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Document Date: November 2, 2006

An Introduction To Derivatives And Risk Management, 7th

Edition

Don Chance and Robert Brooks

Technical Note: Probability of Call Expiring in-the-Money, Ch. 5, p. 138

This technical note supports the material in the Characteristics of the Black-

Scholes-Merton section of Chapter 5 Option Pricing Models: The Black-Scholes-Merton

Model. This note shows that under the assumption that investors are risk neutral the

probability of the call expiring in-the-money is N(d2). If investors are not risk neutral, an

adjustment to the expected return is required, as explained later.

The standard model for capturing movements in the stock price is called a

lognormal diffusion or geometric Brownian motion process:

0 0 0 tdS S dt S dz = +

Where S0 is the current stock price, dt is an infinitesimally short period of time, is the

expected return on the stock, is the volatility, and dzt is a random variable that captures

the uncertainty in the stock price. The variable dzt is driven by a standard normal

variable, t, such that t tdz dt = . Of course, t has an expected value of zero and a

variance of 1. Given that t is normally distributed and is the source of all of the

uncertainty, we should be able to use normal probability theory to derive the probability

of the option expiring in-the-money. We shall need to express the stochastic process in

such a manner that the return on the asset is normally distributed. In Geometric

Brownian Motion the log return on the asset is normally distributed, so we will need the

stochastic process for the log of the asset return.

Define dS0 + S0 as the asset price at an instant, dt, later. Thus, we can write the

stochastic process as dS0 + S0 = S0[1 + dt + dz]. Working with the term in brackets,

note that we can write it in the following, seemingly complex, way:

( ) ( )( )2

2 21 1 2 2t tdt dz dt dz dt dz + + = + + + +

/ 2t .

By multiplying out the terms on the right hand side, it is easy to verify that the above

statement is true. Now define = - 2/2 and the above can be written as 1 + [dt +

dzt + (dt + dzt)2/2]. The term in brackets is equivalent to a second-order Taylor

series expansion of the function . A second-order expansion is sufficient, becausetdt dz

e +

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all terms higher than second order will involve powers of dt greater than 1.0 and by

definition, these terms approach zero in the limit.

Now we can write out the stochastic process as . Dividing by

S

0 0 0tdt dz dS S S e

++ =

0 we obtain . Taking natural logs, we have the stochastic process of

the log return on the asset,

0 0/ 1tdt dz

dS S e

+

+ =

0 0

0

ln tdS S

dt dz S

+

= +

.

This result confirms that the log return is normally distributed with mean and volatility

. For our purposes here, we use the following version,

0 0 0 .tdt dz dS S S e

++ =

The point of this Technical Note is to determine the probability that the option

will expire in-the-money. Noting that the time increment until expiration is T, we have

the asset price at expiration as ST and the stochastic process for z as*

Tz T = . Thus,

*

0

T T

TS S e +=

with zT normally distributed with mean zero and variance T, per the central limit

theorem.

We want to know

( ) ( )Pr .T TX

ob S X f S dS

> = T

Let us first evaluate the second term on the right-hand side. By definition,

[ ]*

0Pr Pr T T

Tob S X ob S e X + > = > .

Note that*

0

T TS e X + > is equivalent to ( ) *0ln S X T T + + or

( )0* ln S X T

T

+ > .

Recall that is the expected simple return on the asset and is the expected logarithmic

return on the asset where = - 2/2. Under the equivalent martingale/risk neutrality

approach, we can let = r so that = r - 2/2 so that

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( ) ( )20* ln 2S X r T T

+ > .

You should recognize this as * > -d2 or*

< d2. So we have

[ ] ( )2Pr

T

ob S X N d > = .

Of course, this expression is one of the terms in the Black-Scholes model. To

interpret N(d2) as the probability that the option expires in-the-money, however, requires

that we assume investors are risk neutral or that we assume the expected return on the

stock is the risk-free rate. These two assumptions are permissible within the Black-

Scholes model. They do not violate any accepted norms of economic theory, and they

guarantee the correct arbitrage-free price. But if what we want is the true probability of

exercise, we must replace the risk-free rate with the expected return on the stock.

References

Hull, J. Options, Futures and Other Derivatives, 6th

ed. Upper Saddle River, New

Jersey: Prentice-Hall (2006), Ch. 13.

Jarrow, R. and A. Rudd. Option Pricing. Homewood, Illinois: R. D. Irwin (1983), Ch. 7.

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