= Conditional power moments of a continuous geometric Asian average
{title2=$\mathbb E_Q[A_T^p\mid\mathcal F_t]=A_t^{pt/T}S_t^{p\tau/T}\exp\{p(r-\sigma^2/2)\tau^2/(2T)+p^2\sigma^2\tau^3/(6T^2)\}$}
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 is
$$
\log A_T=\frac tT\log A_t+\frac\tau T\log S_t+\frac{r-\sigma^2/2}{2T}\tau^2
+\frac\sigma T\int_t^T(W_u-W_t)du.
$$
The 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.
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