Hedging with a square-payoff derivative as traded asset ($\Gamma =s{F}_s/(2U),U=e^{(r+{\sigma }^2)(T-t)}{S}^2$)

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 $\Gamma =s{F}_s/(2U)$ units of $U$ and bank-account value $F-s{F}_s/2$. Substitution in the self-financing equation proves the hedge.