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

At maturity, $P_T^T=1$. If $P_t^T\le 0$ 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

The one-period spot interest rate is defined byand the bank account byA probability measure $Q$ equivalent to the physical measure is a risk-neutral measure when every discounted zero-coupon bond priceis a $Q$-martingale. Equivalently,

If $T\mapsto P_t^T$ is nonincreasing, thenso $1+r_t=(P_t^{t+1}{)}^{-1}\ge 1$ and $r_t\ge 0$.

Conversely, if every spot rate is nonnegative, then $B_{T+1}=B_T(1+r_T)\ge B_T$. Under a risk-neutral measure,Hence

Risk-neutral valuation givesFor $s\ge t$,ThereforeThe future ${\zeta }_j$ are independent and identically distributed under the stated model, sowith an empty product equal to one.