The one-period <spot interest rate> is defined by
$$
1+r_t=\frac1{P_t^{t+1}},
$$
and the <bank account> by
$$
B_0=1,
\qquad
B_t=\prod_{s=0}^{t-1}(1+r_s).
$$
A probability measure $Q$ equivalent to the physical measure is a <risk-neutral measure> when every discounted <zero-coupon bond> price
$$
\frac{P_t^T}{B_t},\qquad0\leq t\leq T,
$$
is a $Q$-martingale. Equivalently,
$$
P_t^T=B_t\mathbb E^Q[B_T^{-1}\mid\mathcal F_t].
$$

Solution

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

The time-$t$ one-period spot interest rate is the rate available for investment from $t$ to $t+1$. For a unit-face-value <zero-coupon bond>,
$$
1+r_t=\frac1{P_t^{t+1}}.
$$

Spot interest rate

{wiki}

A zero-coupon bond makes one payment at maturity and no earlier coupon payments. A unit-face-value bond maturing at $T$ has terminal value $P_T^T=1$.


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