With $P_0^0=1$, the time-zero value of payment $r_t-s$ is
$$
P_0^{t-1}-(1+s)P_0^t.
$$
Summing over $t$ telescopes, so the swap value is
$$
1-P_0^T-s\sum_{t=1}^TP_0^t.
$$
The par swap rate is therefore
$$
s=\frac{1-P_0^T}{\sum_{t=1}^TP_0^t}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 2 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-211/2/d/solution