Financial Simulation Modeling in Excel

BINOMIAL TREES

The most basic method of presenting stock price movement is a binomial tree. It is very similar to the Brownian motion process we discussed in Chapter 3 except the step size can be of an arbitrary magnitude and the probability of an up or down movement does not necessarily have to be equal.

Figure 4.1 represents a basic recombinant binomial tree, a tree where two nodes can transition to the same node, with an initial starting stock price of S₀ that transitions after step 1 to either price S₁ with probability p₁ or S₂ with probability 1 – p₁. Furthermore, you can see that from step 1, the price can take a total of 3 possible values at step 2. For the sake of simplicity let us start by examining the starting branch represented by S₀, S₁, and S₂. Given this diagram, what would be the expected or average price of the stock? From statistics, we know the answer to be the value if the event occurs times the probability of occurrence plus the value if the event does not occur times the probability of the event not occurring (equation 4.1):

FIGURE 4.1 Basic recombinant binomial tree process where at each node the price can transition to two possible states. This is a recombinant tree because two nodes can transition to the same node in the next step.

If we were to sell an option, ...