Pricing Options Using Multi Step Binomial Trees

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The ideas we developed for a single-period binomial model also apply to a multi-period approach. In this video we will look at a two-step, or two time-period binomial tree. In this framework the stock price must follow one of four patterns. For the two periods, the stock can go up-up, up-down, down-up, or down-down. At the moment, we are assuming fixed down and up percentages, the down-up and up-down paths will end with the same final stock price (but you will later see that this restriction is not required, the valuation approach is more flexible than this).

Binomial trees can be used for valuing puts or calls. Consider a two step binomial tree, with each step one year long where at each node the stock moves up or down 20% and the risk-free rate is 5%. Suppose the stock price is now $20 and that we will try to value a put with a strike of $20.

You work from right to left, backward in time, valuing the option node by node, first calculating fu and fd using the above formulas, then value f using the formula.

Now that we have looked at a two-period binomial tree, you can easily see that we can, using the same formulas, produce binomial trees with as many nodes as we want. The more periods that we add, the more realistic our model becomes. A binomial tree with just 20 periods gives more than a million stock price movement patterns.

Clearly working out a series of one-second node one-penny price movement binomial trees would take quite a while, but it is easy to code the approach on your computer, and it is virtually unlimited as to how many nodes you can add. The binomial model assumes that movements in the price follow a binomial distribution. If you increase the number of nodes, and are modeling the stock price evolution over a very short period of time, you begin to approach a very realistic share price trajectory. Each node could be one second in duration, and show the stock’s expected price moves of, as an example, up one penny or down one penny. This begins to approximate real-life stock price movements quite accurately.

At each second during a trading day, it is fairly realistic to assume that a $20 stock will increase or decrease by as little as $0.01 or $0.02. As you increase the number of nodes, this binomial distribution approaches the lognormal distribution assumed by Black–Scholes (see that video).

When analyzed as a numerical procedure, the Cox, Ross, and Rubinstein binomial method can be viewed as a special case of the explicit finite difference method for the Black-Scholes partial differential equation.
The binomial tree approach is very flexible, and can take into account dividends, early exercise opportunities, and even different distributions of stock price movements over the time to maturity of the derivative being valued. This means that you could model low volatility periods of stock price movements, and then higher volatility periods for the stock—perhaps around their earnings announcements—over the duration of an option’s life.

Although computationally slower than the Black–Scholes formula, it is more accurate, particularly for longer-dated options on securities with dividend payments. For these reasons, various versions of the binomial model are widely used by practitioners in the options markets. For options with several sources of uncertainty and for options with complicated features, binomial methods can be less practical due to several difficulties, at which point Monte Carlo option models are used instead.