Binomial Tree, Cox-Ross-Rubinstein, Method¶

Overview¶

The Cox-Ross-Rubinstein Binomial Tree method is an instance of the Binomial Options Pricing Model (BOPM) , published originally by Cox, Ross and Rubinstein in their 1979 paper “Option Pricing: A Simplified Approach” [CRR1979].

In this method, the binomial tree is used to model the propagation of stock price in time towards a set of possibilities at the Expiration date, based on the stock Volatility. For “N” time steps into which the model scenario duration is subdivided, there are N+1 possible stock prices at the expiration time.

Based on the N+1 Call or Put Option values at expiration, option values are backward-propagated to the initial time using step probabilities and the interest-rate, to obtain the Call or Put Option price. Comparing intermediate Call/Put values during back-propagation to stock prices allows American Option prices to be calculated.

Cox-Ross-Rubinstein show that as N tends to ∞, the binomial European Put/Call solutions tend towards the Black-Scholes solutions. (Both models make the same underlying assumptions.) In an example where K = $35.00 and N = 150, they show the difference is less than$0.01.

In a later paper, Leisen & Reimer [LR1995] propose a method to increase the convergence speed of the CRR binomial lattice to converge faster.

The diagram above shows an example of a binomial tree, where the number of time steps is $$n$$. (Note that $$n$$ steps results in $$n + 1$$ separate propagated $$S$$ values after the n-th step.)

At each step the initial stock price $$S_0$$ is propagated in an Up path and a Down path from each node, with Up and Down factors $$u$$ and $$d$$. The “Up” probability is $$p$$; Down is $$1 - p$$.

The equations in the diagram show the derivation, where $$\sigma$$ is the stock volatility, $$r$$ the “risk-free rate”, $$t$$ the scenario duration and $$n$$ the number of time steps. The dividend yield in the above is assumed to be zero and not included in the expression for $$p$$, but may be included when required.

(Diagram source: Wikipedia article Binomial Options Pricing Model (BOPM) .)

References¶

 [CRR1979] Cox, J. C., Ross, S. A., Rubinstein, M., “Option Pricing: A Simplified Approach”, Journal of Financial Economics (1979)
 [LR1995] Leisen, D., Reimer, M., “Binomial Models for Option Valuation - Examining and Improving Convergence”, Rheinische Friedrich-Wilhelms-Universität, Bonn, (1995).