Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 39 / 2 / ii

Suppose $Q_1,Q_2$ are equivalent martingale measures. Fix a finite maturity $T$ and an event $E\in {\mathcal{F}}_T$. Because the market is a complete market, replicate the claim $H=B_T{1}_E$, whose discounted payoff is bounded. For a self-financing strategy, the discounted value satisfiesUnder either pricing measure its conditional mean increment is zero, since $h_t^S$ is known at time $t$ and the discounted stock is a martingale. In the standard admissible discrete-time pricing class these finite-horizon values are integrable; a constant lower bound on discounted wealth also gives integrability successively from the conditional increment formula. Therefore replication and taking expectations giveThe replicating portfolio and its initial cost are the same for both measures, so $Q_1(E)=Q_2(E)$. Since $E$ was arbitrary, the measures agree on ${\mathcal{F}}_T$, for every finite $T$, and hence on the sigma-algebra generated by the market filtration. Thus $\boxed{\text{the equivalent martingale measure is unique}}$. This is the measure version of complete markets have unique state-price densities.