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

Under the physical probability, the stock has mean $\mathbb{E}{S}_t=50e^{0.1t}$. Nonnegativity permits interchanging expectation and integration in the accumulated dividends. HenceTherefore

Let the accumulated bank account from past dividends beHolding one stock share and banking its dividends is a self-financing portfolio with wealth $X_t=S_t+A_t$ and $d{A}_t=0.05A_{t}dt+0.01S_{t}dt$. Doubling that strategy replicates the specified terminal payoff. Its price is thereforeIn particular its initial price is one hundred pounds. This illustrates banking stock dividends versus reinvesting them. The accrued-dividend term is known at time $t$ and belongs to the payoff defined from time zero; it cannot be discarded when $t>0$.

For an ex-dividend stock, holding $\Delta$ shares earns both $\Delta dS$ and the dividend gain $\Delta \delta Sdt$. Freeze $\Delta =V_S$ locally in a self-financing hedge that is long the claim and short these shares. Itô's formula cancels its Brownian exposure, leaving drift $V_t+\frac{1}{2}{\sigma }^2{S}^2{V}_{SS}-\delta S{V}_S$. Its value is $V-S{V}_S$, so absence of arbitrage requires this drift to equal $r(V-S{V}_S)$. Rearrangement gives the Black-Scholes equation with continuous stock dividends:With the given parameters,The physical drift $0.1$ cancels because the hedge removes the stock-price risk; the dividend payment remains in the stock's total gain.

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).

Under the pricing measure, the ex-dividend stock drift is $r-\delta =0.04$. ThusA direct replication holds $2e^{-\delta (T-t)}$ shares and reinvests every dividend into more shares. The share count grows at rate $\delta$, reaching two shares at expiry, with no external cash input. This payoff contains only the terminal ex-dividend stock value, unlike part (b), which also includes the separately accumulated dividends.