Log-mean-reverting price ($d{S}_t=\alpha (\mu -\log S_t)S_{t}dt+\sigma S_{t}d{W}_t$)

For $\alpha >0$, $\sigma >0$, this positive stochastic process is the exponential of an Ornstein-Uhlenbeck process. The Itô formula givesThe mean level of the logarithm differs from $\mu$ by the quadratic variation correction. In a stock model, specifying this under a physical probability measure does not by itself specify risk-neutral pricing. A non-dividend-paying traded stock has drift $rS$ under a risk-neutral measure; a different drift under that measure would require a corresponding dividend or carry convention.

For a log-mean-reverting price, set $a=e^{-\alpha (T-t)}$. The Ornstein-Uhlenbeck transition distribution gives conditional log-mean $a\log S_t+(1-a)\tilde{\mu }$ and log-variance ${\sigma }^2(1-a^2)/(2\alpha )$. The moment-generating function of a normal distribution therefore gives the displayed formula for $F(t,T)=\mathbb{E}[S_T\mid {\mathcal{F}}_t]$. It is a martingale in $t$ for fixed $T$, by the tower property of conditional expectation, and the Itô formula yieldsThis uses the same probability measure as the specified conditional expectation. It describes futures pricing only when that measure is an appropriate pricing measure, with the required cash-flow conventions. For a non-dividend-paying traded stock and constant interest rate, the forward contract delivery price follows the usual carry relation instead.