Past exam of the martingales and financial mathematics course of the University of Oxford / 2010 / B / 6 / c / Solution

The conditional mean solves $m^{\prime }(v)=rm(v)+\delta$ with $m(t)=S_t$. Hence, for $r\ne 0$,The forward contract's terminal payoff is $S_T-K$, so discounting this expectation gives its current valueFor $r=0$, the continuous limits are $S_t+\delta \tau$ for the mean and $S_t+\delta \tau -K$ for the contract value. A fair delivery price, as distinct from the current contract value, is the strike making this value zero: $K_{\ast }=S_t{e}^{r\tau }+\delta (e^{r\tau }-1)/r$.