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
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