= Hedging with a square-payoff derivative as traded asset
{title2=$\Gamma=sF_s/(2U),\quad U=e^{(r+\sigma^2)(T-t)}S^2$}
In the <Black-Scholes model>, the claim paying $S_T^2$ has the displayed positive price $U$ and dynamics $dU/U=(2\mu-r)\,dt+2\sigma\,dW$. Its nonzero volatility lets it replace the stock as a risky hedging asset. A claim with ordinary stock-price function $F(t,s)$ is replicated by $\Gamma=sF_s/(2U)$ units of $U$ and bank-account value $F-sF_s/2$. Substitution in the self-financing equation proves the hedge.
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