Power option ($S_T^p$)

A power option is a European contingent claim whose payoff is a fixed power of the terminal underlying price. In the Black-Scholes model, its value is $S_t^p\exp ((p-1)r(T-t)+\frac{1}{2}p(p-1){\sigma }^2(T-t))$ when the required moments are finite.

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.