We consider the discrete-time Cox-Ross-Rubinstein model with \(N+1\) time instants \(t=0,1, \ldots, N\).

The price \(\pi_{t}\) of the riskless asset evolves as \(\pi_{t}=\pi_{0}(1+r)^{t}, t=0,1, \ldots, N\). The evolution of \(S_{t-1}\) to \(S_{t}\) is given by

\[ S_{t}= \begin{cases}(1+b) S_{t-1} & \text { if } R_{t}=b \\ (1+a) S_{t-1} & \text { if } R_{t}=a\end{cases} \]

with \(-1

\[ R_{t}:=\frac{S_{t}-S_{t-1}}{S_{t-1}}, \quad t=1,2, \ldots, N \]

Let \(\xi_{t}\), resp. \(\eta_{t}\), denote the (possibly fractional) quantities of the risky, resp. riskless, asset held over the time period \([t-1, t]\) in the portfolio with value

\[ \begin{equation*} V_{t}=\xi_{t} S_{t}+\eta_{t} \pi_{t}, \quad t=0,1, \ldots, N \tag{3.47} \end{equation*} \]

a) Show that \[ \begin{equation*}
V_{t}=\left(1+R_{t}ight) \xi_{t} S_{t-1}+(1+r) \eta_{t} \pi_{t-1}, \quad t=1,2, \ldots, N \tag{3.48}
\end{equation*} \]

b) Show that under the probability \(\mathbb{P}^{*}\) defined by \[
\mathbb{P}^{*}\left(R_{t}=a \mid \mathcal{F}_{t-1}ight)=\frac{b-r}{b-a}, \quad \mathbb{P}^{*}\left(R_{t}=b \mid \mathcal{F}_{t-1}ight)=\frac{r-a}{b-a}, \]
where \(\mathcal{F}_{t-1}\) represents the information generated by \(\left\{R_{1}, R_{2}, \ldots, R_{t-1}ight\}\), we have \[
\mathbb{E}^{*}\left[R_{t} \mid \mathcal{F}_{t-1}ight]=r .
\]

c) Under the self-financing condition \[
\begin{equation*}
V_{t-1}=\xi_{t} S_{t-1}+\eta_{t} \pi_{t-1}, \quad t=1,2, \ldots, N \tag{3.49}
\end{equation*}
\]
recover the martingale property \[
V_{t-1}=\frac{1}{1+r} \mathbb{E}^{*}\left[V_{t} \mid \mathcal{F}_{t-1}ight]
\]
using the result of Question (a).

d) Let \(a=5 \%, b=25 \%\) and \(r=15 \%\). Assume that the value \(V_{t}\) at time \(t\) of the portfolio is \(\$ 3\) if \(R_{t}=a\) and \(\$ 8\) if \(R_{t}=b\), and compute the value \(V_{t-1}\) of the portfolio at time \(t-1\).