Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2023 / 3 / d

Keep the maturity $T$ fixed and write $F(t,T)=\exp L(t,x_t)$, whereAs calendar time increases, ${\partial }_{t}a=\alpha a$, so $L_x=a$, $L_{xx}=0$ and $L_t=\alpha a(x-\tilde{\mu })-{\sigma }^2{a}^2/2$. Applying the Itô formula to $L$ and then to its exponential givesThe Itô formula's exponential correction cancels the displayed drift of $L$. The zero drift also follows from the tower property of conditional expectation: for $t\le u\le T$, ${\mathbb{E}}_t[F(u,T)]={\mathbb{E}}_t[S_T]=F(t,T)$. In particular $F(T,T)=S_T$, while the quote's instantaneous spot volatility approaches $\sigma$ as expiry approaches.

This is a martingale under the same measure used in the stipulated conditional expectation. To interpret it as futures pricing, that measure must be a suitable pricing measure. For a non-dividend-paying traded stock at a deterministic interest rate, the absence of arbitrage instead gives a forward contract delivery price $S_t{e}^{r(T-t)}$. A physical-measure expectation under the given mean-reverting drift is not automatically that price; the question's expectation convention is sufficient for the calculation above, but not for this additional identification.