Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2021 / 2 / b / Solution

Buy $1/F_0^T$ shares and take a short forward contract for $1/F_0^T$ shares at delivery price $F_0^T$. The forward has zero initial value, and the combined terminal payoff isTherefore the synthetic zero-coupon bond has initial priceIts implied continuously compounded maturity yield is $r_T=T^{-1}\log (F_0^T/S_0)>0$.

To replicate a constant-rate risk-free asset throughout the interval, the market must support dynamically coherent discounting, not just this one terminal payment. A sufficient condition is that for all $0\le t\le u\le T$ the fair forward prices satisfy $F_t^u=S_t{e}^{r(u-t)}$ for the same deterministic $r>0$, with the required stock/forward positions tradable and no intervening payments. Then $P(t,u)=S_t/F_t^u=e^{-r(u-t)}$, and a suitably scaled fixed-maturity zero-coupon bond grows deterministically as $e^{rt}$. A single initial forward quote does not establish these intermediate-time conditions; even a flat initial yield curve need not imply deterministic future rates.