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

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.