= Solution
Solving the <geometric Brownian motion> equation over $[t,T]$ gives
$$
S_T=S_t\exp\left(\sigma_0(W_T-W_t)-\frac12\sigma_0^2(T-t)\right).
$$
The <Brownian increment> is independent of $\mathcal F_t$ and has a <normal distribution> with variance $T-t$. Its exponential <moment-generating function> therefore gives
$$
C(t,T)=\sqrt{S_t}\,e^{-\sigma_0^2(T-t)/4}
\mathbb E\left[e^{\sigma_0(W_T-W_t)/2}\mid\mathcal F_t\right]
=\sqrt{S_t}\,e^{-\sigma_0^2(T-t)/8}.
$$
The required deterministic price function is
$$
\boxed{F(t,T,s,v)=\sqrt{s}\exp\left(-\frac18v^2(T-t)\right).}
$$
This includes $s=0$, $v=0$ and $t=T$. For $s>0$ and $t<T$, it is continuous and strictly decreasing in $v\geq0$, with range $(0,\sqrt{s}]$.
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