Girsanov's theorem
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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 instrument (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 <math>\{W_t\}<math> be a Wiener process on the Wiener probability space <math>\{\Omega,F,P\}<math>. Let <math>x_t<math> be a measurable process adapted to the natural filtration of the Wiener process <math>\{F^W_t\}<math>, such that
- <math> E_P[\exp(\lambda\int_0^T x_s^2 ds)] < \infty <math>
for some <math>\lambda> 1<math>. Further let Q be a probability measure on <math>\{\sigma,F\}<math> such that that Radon-Nikodym derivative <math>\frac{\delta Q}{\delta P} = SE(x \bullet W)<math>
where SE is the stochastic exponential of x with respect to W, i.e. <math>SE(x\bullet W)<math> is the solution of the integral equation
- <math> SE = 1 + \int_0^t SE_t d[x\bullet W]_t <math>
then
- <math> W_t - \int_0^t \frac{1}{z_s} d
is a Wiener process on the filter probability space <math>\{\Omega,F,Q,\{F^W_t\}\}<math>
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
- <math> \frac{\delta Q}{\delta P} = SE( -\frac{\mu-r}{\sigma}\bullet W )<math>