 Solution

Differentiate the Gaussian price with respect to $s$. The terms involving derivatives of $d$ cancel because ${\varphi }^{\prime }(d)=-d\varphi (d)$ and $s-K{e}^{-r\tau }=\nu d$. Thus the delta hedge isFor every $t<T$, $\nu >0$ and $S_t$ is finite, so $0<{\pi }_t<1$. At maturity its limiting value is the payoff derivative $1_{\{S_T>K\}}$ except at the kink, an event of probability zero under the equivalent Gaussian law. Consequently the stock holding is always nonnegative and never exceeds one. The initial drift $\mu$ does not enter this hedge; it is removed by the change to the risk-neutral measure.