= Solution
{c}
Since $a_\tau$ does not depend on $\sigma$, differentiation of the <digital option> value gives its <vega>:
$$
\boxed{\frac{\partial V}{\partial\sigma}=-\frac{e^{-r\tau}}\sigma\,d\phi(d).}
$$
Thus <vega> is negative for $s>0$, zero at $s=0$, and positive for $s<0$. Increasing volatility spreads the conditional <normal distribution> around its mean: when the current asset value is positive this reduces the chance of ending positive, while for a negative starting value it increases that chance. At zero the distribution stays symmetric and the probability remains one half. A digital payoff need not have the positive <vega> familiar from a <European call option>.
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