We consider the discrete-time Cox-Ross-Rubinstein model with \(N+1\) time instants \(t=0,1, \ldots, N\), with a riskless asset whose price \(A_{t}\) evolves as \(A_{t}=A_{0}(1+r)^{t}\), \(t=0,1, \ldots, N\). The evolution of \(S_{t-1}\) to \(S_{t}\) is given by

with \(-1

and \(\mathcal{F}_{t}\) is generated by \(R_{1}, R_{2}, \ldots, R_{t}, t=1,2, \ldots, N\).

a) What are the possible values of \(R_{t}\) ?

b) Show that, under the probability measure \(\mathbb{P}^{*}\) defined by \[
p^{*}=\mathbb{P}^{*}\left(R_{t+1}=b \mid \mathcal{F}_{t}ight)=\frac{r-a}{b-a}, \quad q^{*}=\mathbb{P}^{*}\left(R_{t+1}=a \mid \mathcal{F}_{t}ight)=\frac{b-r}{b-a}
\]
\(t=0,1, \ldots, N-1\), the expected return \(\mathbb{E}^{*}\left[R_{t+1} \mid \mathcal{F}_{t}ight]\) of \(S\) is equal to the return \(r\) of the riskless asset.

c) Show that under \(\mathbb{P}^{*}\) the process \(\left(S_{t}ight)_{t=0,1, \ldots, N}\) satisfies \[
\mathbb{E}^{*}\left[S_{t+k} \mid \mathcal{F}_{t}ight]=(1+r)^{k} S_{t}, \quad t=0,1, \ldots, N-k, \quad k=0,1, \ldots, N \]

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