The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 462 Wednesday, February 4, 2004 1:08 PM

462 The Mathematics of Financial Modeling and Investment Management

σt

dZt = ------dBˆ t

Vt

Applying Itô’s lemma, given that ZtVt = St, we obtain the fundamen-

tal result:

dSt = rtdt + σtdBˆt

This result states that, under the equivalent martingale measure, all

price processes become Itô processes with the same drift.

Application of Girsanov’s Theorem to Black-Scholes

Option Pricing Formula

To illustrate Girsanov’s Theorem, let’s see how the Black-Scholes option

pricing formula can be obtained from an equivalent martingale mea-

sure. In the previous setting, let’s assume that N = 3, d = 1, rt is a con-

stant and

σt = σSt

with σ constant. Let S be the stock price process and C be the option

price process. The option’s price at time T is

C = max(S^1

T – K)

In this setting, therefore, the following three equations hold:

S

dSt = μt

S

dt + σStdBt

2 c

dCt = μt

c

dt + σtdBt

dVt = rVtdt

Given that CtVt –^1 is a martingale, we can write

2

Ct = VtEQt ------- = Et (

CT Q

[e –rT – t)max(ST – K)]

Vt