 Solution

Since $a_{\tau }$ does not depend on $\sigma$, differentiation of the digital option value gives its vega: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.