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 $0 < p < 1$.

Proposition 3.1 If $S(0) = A(0)$, then: $$S^d < A(1) < S^u,$$ or else an arbitrage opportunity would arise.