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

For $x_t=\log S_t$, the Itô formula uses $(\log S{)}^{\prime }=1/S$ and $(\log S{)}^{\prime \prime }=-1/S^2$. ConsequentlyThe quadratic variation term shifts the mean level. Thus the log-mean-reverting price has logarithm following the Ornstein-Uhlenbeck processIt is the logarithm, rather than the price itself, which reverts linearly to a constant mean level.

Apply the Itô product rule with deterministic integrating factor $e^{\alpha u}$:Integrate from $t$ to $T$ and multiply by $e^{-\alpha T}$ to obtainThis is the explicit solution behind the Ornstein-Uhlenbeck transition distribution.

Write $\tau =T-t$ and $a=e^{-\alpha \tau }$. The stochastic integral in b has a deterministic integrand, so it is a Gaussian random variable independent of the information available at $t$. Its variance, by the Itô isometry, isThus $x_T$ conditionally has a normal distribution with expected value $m=a{x}_t+(1-a)\tilde{\mu }$ and variance $v$. The moment-generating function of a normal distribution gives ${\mathbb{E}}_t[e^{x_T}]=e^{m+v/2}$, so the conditional mean of a log-mean-reverting price isThe correction $v/2$ is essential: exponentiating the conditional mean of the logarithm alone would omit the dispersion of the lognormal distribution.

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.