In finance, the binomial options model provides a generalisable numerical method for the valuation of options. The Binomial model was first proposed by Cox, Ross and Rubinstein (1979). Essentially, the model uses a "discrete-time" model of the varying price over time of the underlying financial instrument. Option valuation is then via application of the risk neutrality assumption over the life of the option, as the price of the underlying instrument evolves. Finance studies and addresses the ways in which individuals, businesses and organizations raise, allocate and use monetary resources over time, taking into account the risks entailed in their projects. ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ... In finance, an option is a contract whereby the contract buyer has a right to exercise a feature of the contract (the option) at future date (the exercise date), and the writer (seller) has the obligation to honour the specified feature of the contract. ... In finance, an underlying is an investment from which a derivative security is derived. ... In Economics, the term risk neutral is used to describe an individual who cares only about the expected outcome of an investment, and not the risk (variance of outcomes or the potential gains or losses). ...

Because it models the underlying over time, as opposed to at a particular point, this approach is able to handle a variety of conditions for which other models cannot easily be applied. (For example, the model is used to value American options which can be exercised at any point and Bermudan options which can be exercised at various points.) The model is also relatively simple, mathematically, and can therefore be readily implemented in a software (or even spreadsheet) environment; its use is therefore widespread in finance. The style or family of a financial option is a general term denoting the class into which the option falls, usually defined by the manner in which the option may be exercised. ... The style or family of a financial option is a general term denoting the class into which the option falls, usually defined by the manner in which the option may be exercised. ... Computer software (or simply software) refers to one or more computer programs and data held in the storage of a computer for some purpose. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

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.

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: In finance, an underlying is an investment from which a derivative security is derived. ... Volatility is the standard deviation of the change in value of a financial instrument with a specific time horizon. ...

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. Conceptually, the value of an option consists of two components, its intrinsic value and its time 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 A call option is a financial contract between two parties, the buyer and the seller of this type of option. ... The strike price, or exercise price, is a key variable in a derivatives contract between two parties. ... A put option (sometimes simply called a put) is a financial contract between two parties, the buyer and the seller of the option. ... The strike price, or exercise price, is a key variable in a derivatives contract between two parties. ...

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 is equal to the discounted expected value of its future payoff. See Risk neutral valuation. In mathematical finance, a risk-neutral measure is a probability measure in which todays fair (i. ... Definition Fair value, also called fair price, is a concept used in finance and economics. ... A derivative is a financial contract whose payoffs over a period of time are derived from the performance of assets, interest rates, exchange rates, or indices. ... In finance, discounting is the process of finding the current value of an amount of cash at some future date, and along with compounding cash form the basis of time value of money calculations. ... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... Rational pricing is the assumption in financial economics that asset prices (and hence asset pricing models) will reflect the arbitrage-free price of the asset as any deviation from this price will be arbitraged away. This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to...

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 risk-free interest rate is the interest rate that it is assumed can be obtained by investing in financial instruments with no risk. ...

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 style or family of a financial option is a general term denoting the class into which the option falls, usually defined by the manner in which the option may be exercised. ...

The Binomial Value is calculated as follows.

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

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. The dividend yield on a company stock is the companys annual dividend payments divided by its market cap, or the dividend per share divided by the price per share. ... In economics, arbitrage is the practice of taking advantage of a state of imbalance between two (or possibly more) markets: a combination of matching deals are struck that exploit the imbalance, the profit being the difference between the market prices. ... Rational pricing is the assumption in financial economics that asset prices (and hence asset pricing models) will reflect the arbitrage-free price of the asset as any deviation from this price will be arbitraged away. This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to... Rational pricing is the assumption in financial economics that asset prices (and hence asset pricing models) will reflect the arbitrage-free price of the asset as any deviation from this price will be arbitraged away. This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to...

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. The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ... The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ... The style or family of a financial option is a general term denoting the class into which the option falls, usually defined by the manner in which the option may be exercised. ...

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.

The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ... Financial mathematics is the branch of applied mathematics concerned with the financial markets. ...

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, Prof. Thayer Watkins
  • Using The Binomial Model to Price Derivatives, Quantnotes
  • American Options and Lattice Model Pricing, Quantnotes
  • Binomial Method (Cox, Ross, Rubinstein), global-derivatives.com
  • Binomial Option Pricing (PDF), Prof. Robert M. Conroy
  • Options pricing using a binomial lattice, The Investment Analysts Society of South Africa
  • Convergence of the Binomial to the Black-Scholes Model (PDF), Prof. Don M. Chance
  • The Binomial Model, Peter Hoadley

• Resources
  • Binomial Tree Option Calculator, Peter Hoadley
  • American Options - Binomial Method (spreadsheet), global-derivatives.com
  • European Options - Binomial Method (spreadsheet), global-derivatives.com