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

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.

For claim dividends proportional to the underlying stock, put $V(S,t)=S\varphi (t)$. Then $V_S=\varphi$, $V_{SS}=0$ and $V_t=S{\varphi }^{\prime }$, so the partial differential equation reduces to the linear ordinary differential equationThe integrating factor $e^{-Dt}$ gives $(e^{-Dt}\varphi {)}^{\prime }=-e^{-Dt}f(t)$. Integrating backwards from the terminal value,The formula assumes the displayed integral is finite. Under the risk-neutral measure, ${\mathbb{E}}_t[S_u]=S{e}^{(r-D)(u-t)}$. Discounting each cash payment at its own payment time gives the same result:The interest rate disappears because every payment is proportional to the stock price.

Setting $f=0$ in b givesThis is a prepaid forward contract: the holder pays now and receives one share at $T$, with no interim dividend payments. Buying $e^{-D(T-t)}$ shares now and reinvesting their continuous dividend yield grows the shareholding to one at expiry. Thus the displayed initial cost replicates the terminal delivery. When $D=0$, holding one share already does so and $V=S$.