Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2023 / 2 / a / Solution

For the non-dividend-paying stock, a self-financing portfolio replicating a smooth European option has $\Delta =V_S$ shares and $V-S{V}_S$ in the bank account. Its gain satisfiesThe Itô formula gives $dV=(V_t+\mu S{V}_S+{\sigma }^2{S}^2{V}_{SS}/2)dt+\sigma S{V}_{S}dW$. Matching its diffusion and drift with the portfolio gains cancels $\mu$, yieldingThe financial payoff supplies the terminal value. This is the Black-Scholes equation; the associated risk-neutral measure replaces the physical stock drift by $r$.