The binomial model is an [[option|option]] valuation method which uses an iterative procedure, allowing for the specification of nodes, or points in time, during the time span between the valuation date and the option’s expiration date.
Consider a portfolio vector \(h(x,y)\) consisting of \(x\) bonds with price process \(B_t\) and \(y\) units of stock with price process \(S_t\) over one period \(t\in\{0,1\}\). The price processes are described by \[ \begin{align}S_{0}=s&,\quad S_1=\begin{cases}us&\text{w.p. }~p_u\\ds&\text{w.p. }~p_d\end{cases}\\B_0=1&,\quad B_1=(1+R)\end{align} \] The value process for the portfolio is given by \[ V_t^h=xB_t+yS_{t}\quad(t\in\{0,1\}) \]
The model is free of [[arbitrage|arbitrage]] if an only if the following condition holds: \[ d\leq (1+R)\leq u \]