Girsanov’s theorem is a fundamental result in the field of stochastic calculus that allows us to change the probability measure under which we model a stochastic process. The main purpose of Girsanov’s theorem is to convert a stochastic process governed by a certain probability measure into another process with a different probability measure.
In quantitative finance, Girsanov’s theorem is often used to change from the real-world probability measure (P) to the risk-neutral probability measure (Q), which is useful for pricing derivatives and other financial instruments. These two measures serve different purposes - the real-world measure represents the actual probability distribution of asset returns, while the risk-neutral measure adjusts the probabilities to eliminate any risk premium, simplifying the computation of derivative prices.
The theorem establishes that, under certain conditions, if Wt is a standard Brownian motion under the probability measure P, then there exists an equivalent probability measure, Q, such that
Wt* = Wt − ∫0tθsds,
is a Brownian motion under Q. Here, Wt* is the Radon-Nikodym derivative or the Girsanov transformation, and θt is a predictable, progressively measurable process that is square-integrable. This transformed Brownian motion is called the ’change of measure,’ and it simplifies the dynamic involving the drift term. Note that the volatility term remains the same under both measures.
For example, consider a Black-Scholes-Merton model for a stock price St under the real-world measure P:
dSt = μStdt + σStdWt,
where μ is the drift or the expected return, σ is the volatility, and Wt is the standard Brownian motion.
By applying Girsanov’s theorem, we can change the measure to the risk-neutral probability measure Q by choosing $\theta_t = \frac{\mu - r}{\sigma}$, where r is the risk-free interest rate. The new Brownian motion under the risk-neutral measure Q is given by
dWt* = dWt − θtdt.
Substituting Wt*, the pricing equation under the risk-neutral measure Q becomes
dSt = rStdt + σStdWt*.
Under measure Q, the expected return of the stock price is now equal to the risk-free interest rate, and the risk premium is eliminated.
In summary, Girsanov’s theorem is an essential tool in quantitative finance and stochastic calculus to change the probability measure under which a stochastic process is modeled. It helps in transitioning from real-world probabilities to risk-neutral probabilities, which simplifies the pricing of derivatives and other financial assets.