Consider a two-step trinomial market model \(\left(S_{t}^{(1)}ight)_{t=0,1,2}\) with \(r=0\) and three possible return rates \(R_{t}=-1,0,1\), and the risk-neutral probability measure \(\mathbb{P}^{*}\) given by

\[ \mathbb{P}^{*}\left(R_{t}=-1ight):=p^{*}, \quad \mathbb{P}^{*}\left(R_{t}=0ight):=1-2 p^{*}, \quad \mathbb{P}^{*}\left(R_{t}=1ight):=p^{*} \]

Taking \(S_{0}^{(1)}=1\), price the European put option with strike price \(K=1\) and maturity \(N=2\) at times \(t=0\) and \(t=1\).