Asian call option with a negative strike price. Consider the asset price process

\[S_{t}=S_{0} \mathrm{e}^{r t+\sigma B_{t}-\sigma^{2} t / 2}, \quad t \geqslant 0\]

where \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion. Assuming that \(K \leqslant 0\), compute the price

\[\mathrm{e}^{-(T-t) r} \mathbb{E}^{*}\left[\left.\left(\frac{1}{T} \int_{0}^{T} S_{u} d u-Kight)^{+} ightvert\, \mathcal{F}_{t}ight]\]

of the Asian option at time \(t \in[0, T]\).