Power put-call parity ($C_{\beta }-P_{\beta }=S^{\beta }{e}^{[{\lambda }_{\beta }-r](T-t)}-K^{\beta }{e}^{-r(T-t)}$)

The difference of the terminal power-call and power-put payoffs is $S_T^{\beta }-K^{\beta }$. Risk-neutral pricing and the geometric Brownian motion moment formula therefore give the displayed parity, with ${\lambda }_{\beta }=\beta (r-q)+{\sigma }^2\beta (\beta -1)/2$. At $\beta =1$ this reduces to dividend-adjusted put-call parity.