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