12.9 Cumulative Normal Distribution Function
12.10 Warrants Issued by a company on its own Stock
12.11 Implied Volatilities
12.12 The Causes of Volatility
12.13 Dividends
12.14 Summary
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12-3
Finance 457
Prospectus:
• In the early 1970s, Fischer Black, Myron
Scholes and Robert Merton made a major
breakthrough in the pricing of options.
• In 1997, the importance of this work was
recognized with the Nobel Prize.
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12-4
Finance 457
12.1 Log-Normal Property of Stock Prices
• This is fully developed in chapter 11.
• Assume that percentage changes in stock price in a
short period of time are normally distributed.
• Let:
m: expected return on the stock
s: volatility on the stock
• The mean of the percentage change in time dt is mdt
• The standard deviation of this percentage change is
σ δt
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12-5
Finance 457
12.1 Log-Normal Property of Stock Prices
• The percentage changes in stock price in a short
period of time are normally distributed:
δS
  ( μδt , σ δt )
S
• Where dS is the change in stock price in time dt, and
(m,s) denotes a normal distribution with mean m
and standard deviation s.
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12-6
Finance 457
12.1 Log-Normal Property of Stock Prices
• As shown in section 11.7, the model implies that
2


s 
T , σ T 
ln S T  ln S 0    μ 
2 


• From this it follows that


ST
s2 
T , σ T 
ln
   μ 
S0
2 


• and
2



s 
T , σ T 
ln S T   ln S 0   μ 
2 



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12-7
Finance 457
12.1 Log-Normal Property of Stock Prices
2



s 
T , σ T 
ln S T   ln S 0   μ 
2 



The above equation shows that lnST is normally
distributed.
This means that ST has a lognormal distribution.
A variable with this distribution can take any value
between zero and infinity.
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Finance 457
Properties of the Log-Normal Distribution
A variable with this
distribution can take any
value between zero and
infinity.
0
Unlike a normal distributions, it is skewed so that the
mean, median, and mode are all different.
E (ST )  S 0 e
McGraw-Hill/Irwin
μT
var( ST )  S e
2 2 μT
0
(e
s 2T
 1)
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12-9
Finance 457
12.2 The Distribution of the Rate of Return
• The lognormal property of stock prices can be used
to provide information on the probability
distribution of the continuously compounded rate of
return earned on a stock between time zero and T.
• Define the continuously compounded rate of return
per annum realized between times zero and T as h.
• It follows that S T  S 0 ehT
1 ST
so that h  ln
T S0
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12-10
Finance 457
12.2 The Distribution of the Rate of Return
2




S
s
T
It follows from: ln
T , σ T 
   μ 
S0
2 



s
σ 
that h    μ 

,

2
T

2
Thus, the continuously compounded rate of
return per annum in normally distributed
with mean μ 
McGraw-Hill/Irwin
s
2
2
and standard deviation
σ
T
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12-11
Finance 457
12.3 The Expected Return
• The expected return, m, required by investors from a
stock depends on the riskiness of the stock.
• The higher the risk, the higher m, ceteris paribus.
• m also depends on interest rates in the economy
• We could spend a lot of time on the determinants of
m, but it turns out that the value of a stock option,
when expressed in terms of the value of the
underlying stock, does not depend on m at all.
• There is however, one aspect of m that frequently
causes confusion and is worth explaining.
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Finance 457
A subtle but important difference
δS
  ( μδt , σ δt )
S
• Shows that mdt is the expected percentage change in
the stock price in a very short period of time d t .
• This means that m is the expected return in a very
short short period of time dt .
• It is tempting to assume that m is also the
continuously compounded return on the stock over a
relatively long period of time.
• However, this is not the case.
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12-13
Finance 457
A subtle but important difference
• The continuously compounded return on the stock
over T years is:
1 ST
ln
T S0
2

s
σ 

Equation (12.7): h    μ 
,
2
T

Shows that the expected value of this is μ 
s2
The distinction between m and μ 
2
is subtle but important.
McGraw-Hill/Irwin
s
2
2
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12-14
Finance 457
A subtle but important difference
Start with
E ( ST )  S 0 e μT
Taking logarithms: ln[ E ( ST )]  ln( S 0 )  μT
Since ln is a nonlinear function, ln[ E (ST )]  E[ln( ST )]
ST
So we cannot say E[ln(
)]  μT
S0
ST
In fact, we have: E[ln(
)]  μT
S0
So the expected return over the whole period T,
2
s
expressed with compounding dt, is close to μ 
2
Not m
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12-15
Finance 457
A subtle but important difference
• The above shows that a simple term like expected
return is ambiguous.
• It can refer to
μ
s
2
2
or m
• For example, if your portfolio has had the
following returns over the last five years:
30%; 20%; 10%; –20%; –40%;
• What is the expected return?
• Unless otherwise stated m will be expected return.
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12-16
Finance 457
12.4 Volatility
• The volatility of a stock, s, is a measure of our
uncertainty about the returns.
• Stocks typically have a volatility between 20% and
50%
2

s
σ 

• From h    μ 
,
2
T

• The volatility of a stock price can be defined as the
standard deviation of the return provided by the
stock in one year when the return is expressed using
continuous compounding.
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12-17
Finance 457
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
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12-18
Finance 457
Estimating Volatility from Historical Data (page 239-41)
1. Take observations S0, S1, . . . , Sn at intervals of t
years
2. Calculate the continuously compounded return in
each interval as:
 Si 

ui  ln
 Si 1 
3. Calculate the standard deviation, s , of the ui´s
4. The historical volatility estimate is:
McGraw-Hill/Irwin
s
sˆ 
t
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12-19
Finance 457
12.5 Concepts Underlying the BlackScholes-Merton Differential Equation
• The arguments are similar to the no-arbitrage
arguments we used to value stock options using
binomial valuation in Chapter 10.
• Set up a riskless portfolio consisting of a position in
the derivative and a position in the stock.
• In the absence of profitable arbitrage, the portfolio
must earn the risk-free rate, r.
• This leads to the Black–Scholes–Merton differential
equation.
• An important difference is the length of time that
the portfolio remains riskless.
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12-20
Finance 457
Assumptions
1. The stock price follows a lognormal process with
m and s constant.
2. Short selling with full use of proceeds permitted.
3. No transactions costs or taxes.
4. No dividends during the life of the derivative.
5. No riskless arbitrage opportunities.
6. Security trading is continuous.
7. The risk-free rate, r, is constant and the same for
all maturities.
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12-21
Finance 457
12.6 Derivation of the Black-ScholesMerton Differential Equation
• The stock price process we are using:
dS  mSdt + sSdz
• Let f be the price of a call option or other derivative
contingent upon S. The variable f must be some
function of S and t. From Itô’s lemma
 f
f 1  2 f 2 2 
f
df   mS 

s S dt  sSdz
2
t 2 S
S
 S

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12-22
Finance 457
12.6 Derivation of the Black-ScholesMerton Differential Equation
• The appropriate portfolio is:
f
short one derivative and long
shares
S
• Define P as the value of the portfolio.
• By definition,
f
P f 
S
S
The change, dP, in the value of the portfolio in time dt
f
dP  df  dS
S
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12-23
Finance 457
12.6 Derivation of the Black-ScholesMerton Differential Equation
f
dP  df  dS
S
Substituting dS  mSdt + sSdz and
 f
f 1  2 f 2 2 
f
df   mS  
s S dt  sSdz
2
t 2 S
S
 S

2


f
1

f 2 2
yields dP   
 t  2 S 2 s S dt


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12-24
Finance 457
12.6 Derivation of the Black-ScholesMerton Differential Equation
 f 1  2 f 2 2 
dP    
s S dt
2
 t 2 S

Because dP does not involve dz, the portfolio must be
riskless during time dt.
The no-arbitrage condition is therefore:
dP  rPdt
Substituting from above yields:
 f 1  2 f 2 2 
f 

dP    
s S dt  r   f  S dt
2
S 

 t 2 S

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12-25
Finance 457
12.6 Derivation of the Black-ScholesMerton Differential Equation
 f 1  2 f 2 2 
f 

dP    
s S dt  r   f  S dt
2
S 

 t 2 S

It’s a short step to:
f
f 1 2 2  2 f
 rS
 s S
 rf
2
t
S 2
S
This is the Black–Scholes–Merton differential
equation. It has many solutions, corresponding to
the different derivatives that can be defined with S
as the underlying variable.
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12-26
Finance 457
12.6 Derivation of the Black-ScholesMerton Differential Equation
f
f 1 2 2  f
 rS
 s S
 rf
2
t
S 2
S
2
The particular solution that is obtained when the
equation is solved depends on the boundary
conditions that are used.
In the case of a European call, the key boundary
condition is:
f = max(S – K, 0) when t = T
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12-27
Finance 457
Example
• Consider a forward contract on a non-dividend
paying stock.
• From chapter 3 we have
f  S  Ke  r (T t )
f
1
S
f
 rKe  r (T t )
t
2 f
0
2
S
Clearly this satisfies the Black–Scholes–Merton
differential equation: f
f 1
2 f
t
McGraw-Hill/Irwin
 rS
 s S
 rf
2
S 2
S
2
2
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12-28
Finance 457
12.7 Risk Neutral Valuation
• Without a doubt, the single most important tool for
the analysis of derivatives.
• Note that the Black–Scholes–Merton differential
equation does not involve any variable that is
affected by the risk preferences of investors.
• The only variables are S0, T, s, and r.
• So any set of risk preferences can be used when
evaluating f. Let’s use risk neutrality.
• Now we can calculate the value of any derivative by
discounting its expected payoff at the risk-free rate.
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12-29
Finance 457
Risk Neutral Valuation of Forwards
• Consider a long forward contract that matures at time
T with delivery price K.
• The payoff at maturity is ST – K
• The value of the forward contract is the expected
value at time T in a risk-neutral world discounted at
the risk-free rate.
f = e–rT Ê(ST – K)
• Since K is constant, f = e–rT [Ê(ST) – K]
• In a risk-neutral world, m becomes r so Ê(ST) = S0 erT
• We have f = S0 – K e–rT which is the no-arbitrage
result we have from chapter 3.
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12-30
Finance 457
12.8 Black–Scholes Pricing Formulae
The Black-Scholes formulae for the price of a
European call and a put written on a non-dividend
paying stock are:
c  S 0  N(d1 )  Ke  rT  N(d 2 )
p  Ke  rT  N(d 2 )  S 0  N(d1 )
σ2
ln( S 0 / K )  (r  )T
2
d1 
s T
d 2  d1  s T
N(d) = Probability that a standardized, normally distributed,
random variable will be less than or equal to d.
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12-31
Finance 457
A Black–Scholes Example
Find the value of a six-month call option on the Microsoft with
an exercise price of $150
The current value of a share of Microsoft is $160
The interest rate available in the U.S. is r = 5%.
The option maturity is 6 months (half of a year).
The volatility of the underlying asset is 30% per annum.
Before we start, note that the intrinsic value of the option is
$10—our answer must be at least that amount.
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12-32
Finance 457
A Black–Scholes Example
Let’s try our hand at using the model. If you have a calculator
handy, follow along.
First calculate d1 and d2
ln( S / E )  (r  .5σ 2 )T
d1 
s T
ln( 160 / 150)  (.05  .5(0.30) 2 ).5
d1 
 0.5282
0.30 .5
Then,
d 2  d1  s T  0.52815  0.30 .5  0.31602
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12-33
Finance 457
A Black–Scholes Example
c  S  N(d1 )  Ke
d1  0.5282
d 2  0.31602
 rT
 N(d 2 )
N(d1) = N(0.52815) = 0.7013
N(d2) = N(0.31602) = 0.62401
c  $160  0.7013  150e .05.5  0.62401
c  $20.92
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12-34
Finance 457
Another Black–Scholes Example
Assume S = $50, K = $45, T = 6 months, r = 10%,
and s = 28%, calculate the value of a call and a put.
2


0
.
28
50
 0.50
ln
  0.10  0 
45 
2 
d1 
 0.884
0.28 0.50


d2  0.884  0.28 0.50  0.686
From a standard normal probability table, look up N(d1) =
0.812 and N(d2) = 0.754 (or use Excel’s “normsdist” function)
C  50e0(0.5) (0.812)  45e0.10(0.50) (0.754)  $8.32
P  $8.32  $50  $45e0.10(0.50)  $1.125
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12-35
Finance 457
12.8 Black–Scholes Pricing Formulae
To provide intuition, rewrite the Black-Scholes call
formula as:
ce
 rT
[ S 0 N(d1 )e  K N(d 2 )]
rT
N(d2) is the probability that the option will be
exercised in a risk-neutral world, so KN(d2) is the
expected value of the cost of exercise.
S0N(d1)erT is the expected value of a variable that
equals ST if ST > K and is zero otherwise in a riskneutral world.
The present value at the risk-free rate is the value of a
call
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12-36
Properties of the Black–Scholes Formulae
Finance 457
Consider what happens when ST becomes large.
The option is almost certain to finish in-the-money,
so the call becomes like a forward contract.
From chapter 3 we have
f = S0 – K e–rT
c  S 0  N(d1 )  Ke  rT  N(d 2 )
When S0 becomes large, d1 and d2 become large, so
N(d2) and N(d1) become close to 1
The Black-Scholes call price reduces to the futures
price.
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12-37
Properties of the Black–Scholes Formulae
Finance 457
Consider what happens when s approaches zero.
Because the stock is riskless, ST = S0 erT
At expiry, the payoff from the call will be
max(S0 erT – K, 0)
If we discount at r
c = e–rTmax(S0 erT – K, 0) = max(S0 – K e–rT, 0)
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12-38
Properties of the Black–Scholes Formulae
Finance 457
If S0 > K e–rT When s approaches zero, d1 and d2 tend
to +, so N(d2) and N(d1) become close to 1.
The Black–Scholes call price is then:
S0– K e–rT
If S0 < K e–rT When s approaches zero, d1 and d2 tend
to –, so N(d2) and N(d1) become close to zero.
The Black–Scholes call price is then
0
So, the Black–Scholes value of a call when s
approaches zero
c = max(S0 – K e–rT, 0)
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12-39
Finance 457
12.9 Cumulative Normal Distribution
Function
• NORMSDIST in Excel rocks.
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12-40
Finance 457
12.10 Warrants Issued by a company on
its own Stock
•
•
There is a dilution effect.
We can use the Black-Scholes formula for the
value of a call if:
1. The stock price S0 is replaced by S0 + (M/N)W
2. The volatility is the volatility of the equity (I.e. the
volatility of the shares plus the warrants, not just
the shares).
3. The formula is multiplied by Ng/(N + Mg)
Ng
M
W
[( S 0  W ) N(d1 )  Ke rT N(d 2 )]
N  Mg
N
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12-41
Finance 457
12.11 Implied Volatilities
• These are the volatilities that are implied by the
observed prices of options in the market.
• It is not possible to solve
c  S 0 N(d1 )  Ke  rT N(d 2 )
• For s
• In practice, use goal seek in Excel.
• It’s best to use near-the-money options to estimate
volatility.
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12-42
Finance 457
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
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12-43
Finance 457
12.12 The Causes of Volatility
• Trading itself can be said to be a cause.
• When implied volatilities are calculated, the life of
an option should be measured in trading days.
• Furthermore, if daily data are used to provide a
historical volatility estimate, day when the exchange
are closed should be ignored and the volatility per
annum should be calculated from the volatility per
trading day using this formula:
volatility per annum  volatilit y per tradin g day  number of trading days per annum
• The normal assumption is that there are 252 trading
days per year.
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Finance 457
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
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Finance 457
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
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12-46
Finance 457
12.13 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 anticipated.
• Elton and Gruber estimate this as 72% of the
dividend.
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Finance 457
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
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Finance 457
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 first European price is for an option
maturing at the same time as the American
option
2. The second European price is for an
option maturing just before the final exdividend date
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Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
12-49
Finance 457
Black’s Approach to Dealing with
Dividends in American Call Options
Set the American price equal to the maximum of two
European prices:
K1
ST
K1 + cAmerican
– c1
Buy a long-lived option strike K1 for c1
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12-50
Finance 457
12.14 Summary
• This chapter covers important material:
–
–
–
–
–
–
The lognormality of stock prices
The calculation of volatility from historical data
Risk-neutral valuation
The Black-Scholes option pricing formulas
Implied volatilities
The impact of dividends
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.