Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 211 / 3

The cash-discounted stock is a positive continuous local martingale, because its dynamics contain no drift. Applying Itô formula to ${\pi }_t=U(t,v_t,S_t)$, the displayed partial differential equation cancels its drift exactly, leaving another local martingale. Since $U$ is bounded, $\pi$ is in fact a true martingale.

Thus the physical measure itself is an equivalent local martingale measure relative to cash for all three traded assets. The continuous-time fundamental theorem of asset pricing rules out arbitrage, more precisely no free lunch with vanishing risk, in the usual admissible class.

The bounded local martingale $\pi$ is a true martingale. Its terminal condition is ${\pi }_T=U(T,v_T,S_T)=g(S_T)$, soThis is also the Feynman-Kac formula for the displayed backward equation.

The explicit stochastic exponential solutions satisfyConditionally on the path generated by $W$, the last stochastic integral is a centered Gaussian random variable with variance $Y_T$, because $W^{\perp }$ is independent of $W$. The conditional expectation of $g(S_T)$ is thereforeTaking expectations and using part b proves the result.

Integrating the variance equation givesThe Doléans-Dade exponential solution of $d{\tilde{S}}_t=\rho {\tilde{S}}_t\sqrt{v_t}d{W}_t$ isSubstitution yields

The pair $(v_t,Y_t)$ is a Markov diffusion with infinitesimal generatorPart d shows that the terminal condition in the equation for $\tilde{U}$ is exactlywhen evaluated at $(v_T,Y_T)$. The Feynman-Kac formula applied to the displayed backward equation therefore givesPart c identifies the right side with ${\pi }_0$.