Girsanov's Theorem

In probability theory, Girsanov's Theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure which describes the probability that an underlying (such as a share price or interest rate) will take a particular value or values to the risk neutral measure which is a very useful tool for evaluating the value of derivatives on the underlying.

We state the theorem first for the special case when the stochastic process of interest is a Wiener process. This special case is sufficient for risk-neutral pricing in the Black-Scholes model.

Let be a Wiener process on the Wiener probability space . Let be a measurable process adapted to the natural filtration of the Wiener process , such that

for some . Further let Q be a probability measure on such that that Radon-Nikodym derivative

where SE is the stochastic exponential of x with respect to W, i.e. is the solution of the integral equation

then

is a Wiener process on the filter probability space

This theorem can be used to show in the Black-Scholes model the unique equilibrium price measure (or risk neutral measure) , i.e. the measure in which the fair value of a derivative is the discounted expected value, Q, is specified by