Forward contracts. Recall that the price \(\pi_{t}(C)\) of a claim payoff \(C=h\left(S_{T}ight)\) of maturity \(T\) can be written as \(\pi_{t}(C)=g\left(t, S_{t}ight)\), where the function \(g(t, x)\) satisfies the Black-Scholes PDE

\[
\left\{\begin{array}{l}
r g(t, x)=\frac{\partial g}{\partial t}(t, x)+r x \frac{\partial g}{\partial x}(t, x)+\frac{1}{2} \sigma^{2} x^{2} \frac{\partial^{2} g}{\partial x^{2}}(t, x) \\
g(T, x)=h(x)
\end{array}ight.
\]
with terminal condition \(g(T, x)=h(x), x>0\).

a) Assume that \(C\) is a forward contract with payoff

\[
C=S_{T}-K
\]
at time \(T\). Find the function \(h(x)\) in (1).

b) Find the solution \(g(t, x)\) of the above PDE and compute the price \(\pi_{t}(C)\) at time \(t \in[0, T]\). Hint: search for a solution of the form \(g(t, x)=x-\alpha(t)\) where \(\alpha(t)\) is a function of \(t\) to be determined.

c) Compute the quantities

\[
\xi_{t}=\frac{\partial g}{\partial x}\left(t, S_{t}ight)
\]

and \(\eta_{t}\) of risky and riskless assets in a self-financing portfolio hedging \(C\), assuming \(A_{0}=\$ 1\).

d) Repeat the above questions with the terminal condition \(g(T, x)=x\).