At maturity, $P_T^T=1$. If $P_t^T\leq0$ on an event $A\in\mathcal F_t$ of positive probability, buying one bond on $A$ has nonpositive cost and certain payoff $1$ on $A$ at $T$; any negative purchase cost can also be consumed or retained. This is an <arbitrage>. Therefore absence of arbitrage implies
$$
\boxed{P_t^T>0\quad\text{almost surely}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 211 / 5 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-211/5/a/solution