Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 29K / d

Suppose an arbitrage existed. Its discounted terminal wealth would satisfy $V_T\ge 0$ $\mathbb{P}$-almost surely and $\mathbb{P}(V_T>0)>0$. Equivalence of $\mathbb{Q}$ and $\mathbb{P}$ gives the same two statements under $\mathbb{Q}$, soBut part (c), after the usual admissibility or localization needed for integrability, givesa contradiction. Hence the existence of the stated martingale measure rules out arbitrage.