Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 5 / a / Solution

Use the Black-Scholes equation with the claim's terminal condition $V(T,s)=g(s)$:For $t<T$, the Itô formula under the physical measure givesChoose the delta hedge and its bank-account holding byTheir portfolio value is $V(t,S_t)$, andThus the portfolio is self-financing and, with initial capital $V(0,S_0)$, has terminal wealth $g(S_T)$.

Localizing to compact stock-price intervals and times below $T$ justifies these stochastic integrals from the classical smoothness of $V$. The extension to maturity also follows from boundedness. Under the Risk-neutral measure for the Black-Scholes model, discounted $V(t,S_t)$ is a bounded local martingale, hence a true square-integrable martingale. Its quadratic variation has finite expectation, so the stock-integral coefficient is square integrable through $T$. Equivalence of measures preserves its almost-sure finiteness, and the physical drift integral is finite by Cauchy-Schwarz. The terminal limit is $g(S_T)$.

Finally, $B_t=B_0{e}^{rt}$ is bounded away from zero on this finite horizon, and bounded $V$ makes $V(t,S_t)/B_t$ bounded below by a deterministic constant. The hedge is therefore an admissible trading strategy. The displayed holdings replicate the payoff admissibly, even if the bounded payoff can be negative.