Consider an underlying asset whose price \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\) is given by a stochastic differential equation of the form

\[
d S_{t}=r S_{t} d t+\sigma\left(S_{t}ight) d B_{t}
\]

where \(\sigma(x)\) is a Lipschitz coefficient, and an option with payoff function \(\phi\) and price

\[
C(x, T)=\mathrm{e}^{-r T} \mathbb{E}\left[\phi\left(S_{T}ight) \mid S_{0}=xight]
\]

where \(\phi(x)\) is a twice continuously differentiable \(\left(\mathcal{C}^{2}ight)\) function, with \(S_{0}=x\). Using the Itô formula, show that the sensitivity

\[
\operatorname{Theta}_{T}=\frac{\partial}{\partial T}\left(\mathrm{e}^{-r T} \mathbb{E}\left[\phi\left(S_{T}ight) \mid S_{0}=xight]ight)
\]

of the option price with respect to maturity \(T\) can be expressed as

\[
\begin{aligned}
\text { Theta }_{T}= & -r \mathrm{e}^{-r T} \mathbb{E}\left[\phi\left(S_{T}ight) \mid S_{0}=xight]+r \mathrm{e}^{-r T} \mathbb{E}\left[S_{t} \phi^{\prime}\left(S_{T}ight) \mid S_{0}=xight] \\
& +\frac{1}{2} \mathrm{e}^{-r T} \mathbb{E}\left[\phi^{\prime \prime}\left(S_{T}ight) \sigma^{2}\left(S_{T}ight) \mid S_{0}=xight]
\end{aligned}
\]