In
finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
, the binomial options pricing model (BOPM) provides a generalizable
numerical method
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.
Mathem ...
for the valuation of
options. Essentially, the model uses a "discrete-time" (
lattice based) model of the varying price over time of the
underlying
In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be use ...
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'' (), and formalized by
Cox
Cox may refer to:
* Cox (surname), including people with the name
Companies
* Cox Enterprises, a media and communications company
** Cox Communications, cable provider
** Cox Media Group, a company that owns television and radio stations
** ...
,
Ross Ross or ROSS may refer to:
People
* Clan Ross, a Highland Scottish clan
* Ross (name), including a list of people with the surname or given name Ross, as well as the meaning
* Earl of Ross, a peerage of Scotland
Places
* RoSS, the Republic of Sou ...
and
Rubinstein in 1979 and by Rendleman and Bartter in that same year.
For binomial trees as applied to
fixed income
Fixed income refers to any type of investment under which the borrower or issuer is obliged to make payments of a fixed amount on a fixed schedule. For example, the borrower may have to pay interest at a fixed rate once a year and repay the prin ...
and
interest rate derivatives see .
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
In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be use ...
over a period of time rather than a single point. As a consequence, it is used to value
American option In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These optionsâ ...
s that are exercisable at any time in a given interval as well as
Bermudan option In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These optionsâ ...
s that are exercisable at specific instances of time. Being relatively simple, the model is readily implementable in computer
software
Software is a set of computer programs and associated documentation and data. This is in contrast to hardware, from which the system is built and which actually performs the work.
At the lowest programming level, executable code consists ...
(including a
spreadsheet
A spreadsheet is a computer application for computation, organization, analysis and storage of data in tabular form. Spreadsheets were developed as computerized analogs of paper accounting worksheets. The program operates on data entered in cel ...
).
Although computationally slower than the
Black–Scholes formula, it is more accurate, particularly for longer-dated options on securities with
dividend
A dividend is a distribution of profits by a corporation to its shareholders. When a corporation earns a profit or surplus, it is able to pay a portion of the profit as a dividend to shareholders. Any amount not distributed is taken to be re-in ...
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 option
Real options valuation, also often termed real options analysis,Adam Borison (Stanford University)''Real Options Analysis: Where are the Emperor's Clothes?''
(ROV or ROA) applies option valuation techniques to capital budgeting decisions.Campbe ...
s) and for options with complicated features (e.g.,
Asian option An Asian option (or ''average value'' option) is a special type of option contract. For Asian options the payoff is determined by the average underlying price over some pre-set period of time. This is different from the case of the usual European op ...
s), binomial methods are less practical due to several difficulties, and
Monte Carlo option model In mathematical finance, a Monte Carlo option model uses Monte Carlo methodsAlthough the term 'Monte Carlo method' was coined by Stanislaw Ulam in the 1940s, some trace such methods to the 18th century French naturalist Buffon, and a question he as ...
s are commonly used instead. When simulating a small number of time steps
Monte Carlo simulation
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
will be more computationally time-consuming than BOPM (cf.
Monte Carlo methods in finance Monte Carlo methods are used in corporate finance and mathematical finance to value and analyze (complex) instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the dist ...
). However, the worst-case runtime of BOPM will be
O(2n), 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 simulation
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
s 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
![Arbre Binomial Options Reelles](https://upload.wikimedia.org/wikipedia/commons/2/2e/Arbre_Binomial_Options_Reelles.png)
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.
Step 1: Create 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
In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be use ...
will move up or down by a specific factor (
or
) per step of the tree (where, by definition,
and
). So, if
is the current price, then in the next period the price will either be
or
.
The up and down factors are calculated using the underlying
volatility,
, and the time duration of a step,
, measured in years (using the
day count convention In finance, a day count convention determines how interest accrues over time for a variety of investments, including bonds, notes, loans, mortgages, medium-term notes, swaps, and forward rate agreements (FRAs). This determines the number of days ...
of the underlying instrument). From the condition that the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
of the log of the price is
, we have:
:
:
Above is the original Cox, Ross, & Rubinstein (CRR) method; there are various other techniques for generating the lattice, such as "the equal probabilities" tree, see.
[Mark s. Joshi (2008)]
The Convergence of Binomial Trees for Pricing the American Put
/ref>
The CRR method ensures that the tree is recombinant, i.e. if the underlying asset moves up and then down (u,d), the price will be the same as if it had moved down and then up (d,u)—here the two paths merge or recombine. This property reduces the number of tree nodes, and thus accelerates the computation of the option price.
This property also allows the value of the underlying asset at each node to be calculated directly via formula, and does not require that the tree be built first. The node-value will be:
:
Where is the number of up ticks and is the number of down ticks.
Step 2: Find 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
In science and engineering, an intrinsic property is a property of a specified subject that exists itself or within the subject. An extrinsic property is not essential or inherent to the subject that is being characterized. For example, mass ...
, or exercise, value:
:, for a call option
In finance, a call option, often simply labeled a "call", is a contract between the buyer and the seller of the call option to exchange a security at a set price. The buyer of the call option has the right, but not the obligation, to buy an ...
:, for a put option
In finance, a put or put option is a derivative instrument in financial markets that gives the holder (i.e. the purchaser of the put option) the right to sell an asset (the ''underlying''), at a specified price (the ''strike''), by (or at) a s ...
,
Where is the strike price
In finance, the strike price (or exercise price) of an option is a fixed price at which the owner of the option can buy (in the case of a call), or sell (in the case of a put), the underlying security or commodity. The strike price may be set b ...
and is the spot price of the underlying asset at the period.
Step 3: Find option value at earlier nodes
Once the above step is complete, the option value is then 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.
In overview: the "binomial value" is found at each node, using the risk neutrality
In economics and finance, risk neutral preferences are preferences that are neither risk averse nor risk seeking. A risk neutral party's decisions are not affected by the degree of uncertainty in a set of outcomes, so a risk neutral party is indif ...
assumption; see Risk neutral valuation. If exercise is permitted at the node, then the model takes the greater of binomial and exercise value at the node.
The steps are as follows:
In calculating the value at the next time step calculated—i.e. one step closer to valuation—the model must use the value selected here, for "Option up"/"Option down" as appropriate, in the formula at the node.
The aside algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
demonstrates the approach computing the price of an American put option, although is easily generalized for calls and for European and Bermudan options:
Relationship with Black–Scholes
Similar assumptions underpin both the binomial model and the Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Blac ...
, and the binomial model thus provides a discrete time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
approximation
An approximation is anything that is intentionally similar but not exactly equality (mathematics), equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very ...
to the continuous process underlying the Black–Scholes model. The binomial model assumes that movements in the price follow a binomial distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...
; for many trials, this binomial distribution approaches the log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
assumed by Black–Scholes. In this case then, for European option In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These optionsâ ...
s without dividends, the binomial model value converges on the Black–Scholes formula value as the number of time steps increases.[Chance, Don M. March 200]
''A Synthesis of Binomial Option Pricing Models for Lognormally Distributed Assets''
. Journal of Applied Finance, Vol. 18
In addition, when analyzed as a numerical procedure, the CRR binomial method can be viewed as a special case
In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case is ...
of the explicit finite difference method for the Black–Scholes PDE; see finite difference methods for option pricing Finite difference methods for option pricing are numerical methods used in mathematical finance for the valuation of options. Finite difference methods were first applied to option pricing by Eduardo Schwartz in 1977.
In general, finite differ ...
.
See also
* Trinomial tree
The trinomial tree is a lattice-based computational model used in financial mathematics to price options. It was developed by Phelim Boyle in 1986. It is an extension of the binomial options pricing model, and is conceptually similar. It can also ...
, a similar model with three possible paths per node.
* Tree (data structure)
In computer science, a tree is a widely used abstract data type that represents a hierarchical tree structure with a set of connected nodes. Each node in the tree can be connected to many children (depending on the type of tree), but must be conn ...
* Lattice model (finance), for more general discussion and application to other underlyings
* Black–Scholes: binomial lattices are able to handle a variety of conditions for which Black–Scholes cannot be applied.
* Monte Carlo option model In mathematical finance, a Monte Carlo option model uses Monte Carlo methodsAlthough the term 'Monte Carlo method' was coined by Stanislaw Ulam in the 1940s, some trace such methods to the 18th century French naturalist Buffon, and a question he as ...
, used in the valuation of options with complicated features that make them difficult to value through other methods.
* Real options analysis
Real options valuation, also often termed real options analysis,Adam Borison ( Stanford University)''Real Options Analysis: Where are the Emperor's Clothes?''
(ROV or ROA) applies option valuation techniques to capital budgeting decisions.Campb ...
, where the BOPM is widely used.
* Quantum finance
Quantum finance is an interdisciplinary research field, applying theories and methods developed by quantum physicists and economists in order to solve problems in finance. It is a branch of econophysics.
Background on instrument pricing
Financ ...
, quantum binomial pricing model.
* Mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
, which has a list of related articles.
* , where the BOPM is widely used.
* Implied binomial tree
In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is r ...
* Edgeworth binomial tree
In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is ...
References
External links
The Binomial Model for Pricing Options
Prof. Thayer Watkins
Binomial Option Pricing
(PDF
Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ...
), Prof. Robert M. Conroy
Binomial Option Pricing Model
by Fiona Maclachlan, The Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
On the Irrelevance of Expected Stock Returns in the Pricing of Options in the Binomial Model: A Pedagogical Note
by Valeri Zakamouline
A Simple Derivation of Risk-Neutral Probability in the Binomial Option Pricing Model
by Greg Orosi
{{Derivatives market
Financial models
Options (finance)
Mathematical finance
Models of computation
Trees (data structures)