Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 39 / 5 / i / Solution

Let $Q$ be the given risk-neutral measure, and write $B_t=B_0{e}^{rt}$. Take $V$ to be classical on $t<T$ and continuous at its prescribed terminal value. By Itô formula and the pricing equation for a local volatility model,soThe discounted added asset is a local martingale, as is $S_t/B_t$, while $B_t/B_t=1$. Consequently $Q$ is an equivalent local martingale measure for all three assets.

Here is the arbitrage argument, including the usual admissible trading strategy condition. If $X$ is the discounted wealth of a self-financing strategy, it is a stochastic integral against the discounted asset prices, plus its initial value, and hence a local martingale. Assume $X_t\ge -c$ for a fixed $c$, as required for an admissible trading strategy. For a localizing sequence ${\tau }_n$, the stopped process $X^{{\tau }_n}$ is a martingale. Apply the conditional Fatou lemma to $X_{t\wedge {\tau }_n}+c\ge 0$: for $s\le t$, continuity and the stopped martingale identity give ${\mathbb{E}}_Q[X_t+c\mid {\mathcal{F}}_s]\le X_s+c$. Thus $X$ is a supermartingale. An arbitrage would have $X_0=0$, $X_T\ge 0$ and positive probability of $X_T>0$. Equivalence of $P,Q$ makes that probability positive under $Q$, contradicting ${\mathbb{E}}_Q[X_T]\le 0$. Therefore the augmented market has no admissible arbitrage.

Nonnegativity of $V$ gives the needed price process, but does not by itself make its discounted local martingale a true martingale. If $V$ is bounded, the latter follows by bounded localization, and its price is also $B_t{\mathbb{E}}_Q[g(S_T)/B_T\mid {\mathcal{F}}_t]$. The arbitrage proof above applies to the stated nonnegative classical solution without assuming that additional bound.