Compute the price [mathrm{e}^{-(T-t) r} mathbb{E}^{*}left[left.left(exp left(frac{1}{T} int_{0}^{T} log S_{u} d u ight)-K ight)^{+} ightvert, mathcal{F}_{t}

Compute the price

\[\mathrm{e}^{-(T-t) r} \mathbb{E}^{*}\left[\left.\left(\exp \left(\frac{1}{T} \int_{0}^{T} \log S_{u} d u\right)-K\right)^{+} \rightvert\, \mathcal{F}_{t}\right], \quad 0 \leqslant t \leqslant T\]

at time \(t\) of the geometric Asian option with maturity \(T\), where \(S_{t}=S_{0} \mathrm{e}^{r t+\sigma B_{t}-\sigma^{2} t / 2}\), \(t \in[0, T]\).

When \(X \simeq \mathcal{N}\left(0, v^{2}\right)\) we have

\[\mathbb{E}^{*}\left[\left(\mathrm{e}^{m+X}-K\right)^{+}\right]=\mathrm{e}^{m+v^{2} / 2} \Phi(v+(m-\log K) / v)-K \Phi((m-\log K) / v) .\]