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Copyright 2001 by Harcourt, Inc. All rights reserved. 1

Chapter 5: Option Pricing Models:

The Black-Scholes Model

[O]nce a model has been developed, we are able to improve

the realism of its assumptions step by step. But unlike

physics, which is a science with constant (if poorly

understood) laws, the laws of economics and finance

change constantly, even as we discover them. Sometimes

they change because we have discovered them.

Charles Sanford

The Risk Manegement Revolution

Proceedings of Symposia in Pure Mathematics, 1997, p. 325


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Important Concepts in Chapter 4

The Black-Scholes option pricing model

The relationship of the models inputs to the option price

How to adjust the model to accommodate dividends andput options

How historical and implied volatility are obtained

Hedging an option position


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Origins of the Black-Scholes Formula

Brownian motion and the works of Einstein, Bachelier,

Wiener, It

Black, Scholes, Merton and the 1997 Nobel Prize


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The Black-Scholes Model as the Limit of the

Binomial Model Recall the binomial model and the notion of a dynamic

risk-free hedge in which no arbitrage opportunities are

available.

Consider the AOL June 125 call option. Figure 5.1, p.

154 shows the model price for an increasing number of

time steps.

The binomial model is in discrete time. As you decrease

the length of each time step, it converges to continuoustime.


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The Assumptions of the Model

Stock Prices Behave Randomly and Evolve According to aLognormal Distribution.

See Figure 5.2a, p. 156, 5.2b, p. 157 and 5.3, p. 158

for a look at the notion of randomness.

A lognormal distribution means that the log(continuously compounded) return is normally

distributed. See Figure 5.4, p. 159.

The Risk-Free Rate and Volatility of the Log Return on the

Stock are Constant Throughout the Options Life There Are No Taxes or Transaction Costs

The Stock Pays

No Dividends

The Options are European


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A Nobel Formula

The Black-Scholes model gives the correct formula fora European call under these assumptions.

The model is derived with complex mathematics but is

easily understandable. The formula is

Tdd

T

/2)T(r/X)ln(S

d

where

)N(dXe)N(dSC

12

2

c0

1

2

Tr

10c


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A Nobel Formula (continued)

where

N(d1), N(d2) = cumulative normal probability

s = annualized standard deviation (volatility) of thecontinuously compounded return on the stock

rc = continuously compounded risk-free rate


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A Digression on Using the Normal Distribution

The familiar normal, bell-shaped curve (Figure 5.5, p.

161)

See Table 5.1, p. 162 for determining the normal

probability for d1 and d2. This gives you N(d1) and

N(d2).


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The Black-Scholes Option Pricing Model

(continued) A Numerical Example

Price the AOL June 125 call

S0 = 125.9375, X = 125, rc = ln(1.0456) = .0446, T =.0959, s = .83.

See Table 5.2, p. 163 for calculations. C = $13.21.

Familiarize yourself with the accompanying software

Excel: bsbin2.xls. See Software Demonstration5.1, p. 164. Note the use of Excels =normsdist()

function.

Windows: bsbwin2.1.exe. SeeAppendix 5B.


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Characteristics of the Black-Scholes Formula

Interpretation of the Formula

The concept of risk neutrality, risk neutral

probability, and its role in pricing optionsThe option price is the discounted expected payoff,

Max(0,ST - X). We need the expected value of ST -

X for those cases where ST > X.


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Characteristics of the Black-Scholes Formula (continued)

Interpretation of the Formula (continued)

The first term of the formula is the expected value of

the stock price given that it exceeds the exerciseprice times the probability of the stock price

exceeding the exercise price, discounted to the

present.

The second term is the expected value of thepayment of the exercise price at expiration.


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The Black-Scholes Formula and the Lower Bound of a

European Call

Recall that the lower bound would be

The Black-Scholes formula always exceeds this

value as seen by letting S0 be very high and then letit approach zero.

)XeSMax(0,Tr

0c


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The Formula When T = 0

At expiration, the formula must converge to the

intrinsic value. It does but requires taking limits since otherwise

would be division by zero.

Must consider the separate cases of ST X and ST

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