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

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.