Let us consider one step of a CRR binomial model with time interval , initial state in , an up-state of in and a down state of in . The probability of an up movement is assumed to be , the probability of a down movement respectively, with . The expected value after the first time step is and has to be the same as the expected value of the two binomial states in :
Let be the expected value of a random variable . Then the variance of equals to With that the variance of the two states of the binomial tree in is:
A stock price, or a project value (for real options analysis) respectively, follows a Geometric Brownian motion (stochastic Wiener process). Let be the expected annual volatility of the process. is defined as standard deviation of the normal distribution of the annual relative returns:
The variance of such a Geometric Brownian motion is:
Our binomial model should match the parameters of the continuous model. Therefore the variance of the binomial model and the variance of the Geometric Brownian motion have to be the same. Hence we get the following equation:
In my point of view this is a very important expression, because it determines the correlation of u and d to match volatility. When terms in and higher powers of are ignored, one solution of this equation is:
These are the values of and proposed by Cox, Ross and Rubinstein in their Binomial Model for matching and . But in principle there are infinite possible solutions of this equations. If you define an up or down movement, you can calculate the other value approximative numerically, e.g. with the goal seek function in Excel, even though there might be no closed solution.
Additionally it can be proofed that the variance does not depend on the expected return , when tends to zero. That means that the volatility is independent from the expected return. This is known as Girsanov’s theorem. When we move from a world with one set of risk preferences to a world with another set of risk preferences, the expected growth rates in variables change, but their volatilities remain the same.