The Binomial Market Model

This article and its associated software are designed to give you some insight into the binomial model of stock price movements and the concept of volatility. The article is a bit mathematical, but if you can do a little algebra you should be able to follow it.

The binomial model is presented in every finance text that discusses the theory of option valuation. It's the most widely used method for computing the "fair value" of American-style options (contracts permitting the holder to exercise them before expiration).

Its theoretical foundation is the famous "random walk" hypothesis, but textbooks don't present it from that perspective. I thought it would be instructive and entertaining to see and test the real implications of this model, so I wrote some software to help visualize the workings and statistical properties of a binomial market.

It's a simple model, driven by a uniformly distributed random number generator, with straightforward statistical properties and no "market logic" or intelligent trading agents. Yet it generates displayed remarkably market-like price behavior, and not just in broad statistics. Long- and short-term price runs and cyclical movements appeared plainly in the graphs, which are much more realistic-looking than I'd expected. As the graph below illustrates, even the long- and short-term moving averages have a surprisingly realistic appearance, at least to the "naked eye". Why do price series derived from random numbers exhibit so much apparent structure?

The answer is that the binomial model inevitably produces a kind of non-periodic, cyclic price motion. Prices in a binomial series are more than random points with a certain statistical distribution, even though a random number generator helps produce them. Like real market prices, each binomial price is related to other, recent prices, and carries an implicit forecast about the rate of change and size of future price movements.

Binomial cycles result from a particular process of price movements, and the binomial process is a reasonable description of some aspects of real markets. To the extent that it reflects reality, it must contribute to the cyclical patterns we see in real stock prices. It may even be possible to distinguish this inherent, non-periodic cyclicality from meaningful changes in a stock’s price behavior.

This mathematical experiment demonstrates that statistics by themselves can be rather weak tools for characterizing market behavior. To really understand markets, we will have to model and understand the underlying processes which produce the price movements we observe.