mcgraw-hill/irwin copyright © 2002 by the mcgraw-hill companies, inc. all rights reserved. 12-0...

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

12-1

Finance 457

12

Chapter Twelve

The Black–Scholes Model


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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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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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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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• This is fully developed in chapter 11.• Assume that percentage changes in stock price in a

short period of time are normally distributed.• Let:

: expected return on the stock

: volatility on the stock • The mean of the percentage change in time t is t • The standard deviation of this percentage change is

tδσ


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Finance 457


• The percentage changes in stock price in a short period of time are normally distributed:

)σ,δ(δ

δttμS

S

• Where S is the change in stock price in time t, and (m,s) denotes a normal distribution with mean m and standard deviation s.


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Finance 457


• As shown in section 11.7, the model implies that

TTμSST σ,

2lnln

2

0

• From this it follows that

TTμ

S

ST σ,2

ln2

0

• and

TTμSST σ,

2lnln

2

0


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Finance 457


The above equation shows that ln ST is normally distributed.

This means that ST has a lognormal distribution.

A variable with this distribution can take any value between zero and infinity.

TTμSST σ,

2lnln

2

0


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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.

μTT eSSE 0)( )1()var(

2220 TμT

T eeSS


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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

• It follows that TT eSS

0

so that0

ln1

S

S

TT


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Finance 457

12.2 The Distribution of the Rate of Return

It follows from:

that

TTμ

S

ST σ,2

ln2

0

Tμ

σ,

2

2

Thus, the continuously compounded rate of return per annum in normally distributed

2

2μ and standard deviation

T

σwith mean


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12.3 The Expected Return

• The expected return, , required by investors from a stock depends on the riskiness of the stock.

• The higher the risk, the higher ceteris paribus.• also depends on interest rates in the economy• We could spend a lot of time on the determinants of

, 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 at all.

• There is however, one aspect of that frequently causes confusion and is worth explaining.


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A subtle but important difference

• Shows that t is the expected percentage change in the stock price in a very short period of time t .

• This means that is the expected return in a very short short period of time t .

• It is tempting to assume that is also the continuously compounded return on the stock over a relatively long period of time.

• However, this is not the case.

)σ,δ(δ

δttμS

S


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Finance 457


• The continuously compounded return on the stock over T years is:

0

ln1

S

S

TT

Equation (12.7):

Tμ

σ,

2

2

Shows that the expected value of this is 2

2μ

The distinction between and

is subtle but important.2

2μ


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Finance 457


Start with

Taking logarithms:

μTT eSSE 0)(

μTSSE T )ln()](ln[ 0

Since ln is a nonlinear function, )][ln()](ln[ TT SESE

So we cannot say μTS

SE T )][ln(

0

In fact, we have: μTS

SE T )][ln(

0So the expected return over the whole period T,

expressed with compounding t, is close to

Not 2

2μ


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• 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?

• It can refer to 2

2μ

• The above shows that a simple term like expected return is ambiguous.

or

• Unless otherwise stated will be expected return.


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12.4 Volatility

• The volatility of a stock, , is a measure of our uncertainty about the returns.

• Stocks typically have a volatility between 20% and 50%

Tμ

σ,

2

2• From

• 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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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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Estimating Volatility from Historical Data (page 239-41)

1. Take observations S0, S1, . . . , Sn at intervals of years

2. 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:

uS

Sii

i

ln1

sˆ


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12.5 Concepts Underlying the Black-Scholes-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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Assumptions

1. The stock price follows a lognormal process with and 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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12.6 Derivation of the Black-Scholes-Merton Differential Equation• The stock price process we are using:

dSSdt + Sdz• 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

SdzS

fdtS

S

f

t

fS

S

fdf

222

2

2

1


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Finance 457

12.6 Derivation of the Black-Scholes-Merton Differential Equation• The appropriate portfolio is:

S

f

short one derivative and long shares

• Define as the value of the portfolio. • By definition,

SS

ff

The change, in the value of the portfolio in time t

SS

ff


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Finance 457


SS

ff

Substituting SSt + Sz and

zSS

ftS

S

f

t

fS

S

ff

222

2

2

1

yields tSS

f

t

f

222

2

2

1


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Finance 457


Because does not involve z, the portfolio must be riskless during time t.

The no-arbitrage condition is therefore:

rt

Substituting from above yields:

tSS

f

t

f

222

2

2

1

tSS

ffrtS

S

f

t

f

222

2

2

1


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Finance 457


It’s a short step to:

tSS

ffrtS

S

f

t

f

222

2

2

1

rfS

fS

S

frS

t

f

2

222

2

1

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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Finance 457


rfS

fS

S

frS

t

f

2

222

2

1

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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Finance 457

Example

• Consider a forward contract on a non-dividend paying stock.

• From chapter 3 we have

)( tTrKeSf

1S

f )( tTrrKet

f

02

2

S

f

rfS

fS

S

frS

t

f

2

222

2

1

Clearly this satisfies the Black–Scholes–Merton differential equation:


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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, , 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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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, 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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Finance 457

12.8 Black–Scholes Pricing FormulaeThe Black-Scholes formulae for the price of a

European call and a put written on a non-dividend paying stock are:

)N()N( 210 dKedSc rT

T

Tσ

rKSd

)2

()/ln(2

0

1

Tdd 12

N(d) = Probability that a standardized, normally distributed, random variable will be less than or equal to d.

)N()N( 102 dSdKep rT


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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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Finance 457


Let’s try our hand at using the model. If you have a calculator handy, follow along.

Then,

T

TσrESd

)5.()/ln( 2

1

First calculate d1 and d2

31602.05.30.052815.012 Tdd

5282.05.30.0

5).)30.0(5.05(.)150/160ln( 2

1

d


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Finance 457


N(d1) = N(0.52815) = 0.7013

N(d2) = N(0.31602) = 0.62401

5282.01 d

31602.02 d

)N()N( 21 dKedSc rT

92.20$

62401.01507013.0160$ 5.05.

c

ec


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Finance 457

Assume S = $50, K = $45, T = 6 months, r = 10%, and = 28%, calculate the value of a call and a put.

125.1$45$50$32.8$ )50.0(10.0 eP

32.8$)754.0(45)812.0(50 )50.0(10.0)5.0(0 eeC

884.0

50.028.0

50.0228.0

010.04550ln

2

1

d

686.050.028.0884.02 d

From a standard normal probability table, look up N(d1) = 0.812 and N(d2) = 0.754 (or use Excel’s “normsdist” function)

Another Black–Scholes Example


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Finance 457

12.8 Black–Scholes Pricing FormulaeTo provide intuition, rewrite the Black-Scholes call

formula as:

)]N()N([ 210 dKedSec rTrT

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 risk-neutral world.

The present value at the risk-free rate is the value of a call


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Finance 457

Properties of the Black–Scholes Formulae

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

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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Finance 457


Consider what happens when 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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If S0 > K e–rT When 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 approaches zero, d1 and d2 tend to –, so N(d2) and N(d1) become close to zero.The Black–Scholes call price is then

0So, the Black–Scholes value of a call when

approaches zero

c = max(S0 – K e–rT, 0)


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12.9 Cumulative Normal Distribution Function• NORMSDIST in Excel rocks.


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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 N/(N + M)

)]N()N()[( 210 dKedWN

MS

MN

NW rT


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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

• For • In practice, use goal seek in Excel. • It’s best to use near-the-money options to estimate

volatility.



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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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12.12 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:

• The normal assumption is that there are 252 trading days per year.

annumper days tradingofnumber day gper tradiny volatilit annumper volatility


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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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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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12.13 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 withDividends 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 ex-dividend date


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Finance 457

Black’s Approach to Dealing withDividends in American Call Options

K1

Buy a long-lived option strike K1 for c1

– c1

ST

Set the American price equal to the maximum of two European prices:

K1 + cAmerican


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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

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