Consider a power call option. This option has an additional parameter, \( \alpha>0 \). The option pays the amount \( S_{T}^{\alpha} \) if exercised on the expiration date \( T \). find the option price given by \[ \exp \left[(\alpha-1)\left(r+\frac{1}{2} \sigma^{2} ight) T+\alpha^{2} \ sigma^{2} T / 2 ight] S_{0}^{\alpha} \Phi\left(\tilde{d}{1} ight)-e^{-r T} K \Phi\left( \tilde{d}_{2} ight) \] where \( \tilde{d}_{i}=\ln \left(e^{r T} S{0} / K^{1 / \alpha } ight) /(\sigma \sqrt{T}) \pm \alpha \sigma \sqrt{T} / 2, i=1,2 \). Hint: The value of the option is given by \( e^{r T} \mathbb{E}\left[\left(S_{T}^{\alpha}-K ight){+} ight] \ )evaluate this integral as in the call case and use the fact that \[ S{T}^{\alpha}=S_{0}^{\alpha} e^{\left[\alpha\left(r -0.5 * \sigma^{2} ight) T+\alpha \sigma \sqrt{T} Z ight]} \] where \( Z \) is distributed as a \( N(0,1) \ ).