In finance, the **binomial options pricing model** (**BOPM**) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a “discrete-time” (lattice based) model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting.

The binomial model was first proposed by William Sharpe in the 1978 edition of *Investments* (ISBN 013504605X),^{[1]} and formalized by Cox, Ross and Rubinstein in 1979^{[2]} and by Rendleman and Bartter in that same year.^{[3]}

For binomial trees as applied to fixed income and interest rate derivatives see Lattice model (finance) #Interest rate derivatives.

Use of the model

The Binomial options pricing model approach has been widely used since it is able to handle a variety of conditions for which other models cannot easily be applied. This is largely because the BOPM is based on the description of an underlying instrument over a period of time rather than a single point. As a consequence, it is used to value American options that are exercisable at any time in a given interval as well as Bermudan options that are exercisable at specific instances of time. Being relatively simple, the model is readily implementable in computer software (including a spreadsheet).

Although computationally slower than the Black–Scholes formula, it is more accurate, particularly for longer-dated options on securities with dividend payments. For these reasons, various versions of the binomial model are widely used by practitioners in the options markets.^{[}

For options with several sources of uncertainty (e.g., real options) and for options with complicated features (e.g., Asian options), binomial methods are less practical due to several difficulties, and Monte Carlo option models are commonly used instead. When simulating a small number of time steps Monte Carlo simulation will be more computationally time-consuming than BOPM (cf. Monte Carlo methods in finance). However, the worst-case runtime of BOPM will be O(2^{n}), where n is the number of time steps in the simulation. Monte Carlo simulations will generally have a polynomial time complexity, and will be faster for large numbers of simulation steps. Monte Carlo simulations are also less susceptible to sampling errors, since binomial techniques use discrete time units. This becomes more true the smaller the discrete units become.

Method

function americanPut(T, S, K, r, sigma, q, n)
{ ‘ T… expiration time ‘ S… stock price ‘ K… strike price ‘ q… dividend yield ‘ n… height of the binomial tree deltaT := T / n; up := exp(sigma * sqrt(deltaT)); p0 := (up*exp(-q * deltaT) – exp(-r * deltaT)) / (up^2 – 1); p1 := exp(-r * deltaT) – p0; ‘ initial values at time T
p[i] := K – S * up^(2*i – n);
} ‘ move to earlier times
‘ binomial value p[i] := p0 * p[i+1] + p1 * p[i]; ‘ exercise value exercise := K – S * up^(2*i – j);
} }
} |

The binomial pricing model traces the evolution of the option’s key underlying variables in discrete-time. This is done by means of a binomial lattice (tree), for a number of time steps between the valuation and expiration dates. Each node in the lattice represents a possible price of the underlying at a given point in time.

Valuation is performed iteratively, starting at each of the final nodes (those that may be reached at the time of expiration), and then working backwards through the tree towards the first node (valuation date). The value computed at each stage is the value of the option at that point in time.

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

- price tree generation,
- calculation of option value at each final node,
- sequential calculation of the option value at each preceding node.

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. The binomial model assumes that movements in the price follow a binomial distribution; for many trials, this binomial distribution approaches the lognormal distribution assumed by Black–Scholes. In this case then, for European options without dividends, the binomial model value converges on the Black–Scholes formula value as the number of time steps increases. ^{[5]} ^{[4]}

In addition, when analyzed as a numerical procedure, the CRR binomial method can be viewed as a special case of the explicit finite difference method for the Black–Scholes PDE; see finite difference methods for option pricing.

References

**^**William F. Sharpe, Biographical, nobelprize.org**^***Cox, J. C.; Ross, S. A.; Rubinstein, M. (1979). “Option pricing: A simplified approach”. Journal of Financial Economics.***7**(3): 229. CiteSeerX 10.1.1.379.7582. doi:10.1016/0304-405X(79)90015-1.**^**Richard J. Rendleman, Jr. and Brit J. Bartter. 1979. “Two-State Option Pricing”.*Journal of Finance*24: 1093-1110. doi:10.2307/2327237- ^ Jump up to:
^{a}Mark s. Joshi (2008). The Convergence of Binomial Trees for Pricing the American Put^{b} - ^ Jump up to:
^{a}Chance, Don M. March 2008^{b}*A Synthesis of Binomial Option Pricing Models for Lognormally Distributed Assets*Archived 2016-03-04 at the Wayback Machine. Journal of Applied Finance, Vol. 18