= Solution
{c}
For deterministic square-integrable <volatility>, apply <Itô's formula> to the exponential to obtain
$$
\boxed{S_t=S_0\exp\left(\mu t-\tfrac12\int_0^t\sigma(u)^2du
+\int_0^t\sigma(u)dW_u\right).}
$$
The <stochastic integral> is centered Gaussian with <variance> $\int_0^t\sigma(u)^2du$.
Under the <risk-neutral measure> the drift is $r$. Conditional on $S_t=s$, put $\tau=T-t$ and $v=\int_t^T\sigma(u)^2du$. Then
$$
S_T=s\exp(r\tau-v/2+\sqrt v\,Z),\qquad Z\sim N(0,1).
$$
Integrating the discounted <call option> against this <normal distribution> yields
$$
\boxed{C(t,s)=s\Phi(d_1)-Ke^{-r\tau}\Phi(d_2),}
$$
$$
d_1=\frac{\log(s/K)+r\tau+v/2}{\sqrt v},
\qquad d_2=d_1-\sqrt v.
$$
Thus the total remaining <variance> replaces $\sigma^2\tau$ throughout the ordinary formula; the corresponding effective constant <volatility> is $\sqrt{v/\tau}$. If $v=0$, use the limiting deterministic value $(s-Ke^{-r\tau})_+$.
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