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

Let $B_t=e^{rt}$ and let ${\psi }_t$ be the bank-account holding. Then $X_t={\psi }_t{B}_t+{\theta }_t{S}_t$. The self-financing portfolio condition gives

SetBy the Girsanov theorem, $W_t^Q=W_t+\lambda t$ is Brownian motion under $Q$. Consequentlyso discounted stock price is a martingale and $Q$ is the Risk-neutral measure for the Black-Scholes model.

Define the risk-neutral claim valueThen $V(0,S_0)=x$, $V(T,s)=g(s)$, and the Black-Scholes equation holds. Differentiation under the integral gives the deltaA Gaussian shift $Z\mapsto Z+\sigma \sqrt{\tau }$ rewrites this aswhich is exactly the stated ${\theta }_t$ at $s=S_t$.

Apply Itô formula to $V(t,S_t)$. The PDE givesThis is the same wealth equation as part a, with the same initial value $x$. Uniqueness therefore gives $X_t^{x,\theta }=V(t,S_t)$ and hence