Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 211 / 1 / b / Solution

A numéraire portfolio $\eta$ satisfies $\eta \cdot P_0>0$ and $\eta \cdot P_1>0$ almost surely. If an arbitrage $H$ already has zero initial cost, it is a terminal-consumption arbitrage. Otherwise $H\cdot P_0<0$; setThen $\tilde{H}\cdot P_0=0$, while its terminal payoff is the nonnegative payoff of $H$ plus a strictly positive multiple of $\eta \cdot P_1$. Hence it is strictly positive almost surely and is a terminal-consumption arbitrage.