Power option. Let \(\left(S_{n}ight)_{n \in \mathbb{N}}\) denote a binomial price process with returns \(-50 \%\) and \(+50 \%\), and let the riskless asset be valued \(A_{k}=\$ 1, k \in \mathbb{N}\). We consider a power option with payoff \(C:=\left(S_{N}ight)^{2}\), and a predictable self-financing portfolio strategy \(\left(\xi_{k}, \eta_{k}ight)_{k=1,2, \ldots, N}\) with value

\[ V_{k}=\xi_{k} S_{k}+\eta_{k} A_{0}, \quad k=1,2, \ldots, N \]

a) Find the portfolio allocation \(\left(\xi_{N}, \eta_{N}ight)\) that matches the payoff \(C=\left(S_{N}ight)^{2}\) at time \(N\), i.e. that satisfies

\[ V_{N}=\left(S_{N}ight)^{2} \]

Hint: We have \(\eta_{N}=-3\left(S_{N-1}ight)^{2} / 4\).

b) In the following questions we use the risk-neutral probability \(p^{*}=1 / 2\) of a \(+50 \%\) return.

i) Compute the portfolio value

\[ V_{N-1}=\mathbb{E}^{*}\left[C \mid \mathcal{F}_{N-1}ight] \]

ii) Find the portfolio allocation \(\left(\eta_{N-1}, \xi_{N-1}ight)\) at time \(N-1\) from the relation

\[ V_{N-1}=\xi_{N-1} S_{N-1}+\eta_{N-1} A_{0} . \]

We have \(\eta_{N-1}=-15\left(S_{N-2}ight)^{2} / 16\).

iii) Check that the portfolio satisfies the self-financing condition

\[ V_{N-1}=\xi_{N-1} S_{N-1}+\eta_{N-1} A_{0}=\xi_{N} S_{N-1}+\eta_{N} A_{0} \]