Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2021 / 1 / a

The portfolio value is ${\Pi }_t={\varphi }_t{S}_t+{\psi }_t{B}_t$. The self-financing condition means that changes in holdings are financed by trading within the portfolio, with no external infusion or withdrawal. ThusThe holdings are predictable and satisfy the usual stochastic-integrability and admissibility conditions; without admissibility, doubling strategies would spoil the intended no-arbitrage interpretation.

For $V\in C^{1,2}$, Itô's formula givesMatch the Brownian coefficient with the self-financing portfolio by choosing the option delta $\varphi =V_s$ and bond value $\psi B=V-s{V}_s$. Equating drifts cancels the physical drift $\mu$ and givesThis is the pricing equation for a linear asset with additive noise. Its spatial domain is the whole real line, because the asset can be negative. In particular the diffusion term is ${\sigma }^2{V}_{ss}/2$, not ${\sigma }^2{s}^2{V}_{ss}/2$. The derivation is delta hedging and does not impose a boundary condition at $s=0$.