= Conditional geometric-average Asian option formula
{title2=$m_t=T^{-1}[I_t+(T-t)\log S_t+\tfrac12(\rho-\sigma^2/2)(T-t)^2],\quad q_t=\sigma^2(T-t)^3/(3T^2)$}
For $I_t=\int_0^t\log S_u\,du$ and $G_T=\exp(T^{-1}I_T)$ in the <Black-Scholes model>, the conditional law of $\log G_T$ under the <risk-neutral measure> has a <normal distribution> with mean $m_t$ and variance $q_t$. The future Brownian integral equals $\int_t^T(T-u)\,dW_u$, so the <Itô isometry> gives the variance. For strike $K>0$ and $t<T$, the call price is $e^{-\rho(T-t)}[e^{m_t+q_t/2}\Phi(d_1)-K\Phi(d_2)]$, where $d_2=(m_t-\log K)/\sqrt{q_t}$ and $d_1=d_2+\sqrt{q_t}$. This follows by completing the square in the truncated normal exponential moment; the known past integral must be included in the mean.
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