Risk-neutral valuation gives
$$
P_t^T
=\mathbb E^Q\left[
\prod_{s=t}^{T-1}(1+r_s)^{-1}
\,\middle|\,\mathcal F_t\right].
$$
For $s\geq t$,
$$
1+r_s=(1+r_t)\prod_{j=t}^{s-1}\zeta_j.
$$
Therefore
$$
\prod_{s=t}^{T-1}(1+r_s)^{-1}
=(1+r_t)^{-(T-t)}
\prod_{j=t}^{T-2}\zeta_j^{-(T-1-j)}.
$$
The future $\zeta_j$ are independent and identically distributed under the stated model, so
$$
\boxed{
P_t^T=(1+r_t)^{-(T-t)}
\prod_{k=1}^{T-t-1}M(-k),}
$$
with an empty product equal to one.


Solution

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

 Solution

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