Binomial model

This model is widely used in modelling stock prices. The following are the basic conditions that this model has to satisfy

• The one step returns \(K(n)\) on a stock are identically distributed independent random variables such that
  \[ K(n) = \left\{ \begin{array}{cc} u & \text{with probability } p \\ d & \text{with probability } 1-p \\ \end{array} \right.\]
• The one-step return \(r\) on a risk-free investment is the same at each time step and \(d<r<u\).

One can show that \(S(n)\) can have values \(S(0)(1+u)^i(1+d)^{n-i}\) with probaility \(\binom{n}{i} p^i(1-p)^{n-i}\). The number \(i\) is the number of upward price movements in a random variable and \(n-i\) is the number of downward movements.

Risk Neutral probability

One can show that \(E(S(1)) = S(0)(1+E(K(1))\) where \(E(K(1))= pu+(1-p)d\) and generalize the result for \(n\) stocks are follows

\[E(S(n))=S(0)(1+E(K(1))^n. \]

If \(r\) represents the risk-free rate of return, then one can expect that \(E(K(1))\) is greater than \(r\). But there are cases in which \(E(K(1)) = r\), and we refer this case as risk-neutral. In this case, we call the probability as \(p_{*}\). One can easily show that

\[p_{*} = \frac{r-d}{u-d}.\]