Set
$$
\lambda=\frac{\mu-r}{\sigma},
\qquad
\frac{dQ}{dP}
=\exp\left(-\lambda W_T-\frac12\lambda^2T\right).
$$
By the <Girsanov theorem>, $W_t^Q=W_t+\lambda t$ is Brownian motion under $Q$. Consequently
$$
dS_t=rS_t\,dt+\sigma S_t\,dW_t^Q,
$$
so discounted stock price is a martingale and $Q$ is the <Risk-neutral measure for the Black-Scholes model>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 211 / 4 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-211/4/b/solution