Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2023 / 1 / a / Solution

Use a self-financing portfolio which holds $\Delta =V_S$ shares and $V-S{V}_S$ in the bank account, withdrawing the claim's cash flow $Cdt$. A share's total gain includes the continuous dividend yield, so its gain is $dS+DSdt$. Matching the claim's total gain givesBy the Itô formula,The delta hedge cancels the Brownian motion term. Comparing the remaining drifts yields the Black-Scholes equation with claim dividends, with the stock's own dividend yield included:This gain calculation accounts for rebalancing through the self-financing portfolio condition. The physical expected-return parameter $\mu$ cancels, as required by risk-neutral pricing.