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

The discrete-time fundamental theorem of asset pricing supplies a strictly positive martingale deflator $Y$. Since the maturity-$T$ bond pays one unit at $T$, its deflated price is a martingale:Division by $Y_t>0$ gives the formula.

For the bond maturing one period later,If $r_t\ge 0$, this is at most $Y_{t-1}$, which is exactly the supermartingale property.

Buy one maturity-$(T-1)$ bond. When it pays one at $T-1$, use all proceeds to buy $1/P_{T-1}^T$ maturity-$T$ bonds. Short $1+f$ maturity-$T$ bonds. The terminal payoff isIts initial replication cost iswhich vanishes for $f=P_0^{T-1}/P_0^T-1$.

With $P_0^0=1$, the time-zero value of payment $r_t-s$ isSumming over $t$ telescopes, so the swap value isThe par swap rate is therefore