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

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.