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