The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 464 Wednesday, February 4, 2004 1:08 PM

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464 The Mathematics of Financial Modeling and Investment Management

EQUIVALENT MARTINGALE MEASURES AND STATE PRICES

We will now show that equivalent martingale measures and state prices

are the same concept. We use the same setting as in the previous sec-

tions. Suppose that Q is an equivalent martingale measure after defla-

tion by the process

t

• r ud

1 ∫ 0 u

-------= e

1

Vt

where r is a bounded short-rate process. The density process ξt for Q is

defined as

ξt = Er--------- , t ∈[0,T]

dQ

dP

where

dQ

dP

is the Radon-Nikodym derivative of Q with respect to P. As in the dis-

crete-state setting, the Radon-Nikodym derivative of Q with respect to

P is a random variable

ξ= dQ---------

dP

with average value on the entire space equal to 1 and such that, for

every event A, the probability of A under Q is the average of ξ:

P ()= EA ξ

Q

A []

It can be demonstrated that, given any ℑt-measurable random vari-

able W, the density process ξt for Q has the following property:

EQW Et[Wξt]

t []= ---------------------

ξt