Past exam of the martingales and financial mathematics course of the University of Oxford / 2008 / B / 7

A self-financing replicating strategy must fund both the claim's price change and its outgoing cash flow:Apply Itô's formula to $V(t,S_t)$. Matching the diffusion gives $\Delta =V_s$. Matching drift and eliminating $\beta B=V-s{V}_s$ gives the Black-Scholes equation with claim dividendsThe physical drift cancels because stock risk is hedged. The cash-flow term has a plus sign because it is a payment to the holder.

Invest each cash payment $q_v{S}_{v}dv$ received at time $v$ in the bank account until $T$. Its accumulated terminal amount is $q_v{S}_v{e}^{r(T-v)}dv$, so the entire future stream is equivalent, path by path, to the terminal payment ${\int }_t^T{q}_v{S}_v{e}^{r(T-v)}dv$.

The risk-neutral pricing formula, including this reinvested payoff, iswhere $d{\tilde{S}}_v=r{\tilde{S}}_{v}dv+\sigma {\tilde{S}}_{v}d{W}_v^{\mathbb{Q}}$. Equivalently, discount each cash flow directly to time $t$. This is the discounted-gains martingale principle underlying the PDE in part (a).

Use the auxiliary diffusion $d{X}_v=\mu X_{v}dv+\sigma X_{v}d{W}_v$, started at $X_t=x$. Under sufficient regularity and integrability for the Feynman-Kac formula,To verify the signs, apply Itô's formula to $e^{-r(v-t)}F(v,X_v)$. Its drift is $e^{-r(v-t)}(F_t+\mu X_v{F}_x+{\sigma }^2{X}_v^2{F}_{xx}/2-rF)=-e^{-r(v-t)}d(v,X_v)$. Adding the accumulated source integral makes a martingale; taking expectations through $T$ proves the formula.

Here the auxiliary drift is the printed $\mu$, rather than automatically $r$: this part is a general terminal-value PDE, whose diffusion must match its generator.

Linearity separates the terminal call payoff from the running payment stream. The first contributes the supplied European call price. Under the risk-neutral measure, $\mathbb{E}(S_v\mid S_t=s)=s{e}^{r(v-t)}$, so the stream contributesConsequentlyThis is the claim dividends proportional to the underlying stock identity. The underlying itself is still non-dividend-paying; the cash flow belongs to the derivative.