The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 465 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Continuous-State, Continuous-Time Models 465

To gain an intuition for the Radon-Nikodym derivative in a contin-

uous-state setting, let’s assume that the probability space is the real line

equipped with the Borel σ-algebra and with a probability measure P. In

this case, ξ= ξ(x), R →R and we can write

QA()= ∫ ξ Pd

A

or, dQ = ξdP. Given any random variable X with density f under P and

density q under Q, we can then write

E []= xq x()xd = xξ()xfx()xd

Q

X ∫ ∫

R R

In other words, the random variable ξis a function that multiplies the

density f to yield the density q.

We can now show the following key result. Given an equivalent

martingale measure with density process ξt a state-price deflator is given

by the process

t

∫–ruud

= ξ^0

πt te

Conversely, given a state-price deflator πt, the density process

t

∫ruud

ξ = e^0 ------πt

t

π 0

defines an equivalent martingale measure. In fact, suppose that Q is an

equivalent martingale measure for XY with πt = ξtYt where

t

∫–ruud

Y^0

t = e

Then, using the above relationship we can write:

E Y Y Y

t[πtXt] = Et[ξtXt ]= ξtE

Q[ξ

t tXt ]= ξtXt = πtXt