= Delta of a squared positive-part call
{title2=$\Delta=2S e^{(r+\sigma^2)\tau}\Phi(d_1+\sigma\sqrt\tau)-2K\Phi(d_1)$}
For <contingent claim payoff> $(S_T-K)_+^2$ in the non-dividend <Black-Scholes model>, differentiating the discounted <expected value> gives $\Delta=2e^{-r\tau}S^{-1}\mathbb E[(S_T^2-KS_T)\mathbf1_{\{S_T\geq K\}}]$. Completing the square in the Gaussian exponent computes each truncated <lognormal> moment and yields the displayed formula, where $d_1=(\log(S/K)+(r+\sigma^2/2)\tau)/(\sigma\sqrt\tau)$. Its terminal limit is $2(S-K)_+$.
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