Consider a two-step binomial random asset model \(\left(S_{k}ight)_{k=0,1,2}\) with possible returns \(a=-50 \%\) and \(b=150 \%\), 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 \(p^{*}=\) \((r-a) /(b-a)=3 / 4\) and \(q^{*}=(b-r) /(b-a)=1 / 4\).

a) Draw a binomial tree for the values of \(\left(S_{k}ight)_{k=0,1,2}\).

b) Compute the values \(V_{k}\) at times \(k=0,1,2\) of the hedging portfolio of the European put option with strike price \(K=5 / 4\) and maturity \(N=2\) on \(S_{N}\).

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

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

hedging the European put option on \(S_{N}\) with strike price \(K:=5 / 4\) and maturity \(N:=2\).