Binomial options model

For other topics using the name "binomial", see binomial (disambiguation).

In finance, 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. Essentially, option valuation here is via application of the risk neutrality assumption over the life of the option, as the price of the underlying instrument evolves. The Binomial model was first proposed by Cox, Ross and Rubinstein (1979).

Contents

1 Methodology 2 Relationship with Black-Scholes 3 See also 4 References 5 External links

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 valuation date and option expiration. Each node in the lattice, represents a possible price of the underlying, at a particular point in time. This price evolution 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 the first node (valuation date), where the calculated result is the value of the option.

Option valuation using this method is, as described, a three step process:

1) price tree generation

2) calculation of option value at each final node

3) progressive calculation of option value at each earlier node; the value at the first node is the value of the option.

The methodology is best illustrated via example. Link here for a graphical Binomial Tree Option Calculator (http://www.hoadley.net/options/binomialtree.aspx?tree=B).

1) The binomial price tree

The tree of prices is produced by working forward from valuation date to expiration. At each step, it is assumed that the underlying instrument will move up or down by a specific factor - u or d - per step of the tree. (The Binomial model allows for only two states.) If S is the current price, then in the next period the price will either be S up or S down, where S up =S x u and S down =S x d. The up and down factors are calculated using the underlying volatility, σ, and years per time step, t:

<math>u = e^{\sigma\sqrt t}<math>

<math>d = e^{-\sigma\sqrt t} = \frac{1}{u}.<math>

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 value of the option is calculated using the risk neutrality assumption. Under this assumption, today's fair price of a derivative security is equal to the discounted expected value of its future payoff. See Risk neutral valuation.

Expected value here is calculated using the option values from the later two nodes (Option up and Option down) weighted by their respective probabilities -- "probability" p of an up move in the underlying, and "probability" (1-p) of a down move. The expected value is then discounted at r, the risk free rate corresponding to the life of the option. This result, the "Binomial Value", is thus the fair price of the derivative at a particular point in time (i.e. at each node), given the evolution in the price of the underlying to that point.

The Binomial Value is found for each node, starting at the penultimate time step, and working back to the first node of the tree, the valuation date, where the calculated result is the value of the option. For an American option, since the option may either be held or exercised prior to expiry, the value at each node is: Max ( Binomial Value, Exercise Value).

The Binomial Value is calculated as follows.

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

<math>p = \frac{e^{(r-q)t} - d}{u - d}<math>

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

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

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.

See also

• Black-Scholes: binomial lattices are able to handle a variety of conditions for which Black-Scholes cannot be applied.
• financial mathematics, which has a list of related articles.

References

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

External links

• Discussion
  • The Binomial Model for Pricing Options (http://www.sjsu.edu/faculty/watkins/binomial.htm), Prof. Thayer Watkins
  • Using The Binomial Model to Price Derivatives (http://www.quantnotes.com/fundamentals/options/binomial.htm), Quantnotes
  • American Options and Lattice Model Pricing (http://www.quantnotes.com/fundamentals/options/americanoptions.htm), Quantnotes
  • Binomial Method (Cox, Ross, Rubinstein) (http://www.global-derivatives.com/options/european-options.php#binomial), global-derivatives.com
  • Options pricing using a binomial lattice (http://www.moneymax.co.za/articles/displayarticlewide.asp?ArticleID=272347), The Investment Analysts Society of South Africa
  • Convergence of the Binomial to the Black-Scholes Model (http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN00-08.pdf) (PDF), Prof. Don M. Chance
  • The Binomial Model (http://www.hoadley.net/options/BS.htm#Binomial), Peter Hoadley

• Resources
  • Binomial Tree Option Calculator (http://www.hoadley.net/options/binomialtree.aspx?tree=B), Peter Hoadley
  • American Options - Binomial Method (http://www.global-derivatives.com/xls/American-Binomial.xls) (spreadsheet), global-derivatives.com
  • European Options - Binomial Method (http://www.global-derivatives.com/xls/European-Binomial.xls) (spreadsheet), global-derivatives.com