Frequently Asked Questions In Quantitative Finance

Chapter 4: Ten Different Ways to Derive Black–Scholes 255

Martingales

The martingale pricing methodology was formalized

by Harrison and Kreps (1979) and Harrison and Pliska

(1981).^1

We start again with

dSt=μSdt+σSdWt

TheWtis Brownian motion with measureP. Now intro-

duce a new equivalent martingale measureQsuch that

W ̃t=Wt+ηt,

whereη=(μ−r)/σ.

UnderQwe have

dSt=rS dt+σSdW ̃t.

Introduce

Gt=e−r(T−t)E

Q

t[max(ST−K,0)].

The quantityer(T−t)Gtis aQ-martingale and so

d

(

er(T−t)Gt

)

=αter(T−t)GtdW ̃t

for some processαt. Applying Ito’s lemma,ˆ

dGt=(r+αη)Gtdt+αGtdWt.

This stochastic differential equation can be rewritten

as one representing a strategy in which a quantity

αGt/σSof the stock and a quantity (G−αGt/σ)er(T−t)

(^1) If my notation changes, it is because I am using the notation
most common to a particular field. Even then the changes are
minor, often just a matter of whether one puts a subscriptton
adWfor example.