a) We consider a forward contract on \(S_{N}\) with strike price \(K\) and payoff

\[ C:=S_{N}-K \]

Find a portfolio allocation \(\left(\eta_{N}, \xi_{N}ight)\) with value

\[ V_{N}=\eta_{N} \pi_{N}+\xi_{N} S_{N} \]

at time \(N\), such that

\[ \begin{equation*}

V_{N}=C, \tag{3.46}

\end{equation*} \]

by writing Condition (3.46) as a \(2 \times 2\) system of equations.

b) Find a portfolio allocation \(\left(\eta_{N-1}, \xi_{N-1}ight)\) with value \[
V_{N-1}=\eta_{N-1} \pi_{N-1}+\xi_{N-1} S_{N-1}
\]
at time \(N-1\), and verifying the self-financing condition \[
V_{N-1}=\eta_{N} \pi_{N-1}+\xi_{N} S_{N-1} \text {. }
\]
Next, at all times \(t=1,2, \ldots, N-1\), find a portfolio allocation \(\left(\eta_{t}, \xi_{t}ight)\) with value \(V_{t}=\eta_{t} \pi_{t}+\xi_{t} S_{t}\) verifying (3.46) and the self-financing condition \[
V_{t}=\eta_{t+1} \pi_{t}+\xi_{t+1} S_{t}
\]
where \(\eta_{t}\), resp. \(\xi_{t}\), represents the quantity of the riskless, resp. risky, asset in the portfolio over the time period \([t-1, t], t=1,2, \ldots, N-1\).

c) Compute the arbitrage-free price \(\pi_{t}(C)=V_{t}\) of the forward contract \(C\), at time \(t=\) \(0,1, \ldots, N\).

d) Check that the arbitrage-free price \(\pi_{t}(C)\) satisfies the relation \[
\pi_{t}(C)=\frac{1}{(1+r)^{N-t}} \mathbb{E}^{*}\left[C \mid \mathcal{F}_{t}ight], \quad t=0,1, \ldots, N \]