Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 6

An $x$-admissible strategy is a predictable vector of holdings $\varphi =({\varphi }^0,\dots ,{\varphi }^d)$ that is integrable against the asset prices, is self-financing, has initial wealthand whose wealth process $V_t={\sum }_i{\varphi }_t^i{S}_t^i$ remains nonnegative.

An arbitrage is a zero-initial-wealth admissible self-financing strategy with $V_T\ge 0$ almost surely and $\mathbb{P}(V_T>0)>0$ for some finite horizon $T$.

Write ${\mu }_t=({\mu }_t^{(1)},\dots ,{\mu }_t^{(d)}{)}^T$ and define the market price of riskIt is bounded by hypothesis. The stochastic exponentialis a true martingale by the Novikov condition. Define the equivalent measure $Q$ by $dQ=Z_{T}dP$. The Girsanov theorem makesa Brownian motion under $Q$. After discounting by the bank account, every risky price has zero drift and is a $Q$-local martingale.

The discounted wealth of an admissible self-financing strategy is a nonnegative local martingale and hence a supermartingale. If an arbitrage existed, its zero initial value would imply nonpositive expected terminal discounted wealth under $Q$, while that wealth is nonnegative and positive with positive $Q$-probability. This contradiction proves that the market has no arbitrage; it is the needed direction of the equivalent local martingale measure criterion.

Let $X$ be a positive strict local martingale solvingand fix $T>0$. Use the bank account $S^0=1$ and two risky assetsBoth discounted prices are nonnegative local martingales under the physical measure itself, so the same supermartingale argument as in part c rules out arbitrage. At maturity,But strictness means $\mathbb{E}[X_T\mid {\mathcal{F}}_t]<X_t$ for some earlier $t$ on a set of positive probability, so the two prices are not indistinguishable before $T$. This no-arbitrage market violates the Law of One Price.