Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 4 / a / Solution

On a fixed horizon $T$, an arbitrage is an admissible trading strategy that is self-financing, has $X_0=0$, and satisfies $X_T\ge 0$ almost surely with $\mathbb{P}(X_T>0)>0$. Here admissibility means that wealth in units of the strictly positive riskless numéraire has a deterministic lower bound throughout the horizon. An equivalent martingale measure $Q$ is equivalent to the physical measure on ${\mathcal{F}}_T$ and makes each discounted risky price ${\tilde{S}}^i=S^i/B$ a martingale.

For a predictable, stochastically integrable risky holding vector $\pi$, the discounted self-financing identity isIndeed, since the riskless account is a finite-variation process, its quadratic covariations vanish, and the Itô formula for $X/B$, combined with $X=\varphi B+\pi \cdot S$ and $dX=\varphi dB+\pi \cdot dS$, gives $d(X/B)=\pi \cdot d(S/B)$.

Use two standard facts from stochastic calculus: a stochastic integral against a continuous local martingale is a local martingale, and a local martingale bounded below by a deterministic constant is a supermartingale. The latter follows by adding the constant to make the process nonnegative, stopping at a localizing sequence, and applying the conditional Fatou lemma. Thus admissible discounted wealth is a $Q$-supermartingale, and an arbitrage would satisfyIt follows that $X_T=0$ $Q$-almost surely, hence also physically almost surely by equivalence, a contradiction. Therefore an equivalent martingale measure rules out arbitrage among admissible self-financing strategies. The lower-bound restriction is essential to this argument; stochastic integrals need not be true martingales for arbitrary unbounded holdings.