The Cox-Ross-Rubinstein binomial option pricing model (CRR model) is a variation of the original Black-Scholes option pricing model. It was first proposed in 1979 by financial economists/engineers John Carrington Cox, Stephen Ross and Mark Edward Rubinstein. The model is popular because it considers the underlying instrument over a period of time, instead of just at one point in time. It does this by using a lattice-based model, which takes into account expected changes in various parameters over an option's life, thereby producing a more accurate estimate of option prices than created by models that consider only one point in time. Because of this, the CRR model is especially useful for analyzing American style options, which can be exercised at any time up to expiration (European style options can only be exercised upon expiration). And, unlike the original Black-Scholes option pricing model, the CRR model has the ability to take into account the effect of dividends paid out by a stock during the life of an option.
The CRR model uses a risk-neutral valuation method. Its underlying principal affirms that when determining option prices, it can be assumed that the world is risk neutral and that all individuals (and investors) are indifferent to risk. In a risk neutral environment, expected returns are equal to the risk-free rate of interest. Like the Black-Scholes model, the CRR model makes certain assumptions, including:
- No possibility of arbitrage; a perfectly efficient market
- At each time node, the underlying price can only take an up or a down move and never both simultaneously
The CRR model employs and iterative structure that allows for the specification of nodes (points in time) between the current date and the option's expiration date. The model is able to provide a mathematical valuation of the option at each specified time, creating a "binomial tree" - a graphical representation of possible values at different nodes.
The CRR model is a two-state (or two-step) model in that it assumes the underlying price can only either increase (up) or decrease (down) with time until expiration. Valuation begins at each of the final nodes (at expiration) and iterations are performed backwards through the binomial tree up to the first node (date of valuation). In very basic terms, the model involves three steps:
- The creation of the binomial price tree
- Option value calculated at each final node
- Option value calculated at each preceding node
While the math behind the Cox-Ross-Rubinstein model is considered less complicated than the Black-Scholes model, you can use online calculators and trading platform-based analysis tools to determine option pricing values. Figure 6 shows an example of the Cox-Ross-Rubinstein model applied to an American-style options contract. The calculator produces both put and call values based on variables the user inputs.
|Figure 6: The Cox-Ross-Rubinstein model applied to an American-style options contract, using the Options Industry Council's online pricing calculator.|
Options Pricing: Put/Call Parity
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