Binomial options pricing model

The Binomial options model provides a generalisable numerical method for the valuation of options. The model differs from other option pricing models, in that it uses a “discrete-time” model of the varying price over time of financial instruments; the model is thus able to handle a variety of conditions for which other models cannot be applied. The Binomial model was first proposed by Cox, Ross and Rubinstein (1979).

Methodology

The binomial pricing model uses a "discrete-time framework" to trace the evolution of the option's key underlying variable via a binomial lattice (tree) for a given number of time steps between t = 0 and option expiration. The resultant evolution then forms the basis for the option valuation. The valuation process is iterative, starting at each final node, and then working backwards through the tree to t = 0, where the calculated value is the value of the option in question.

The methodology is best illustrated via example. Link here for an online, graphical Binomial Tree Option Calculator.

1) The binomial tree

The tree of prices is produced by working forward from the present to expiration. At each step it is assumed that the underlying instrument’s price will move up or down by a specific factor - u or d - per step of the tree. If S is the current price, then in the next period the price will be: either S up =S x u or S down =S x d. The factors are calculated as below, using the underlying volatility, σ , and years per time step, t:

u = exp ( σ √ t )

d = exp ( - σ √ t ) = 1 / u

The above is the original Cox, Ross, & Rubinstein (CRR) method; there are other techniques for generating the lattice, such as "the equal probabilities" tree.

2) Option value at each final node

At each final node of the tree -- i.e. at expiration of the option -- the option value is simply its intrinsic, or exercise, value.

For a call, value = max (S – Exercise price , 0)

For a put, value = max (Exercise price – S , 0)

3) Option value at earlier nodes

At each earlier node, the “Binomial Value” of the option is calculated using the risk neutrality assumption. Here, the “probability weighted value” of the option at the next two nodes (Option up and Option down) is determined with p the “probability” of an up move in the underlying, and (1-p) the “probability” of a down move. This result is then discounted at r, the risk free rate corresponding to the life of the option. For an American option, the option may be held or exercised prior to expiry; the value at each node is therefore, Max ( Binomial Value, Exercise Value). The Binomial Value is calculated as follows.

Binomial Value = [ p × Option up + (1-p)× Option down] × exp (- r × t)

p = [ exp((r-q) × t) - d ] ÷ [ u - d ]

q is the dividend yield of the underlying corresponding to the life of the option.

Note that the alternative approach, arbitrage-free pricing ("delta-hedging"), yields identical results.

Relationship with Black-Scholes

Similar assumptions underpin both the binomial model and the Black-Scholes model, and the binomial model thus provides a discrete time approximation to the continuous process underlying the Black-Scholes model. In fact, for European options, the binomial model value converges on the Black-Scholes formula value as the number of time steps increases.

References

Cox JC, Ross SA and Rubinstein M. 1979. Options pricing: a simplified approach, Journal of Financial Economics, 7:229-263.

External links

The Binomial Model for Pricing Options , Thayer Watkins
Using Binomial Model to Price Derivatives, Quantnotes
American Options and Lattice Model Pricing , Quantnotes
Options pricing using a binomial lattice, Investment Basics: XXXVIII., The Investment Analysts Society of South Africa
Optiontutor, Chapter 2, Binomial Model, OS Financial Trading Systems
The Binomial Model, Peter Hoadley