Two-Period Model

One period binomial

Ω = \{0,1\}.

D_0 = 1, D_j(\}ω\}) = d_ω.

X_0 = (1, s, n),

X_1(ω) = (R, S(ω), 0), C_1 = (0,0,ν(S(ω))).

\mathcal{A}_0 = \{\{0,1\}\}, \mathcal{A}_1 = \{\{0\},\{1\}\}.

X_0D_0 = (X_1 + C_1)D_1|_{\mathcal{A}_0}

(1, s, n) = (R, S(0), ν(S(ω)) d_0 + (R, S(1), ν(S(1)) d_1

\begin{aligned} 1 &= R d_0 + R d_1 \\ s &= S(0) d_0 + S(1) d_1 \\ n &= ν(S(1))d_0 + ν(S(1))d_1 \\ \end{aligned}

Two period binomial

Ω = \{0,1\}^2.

X_0 = (1, s, n),

X_1(ω) = (R, S_1(ω), 0), C_1 = (0,0,0).

X_2(ω) = (R^2, S_2(ω), 0), C_2 = (0,0,ν(S_2(ω)))

\mathcal{A}_0 = \{\{00, 01, 10, 11\}\}

\mathcal{A}_1 = \{\{00, 01\},\{10, 11\}\}.

X_1 D_1 = (X_2 + C_2) D_2|_{\mathcal{A}_1}

\begin{aligned} R &= R^2(d_{00} + d_{01}) \\ S_1(0*) &= S_2(00)(d_{00} + S_2(01) d_{01}) \\ V_1(0*) &= ν(S_2(00))d_{00} + ν(S_2(01)) d_{01}) \\ \end{aligned} \begin{aligned} R &= R^2(d_{10} + d_{01}) \\ S_1(1*) &= S_2(10)(d_{10} + S_2(11) d_{11}) \\ V_1(1*) &= ν(S_2(10))d_{10} + ν(S_2(11)) d_{11}) \\ \end{aligned}

X_0D_0 = (X_1 + C_1)D_1|_{\mathcal{A}_0}

(1, s, n) = (R, S(0), ν(S(ω)) d_0 + (R, S(1), ν(S(1)) d_1

\begin{aligned} 1 &= R d_0 + R d_1 \\ s &= S(0) d_0 + S(1) d_1 \\ n &= ν(S(1))d_0 + ν(S(1))d_1 \\ \end{aligned} \begin{aligned} \end{aligned}