= Solution
{c}
The conditional mean solves $m'(v)=rm(v)+\delta$ with $m(t)=S_t$. Hence, for $r\ne0$,
$$
\boxed{\mathbb E[\widehat S_T\mid\widehat S_t=S_t]
=S_te^{r\tau}+\frac\delta r(e^{r\tau}-1).}
$$
The forward contract's terminal payoff is $S_T-K$, so discounting this expectation gives its current value
$$
\boxed{F(t,S_t)=S_t+\frac\delta r(1-e^{-r\tau})-Ke^{-r\tau}.}
$$
For $r=0$, 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: $K_*=S_te^{r\tau}+\delta(e^{r\tau}-1)/r$.
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