a) Show that for every fixed value of \(S\), the function

\[ d \longmapsto h(S, d):=S \Phi(d+|\sigma| \sqrt{T})-K \mathrm{e}^{-r T} \Phi(d), \]

reaches its maximum at \(d_{*}(S):=\frac{\log (S / K)+\left(r-\sigma^{2} / 2ight) T}{|\sigma| \sqrt{T}}\).

The maximum is reached when the partial derivative \(\frac{\partial h}{\partial d}\) vanishes.

b) By the differentiation rule

\[
\frac{d}{d S} h\left(S, d_{*}(S)ight)=\frac{\partial h}{\partial S}\left(S, d_{*}(S)ight)+d_{*}^{\prime}(S) \frac{\partial h}{\partial d}\left(S, d_{*}(S)ight)
\]
recover the value of the Black-Scholes Delta.