Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2017 / 1 / b

Use a self-financing portfolio with ${\Delta }_t$ units of the traded asset and the remainder in the continuous-time bank account. If its value equals $V(S_t,t)$, its self-financing condition isMeanwhile Itô's formula givesMatching the Brownian motion coefficients gives the delta hedge ${\Delta }_t=V_S$. Matching drifts then cancels the physical drift $\alpha$, leaving the pricing equation for a linear asset with additive noise:The domain is the full real line: additive noise can take the asset through zero. The diffusion coefficient is $\sigma$, so the second-derivative term has no factor $S^2$. As usual, the replication argument assumes suitable differentiability and an admissible trading strategy.