We consider a riskless asset valued \(S_{1}^{(0)}=S_{0}^{(0)}\), at times \(k=0\), 1 , with risk-free interest rate is \(r=0\), and a risky asset \(S^{(1)}\) whose return \(R_{1}:=\left(S_{1}^{(1)}-S_{0}^{(1)}ight) / S_{0}^{(1)}\) can take three values \((-b, 0, b)\) at each time step, with \(b>0\) and

a) Determine all possible risk-neutral probability measures \(\mathbb{P}^{*}\) equivalent to \(\mathbb{P}\) in the sense of Definition 1.5 in terms of the parameter \(\theta^{*} \in(0,1)\), from the condition \(\mathbb{E}^{*}\left[R_{1}ight]=0\).

b) We assume that the variance \(\operatorname{Var}^{*}\left[\frac{S_{1}^{(1)}-S_{0}^{(1)}}{S_{0}^{(1)}}ight]=\sigma^{2}>0\) of the asset return is known to be equal to \(\sigma^{2}\). Show that this condition provides a way to select a unique risk-neutral probability measure \(\mathbb{P}_{\sigma}^{*}\) under a certain condition on \(b\) and \(\sigma\).

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p* : P* (R = b) > 0, 0* =P* (R = 0)>0, q* =P* (R = b) > 0,