Past exam of the martingales and financial mathematics course of the University of Oxford / 2011 / B / 6 / d / Solution

The diffusion associated with the displayed pricing equation is $d{\tilde{S}}_s=0.04{\tilde{S}}_{s}ds+0.2{\tilde{S}}_{s}d{W}_s$. For $Y_s=e^{-0.05s}V(s,{\tilde{S}}_s)$, Itô's formula givesThe drift vanishes by the PDE. Under the usual integrability conditions for the payoff and solution, localization and conditional expectation make $Y$ a true martingale; for polynomial-growth data the lognormal moment bounds provide the needed integrability. Applying the terminal condition yields the Feynman-Kac formulaThe printed diffusion contains an additional undefined factor $\sigma$ multiplying $0.2{\tilde{S}}_{s}d{W}_s$. If literally retained, its second-order generator coefficient would be $0.02{\sigma }^2{S}^2$, so the requested representation would correspond to that different PDE unless ${\sigma }^2=1$. The formula above uses the volatility $0.2$ specified by the stock model and by part (c).