Consider a two-step binomial random asset model \(\left(S_{k}ight)_{k=0,1,2}\) with possible returns \(a=0\) and \(b=200 \%\), and a riskless asset \(A_{k}=A_{0}(1+r)^{k}, k=0,1,2\) with interest rate \(r=100 \%\), and \(S_{0}=A_{0}=1\), under the risk-neutral probabilities \(p^{*}=\) \((r-a) /(b-a)=1 / 2\) and \(q^{*}=(b-r) /(b-a)=1 / 2\).

a) Draw a binomial tree for the possible values of \(\left(S_{k}ight)_{k=0,1,2}\), and compute the values \(V_{k}\) at times \(k=0,1,2\) of the portfolio hedging the European call option on \(S_{N}\) with strike price \(K=8\) and maturity \(N=2\).

Hint: Consider three cases when \(k=2\), and two cases when \(k=1\).

b) Price, then hedge. Compute the self-financing hedging portfolio strategy \(\left(\xi_{k}, \eta_{k}ight)_{k=1,2}\) with values

\[ V_{0}=\xi_{1} S_{0}+\eta_{1} A_{0}, \quad V_{1}=\xi_{1} S_{1}+\eta_{1} A_{1}=\xi_{2} S_{1}+\eta_{2} A_{1}, \quad \text { and } \quad V_{2}=\xi_{2} S_{2}+\eta_{2} A_{2} \]

hedging the European call option with strike price \(K=8\) and maturity \(N=2\).

Consider two separate cases for \(k=2\) and one case for \(k=1\).

c) Hedge, then price. Compute the hedging portfolio strategy \(\left(\xi_{k}, \eta_{k}ight)_{k=1,2}\) from the selffinancing condition, and use it to recover the result of part (a).