Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 211 / 3 / a

The bond pays one at maturity. If $\tau =\inf \{t<T:P_t^T\le 0\}$ had positive probability of being finite, buy one bond at $\tau$. A negative price supplies immediate consumption and a positive terminal payoff; a zero price supplies a free positive terminal payoff. Trading only on the stopping event gives an arbitrage. Therefore $P_t^T>0$ almost surely for every $t<T$.