= Geometric Asian option
{c}
{title2=$\left(\exp(T^{-1}\int_0^T\log S_t\,dt)-K\right)^+$}
An <Asian option> whose averaging uses the <geometric mean>. In the constant-coefficient <Black-Scholes model>, the logarithm of the continuous geometric average is Gaussian with mean $\log S_0+(r-\sigma^2/2)T/2$ and variance $\sigma^2T/3$, because $\operatorname{Var}(\int_0^TW_t\,dt)=T^3/3$. Its forward mean is $S_0e^{rT/2-\sigma^2T/12}$. The discounted lognormal call formula gives its price. It is strictly cheaper than the terminal European call when $\sigma,K,T>0$ and $r\geq-\sigma^2/6$, since both its forward mean and log-variance are no larger, with strictly smaller variance. No unconditional comparison holds for arbitrary negative interest rates: for $r<-\sigma^2/6$, sufficiently small positive strikes reverse the inequality.
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