In a Black and Scholes framework, prove that the price at time \(t\) of the contingent claim \(h\left(S_{T}\right)\) is

\[C_{h}(x, T-t)=e^{-r(T-t)} \mathbb{E}_{\mathbb{Q}}\left(h\left(S_{T}\right) \mid S_{t}=x\right)=e^{-r(T-t)} \mathbb{E}_{\mathbb{Q}}\left(h\left(S_{T}^{t, x}\right)\right)\]

where \(S_{s}^{t, x}\) is the solution of the SDE

\[d S_{s}^{t, x}=S_{s}^{t, x}\left(r d s+\sigma d W_{s}\right), S_{t}^{t, x}=x\]

and the hedging strategy consists of holding \(\partial_{x} C_{h}\left(S_{t}, T-t\right)\) shares of the underlying asset.

Assuming some regularity on \(h\), and using the fact that \(S_{T}^{t, x} \stackrel{\text { law }}{=} x e^{\sigma X_{T-t}}\), where \(X_{T-t}\) is a Gaussian r.v., prove that

\[\partial_{x} C_{h}(x, T-t)=\frac{1}{x} \mathbb{E}_{\mathbb{Q}}\left(h^{\prime}\left(S_{T}^{t, x}\right) S_{T}^{t, x}\right) e^{-r(T-t)} .\]