Matching Volatility in Binomial Model

Let us consider one step of a CRR binomial model with time interval $\Delta t$ , initial state $S_0$ in $t=0$ , an up-state of $S^+=S_0u$ in $t = \Delta t$ and a down state of $S^-=S_0d$ in $t= \Delta t$ . The probability of an up movement is assumed to be $p$ , the probability of a down movement $1-p$ respectively, with $0<p<1$ . The expected value after the first time step $\Delta t$ is $S_0 exp( \mu \Delta t)$ and has to be the same as the expected value of the two binomial states in $\Delta t$ :

$p S_0 u + (1-p) S_0 d=e^{\mu \Delta t}$

Let $E(X)$ be the expected value of a random variable $X$ . Then the variance $var(X)$ of $X$ equals to $var(X)=E(X^2)-[E(X)]^2.$ With that the variance of the two states of the binomial tree in $\Delta t$ is:

$pS_0^2 u^2+(1-p) S_0^2 d^2-(puS_0+(1-p)dS_0 )^2$

A stock price, or a project value (for real options analysis) respectively, follows a Geometric Brownian motion (stochastic Wiener process). Let $\sigma$ be the expected annual volatility of the process. $\sigma$ is defined as standard deviation of the normal distribution $\Phi$ of the annual relative returns:

$\frac{ \Delta S}{S} \thickapprox \Phi ( \mu \Delta t, \sigma \sqrt{\Delta t})$

The variance of such a Geometric Brownian motion is:

$var(S)=S^2 \sigma ^2 \Delta t$

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:

$p u^2 + (1-p) d^2 - ( p u+ (1-p) d)^2 = \sigma ^2 \Delta t$

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 $\Delta t^2$ and higher powers of $\Delta t$ are ignored, one solution of this equation is:

$u=e^ {( \sigma \sqrt {\Delta t})}$

$d=e^ {( - \sigma \sqrt {\Delta t})}$

These are the values of $u$ and $d$ proposed by Cox, Ross and Rubinstein in their Binomial Model for matching $u$ and $d$ . 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 $\mu$ , when $\Delta t$ 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.