Chapter 3 One-Step Binomial Model
Suppose that \(S(0) = 100\) dollars and \(S(1)\) can take two values,
\[ S(1) = \begin{cases} 125, &\text{with probability }p\\ 105, &\text{with probability }1-P \end{cases} \] where \(0 < p < 1\), while the bond prices are \(A(0) = 100\) and \(A(1) = 110\) dollars. Thus, the return \(K_S\) on stock will be 25% if stock goes up, or 5% if stock goes down. (Observe that both stock prices at time 1 happen to be higher than that at time 0; ‘going up’ or ‘down’ is relative to the other price at time 1.) The risk-free return will be \(K_A = 10%\).
In general, the choice of stock and bond prices in a binomial model is constrained by the No-Arbitrage Principle. Suppose that the possible up and down stock prices at time 1 are
\[ S(1) = \begin{cases} S^u, &\text{with probability }p\\ S^d, &\text{with probability }1-P \end{cases} \]
where \(S^d < S^u\) and \(0 < p < 1\).