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

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