The bond pays one at maturity. If $\tau=\inf\{t<T:P_t^T\leq0\}$ 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 $t<T$.

Solution

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

 Solution (source code)