Conditional power moments of a continuous geometric Asian average (${\mathbb{E}}_Q[A_T^p\mid {\mathcal{F}}_t]=A_t^{pt/T}{S}_t^{p\tau /T}\exp \{p(r-{\sigma }^2/2){\tau }^2/(2T)+p^2{\sigma }^2{\tau }^3/(6T^2)\}$)

Let $A_t=\exp (t^{-1}{\int }_0^t\log S_{u}du)$ for a positive geometric Brownian motion in the non-dividend-paying Black-Scholes model, and put $\tau =T-t$. Under the risk-neutral measure, the future log-average isThe final integral is a centred normal variable independent of the past, with variance ${\tau }^3/3$. Applying the moment-generating function of a normal distribution proves the displayed identity for every real $p$. Multiplication by $e^{-r\tau }$ prices the payoff $A_T^p$. The historical average must remain a state variable; terminal stock price alone does not determine this payoff.