Finite difference methods for option pricing
Encyclopedia
Finite difference methods for option pricing are numerical methods
used in mathematical finance
for the valuation of options
. Finite difference method
s were first applied to option pricing
by Eduardo Schwartz
in 1977.
Finite difference methods can solve derivative pricing problems that have, in general, the same level of complexity as those problems solved by tree approaches
, and are therefore usually employed only when other approaches are inappropriate. At the same time, like tree-based methods, this approach is limited in terms of the number of underlying variables, and for problems with multiple dimensions, Monte Carlo methods for option pricing are usually preferred.
The approach arises since the evolution of the option value can be modelled via a partial differential equation
(PDE), as a function
of (at least) time and price of underlying; see for example Black–Scholes PDE. Once in this form, a finite difference model can be derived, and the valuation obtained. Here, essentially, the PDE is expressed in a discretized form, using finite difference
s, and the evolution in the option price is then modelled using a lattice with corresponding dimension
s; time runs from 0 to maturity and price runs from 0 to a "high" value, such that the option is deeply in or out of the money
.
The option is valued as follows:
As above, these methods and tree-based methods are able to handle problems which are equivalent in complexity. In fact, when standard assumptions are applied it can be shown that the explicit technique encompasses the binomial and trinomial tree
methods. Tree based methods, then, suitably parameterized, are a special case
of the explicit finite difference method.
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
used in mathematical finance
Mathematical finance
Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical...
for the valuation of options
Option (finance)
In finance, an option is a derivative financial instrument that specifies a contract between two parties for a future transaction on an asset at a reference price. The buyer of the option gains the right, but not the obligation, to engage in that transaction, while the seller incurs the...
. Finite difference method
Finite difference method
In mathematics, finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.- Derivation from Taylor's polynomial :...
s were first applied to option pricing
Valuation of options
In finance, a price is paid or received for purchasing or selling options. This price can be split into two components.These are:* Intrinsic Value* Time Value-Intrinsic Value:...
by Eduardo Schwartz
Eduardo Schwartz
Eduardo Saul Schwartz is a professor of finance at the Anderson School of Management, University of California, Los Angeles, where he holds the California Chair in Real Estate & Land Economics...
in 1977.
Finite difference methods can solve derivative pricing problems that have, in general, the same level of complexity as those problems solved by tree approaches
Binomial options pricing model
In finance, the binomial options pricing model provides a generalizable numerical method for the valuation of options. The binomial model was first proposed by Cox, Ross and Rubinstein in 1979. Essentially, the model uses a “discrete-time” model of the varying price over time of the underlying...
, and are therefore usually employed only when other approaches are inappropriate. At the same time, like tree-based methods, this approach is limited in terms of the number of underlying variables, and for problems with multiple dimensions, Monte Carlo methods for option pricing are usually preferred.
The approach arises since the evolution of the option value can be modelled via a partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
(PDE), as a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of (at least) time and price of underlying; see for example Black–Scholes PDE. Once in this form, a finite difference model can be derived, and the valuation obtained. Here, essentially, the PDE is expressed in a discretized form, using finite difference
Finite difference
A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
s, and the evolution in the option price is then modelled using a lattice with corresponding dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
s; time runs from 0 to maturity and price runs from 0 to a "high" value, such that the option is deeply in or out of the money
Moneyness
In finance, moneyness is a measure of the degree to which a derivative is likely to have positive monetary value at its expiration, in the risk-neutral measure. It can be measured in percentage probability, or in standard deviations....
.
The option is valued as follows:
- Maturity values are simply the difference between the exercise price of the option and the value of the underlying at each point.
- Values at the boundary prices are set based on moneynessMoneynessIn finance, moneyness is a measure of the degree to which a derivative is likely to have positive monetary value at its expiration, in the risk-neutral measure. It can be measured in percentage probability, or in standard deviations....
or arbitrage bounds on option prices.
- Values at other lattice points are calculated recursivelyRecursionRecursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...
, starting at the time step preceding maturity and ending at time = 0. Here, using a technique such as Crank–Nicolson or the explicit method:
- the PDE is discretized per the technique chosen, such that the value at each lattice point is specified as a function of the value at later and adjacent points; see Stencil (numerical analysis)Stencil (numerical analysis)In mathematics, especially the areas of numerical analysis concentrating on the numerical solution of partial differential equations, a stencil is a geometric arrangement of a nodal group that relate to the point of interest by using a numerical approximation routine. Stencils are the basis for...
; - the value at each point is then found using the technique in question.
- The value of the option today, where the underlyingUnderlyingIn finance, the underlying of a derivative is an asset, basket of assets, index, or even another derivative, such that the cash flows of the derivative depend on the value of this underlying...
is at its spot priceSpot priceThe spot price or spot rate of a commodity, a security or a currency is the price that is quoted for immediate settlement . Spot settlement is normally one or two business days from trade date...
, (or at any time/price combination,) is then found by interpolationInterpolationIn the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
.
As above, these methods and tree-based methods are able to handle problems which are equivalent in complexity. In fact, when standard assumptions are applied it can be shown that the explicit technique encompasses the binomial and trinomial tree
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...
methods. Tree based methods, then, suitably parameterized, are a special case
Special case
In logic, especially as applied in mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B, or equivalently, B is a generalization of A. For example, all circles are ellipses ; therefore the circle is a special case of the...
of the explicit finite difference method.
External links
- Prof. Don M. Chance, Louisiana State UniversityLouisiana State UniversityLouisiana State University and Agricultural and Mechanical College, most often referred to as Louisiana State University, or LSU, is a public coeducational university located in Baton Rouge, Louisiana. The University was founded in 1853 in what is now known as Pineville, Louisiana, under the name...
: Option Pricing Using Finite Difference Methods - Tom Coleman, Cornell UniversityCornell UniversityCornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
: Finite Difference Approach to Option Pricing (includes MatlabMATLABMATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...
Code); Numerical Solution of Black–Scholes Equation - Prof. R. Jones, Simon Fraser UniversitySimon Fraser UniversitySimon Fraser University is a Canadian public research university in British Columbia with its main campus on Burnaby Mountain in Burnaby, and satellite campuses in Vancouver and Surrey. The main campus in Burnaby, located from downtown Vancouver, was established in 1965 and has more than 34,000...
: Numerically Solving PDE’s: Crank-Nicholson Algorithm - Jouni Kerman, Courant Institute of Mathematical SciencesCourant Institute of Mathematical SciencesThe Courant Institute of Mathematical Sciences is an independent division of New York University under the Faculty of Arts & Science that serves as a center for research and advanced training in computer science and mathematics...
, New York UniversityNew York UniversityNew York University is a private, nonsectarian research university based in New York City. NYU's main campus is situated in the Greenwich Village section of Manhattan...
: Numerical Methods for Option Pricing: Binomial and Finite-difference Approximations - Claus Munk, University of AarhusUniversity of AarhusAarhus University , located in the city of Aarhus, Denmark, is Denmark's second oldest and second largest university...
: Introduction to the Numerical Solution of Partial Differential Equations in Finance - D.B. Ntwiga, University of the Western CapeUniversity of the Western CapeThe University of the Western Cape is a public university located in the Bellville suburb of Cape Town, South Africa. It was established in 1960 by the South African government as a university for Coloured people only...
: Numerical Methods for the Valuation of Financial Derivatives - Katia Rocha, Instituto de Pesquisa Econômica Aplicada: The Finite Difference Method