The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 458 Wednesday, February 4, 2004 1:08 PM

458 The Mathematics of Financial Modeling and Investment Management

sure for the process X if X is a martingale with respect to Q and if the

Radon-Nikodym derivative

ξ = dQ---------

dP

has finite variance. The definition of the Radon-Nikodym derivative is

the same here as it is in the finite-state context. The Radon-Nikodym

derivative is a random variable ξ such that Q(A) = EP[ξIA] for every

event A where IA is the indicator function of the event A.

To develop an intuition for this definition, consider that any sto-

chastic process X is a time-dependent random variable Xt. The latter is

a family of functions Ω → R from the set of states to the real numbers

indexed with time such that the sets {Xt(ω) ≤ x} are events for any real

x. Given the probability measure P, the finite-dimension distributions of

the process X are determined. The equivalent measure Q determines

another set of finite-dimension distributions. However, the correspon-

dence between the process paths and the states remains unchanged.

The requirement that P and Q are equivalent is necessary to ensure

that the process is effectively the same under the two measures. There is

no assurance that given an arbitrary process an equivalent martingale

measure exists. Let’s assume that an equivalent martingale measure does

exist for the N-dimensional price process X = (X^1 ,...,XN). It can be dem-

onstrated that if the price process X = (X^1 ,...,XN) admits an equivalent

martingale measure then there is no arbitrage.

The proof is similar to that for state-price deflators as discussed

above. Under the equivalent martingale measure Q, which we assume

exists, every price process and every self-financing trading strategy

becomes a martingale. Using the same reasoning as above it is easy to

see that there is no arbitrage.

This result can be generalized; here is how. If there is a regular defla-

tor Y such that the deflated price process XY = (Y^1 N

tXt , ..., YtXt ) admits

an equivalent martingale measure, then there is no arbitrage. The proof

hinges on the result established in the previous section that, if there is a

regular deflator Y, the price process X admits no arbitrage if and only if

the deflated price process XY admits no arbitrage.

Note that none of these results is constructive. They only state that

the existence of an equivalent martingale measure with respect to a price

process ensures the absence of arbitrage. Conditions to ensure the exist-

ence of an equivalent martingale measure with respect to a price process

are given in the next section.