= Solution
{c}
The <contingent claim payoff> derivative is $\Psi'(s)=2(s-K)_+$. Differentiating the <risk-neutral valuation> is justified by the finite <lognormal distribution>, and $S_T$ is proportional to the starting stock value $s$. Thus
$$
\Delta(t,s)=\frac{2e^{-r\tau}}s
\mathbb E\bigl[(S_T^2-KS_T)\mathbf1_{\{S_T\geq K\}}\bigr],
\qquad\tau=T-t.
$$
For a <lognormal> terminal stock, completing the square in its Gaussian exponent gives
$$
\mathbb E[S_T^p\mathbf1_{\{S_T\geq K\}}]
=s^p e^{pr\tau+p(p-1)\sigma^2\tau/2}
\Phi\left(\frac{\log(s/K)+(r+(p-\tfrac12)\sigma^2)\tau}
{\sigma\sqrt\tau}\right).
$$
Use $p=2$ and $p=1$, and let $d_1=[\log(s/K)+(r+\sigma^2/2)\tau]/(\sigma\sqrt\tau)$. The result is
$$
\boxed{\Delta(t,s)=
2s e^{(r+\sigma^2)\tau}\Phi(d_1+\sigma\sqrt\tau)
-2K\Phi(d_1).}
$$
As $\tau\downarrow0$ this tends to $2(s-K)_+$, including value zero at $s=K$, as required. The extra $\sigma^2$ in the <option delta> PDE's stock drift gives the same shifted Gaussian arguments. This is <delta of a squared positive-part call>.
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