{{Short description|Numerical method in mathematical finance}} '''Finite difference methods for option pricing''' are numerical methods used in mathematical finance for the valuation of options.<ref name="JHull">{{cite book |last=Hull |first=John C. |authorlink=John C. Hull (economist) |edition=5th |title=Options, Futures and Other Derivatives |year=2002 |publisher=Prentice Hall |isbn=978-0-13-009056-0 |url=https://archive.org/details/optionsfuturesot00hull_1 }}</ref> Finite difference methods were first applied to option pricing by Eduardo Schwartz in 1977.<ref name="Schwartz">{{cite journal |last=Schwartz |first=E. |authorlink=Eduardo Schwartz |date=January 1977 |title=The Valuation of Warrants: Implementing a New Approach|journal=Journal of Financial Economics |volume=4 |pages=79–94 |url=http://ideas.repec.org/a/eee/jfinec/v4y1977i1p79-93.html |doi=10.1016/0304-405X(77)90037-X |url-access=subscription }}</ref><ref name="Boyle"> {{cite book |last=Boyle |first=Phelim |author-link=Phelim Boyle|author2=Feidhlim Boyle |title=Derivatives: The Tools That Changed Finance |year=2001| publisher=Risk Publications |isbn=978-1899332885}}</ref>{{rp|180}}
In general, finite difference methods are used to price options by approximating the (continuous-time) differential equation that describes how an option price evolves over time by a set of (discrete-time) difference equations. The discrete difference equations may then be solved iteratively to calculate a price for the option.<ref name="Goddard"> Phil Goddard (N.D.). [http://www.goddardconsulting.ca/option-pricing-finite-diff-index.html ''Option Pricing – Finite Difference Methods'']</ref> 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 the Black–Scholes PDE. Once in this form, a finite difference model can be derived, and the valuation obtained.<ref name="Schwartz" />
The approach can be used to solve derivative pricing problems that have, in general, the same level of complexity as those problems solved by tree approaches.<ref name="JHull" />
==Method== As above, the PDE is expressed in a discretized form, using finite differences, and the evolution in the option price is then modelled using a lattice with corresponding dimensions: 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 then valued as follows:<ref>{{cite book |title=The Mathematics of Financial Derivatives: A Student Introduction |author1=Wilmott, P. |authorlink1=Paul Wilmott |author2=Howison, S. |author3=Dewynne, J. |publisher=Cambridge University Press |year=1995 |isbn=978-0-521-49789-3 |url=https://archive.org/details/mathematicsoffin00wilm }}</ref> # Maturity values are simply the difference between the exercise price of the option and the value of the underlying at each point (for a call, e.g., <math> C(S, T) =max\{S - K, 0\}</math>). # Values at the boundaries – i.e. at each earlier time where spot is at its highest or zero – are set based on moneyness or arbitrage bounds on option prices (for a call, <math>C(0, t) = 0</math> for all t and <math>C(S, t) = S - K e^{-r(T-t)}</math> as <math>S \rightarrow \infty</math>). # Values at other lattice points are calculated recursively (iteratively), 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); ::* the value at each point is then found using the technique in question; working backwards in time from maturity, and inwards from the boundary prices. :4. The value of the option today, where the underlying is at its spot price, (or at any time/price combination,) is then found by interpolation.
==Application== As above, these methods can solve derivative pricing problems that have, in general, the same level of complexity as those problems solved by tree approaches,<ref name="JHull" /> but, given their relative complexity, are usually employed only when other approaches are inappropriate; an example here, being changing interest rates and / or time linked dividend policy. 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. <ref name="Boyle" />{{rp|182}} Note that, when standard assumptions are applied, the explicit technique encompasses the binomial- and trinomial tree methods.<ref>{{cite journal |last=Brennan |first=M. |authorlink=Michael Brennan (finance) |author2=Schwartz, E. |authorlink2=Eduardo Schwartz |date=September 1978 |title=Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis |journal=Journal of Financial and Quantitative Analysis|volume=13|issue=3|pages=461–474 |jstor=2330152 |doi=10.2307/2330152 |s2cid=250121477}}</ref> Tree based methods, then, suitably parameterized, are a special case of the explicit finite difference method.<ref>{{cite journal |last=Rubinstein |first=M. |authorlink=Mark Rubinstein |year=2000 |title=On the Relation Between Binomial and Trinomial Option Pricing Models |journal=Journal of Derivatives |volume=8 |issue=2 |pages=47–50 |url=http://www.in-the-money.com/pages/author.htm |doi=10.3905/jod.2000.319149 |url-status=dead |archive-url=https://web.archive.org/web/20070622150346/http://www.in-the-money.com/pages/author.htm |archive-date=June 22, 2007 |citeseerx=10.1.1.43.5394 |s2cid=11743572 }}</ref>
==References== {{reflist}}
==External links== <!-- alphabetical by author --> * [http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN97-02.pdf Option Pricing Using Finite Difference Methods] {{Webarchive|url=https://web.archive.org/web/20100720231957/http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN97-02.pdf |date=2010-07-20 }}, Prof. Don M. Chance, Louisiana State University * [https://www.cs.cornell.edu/Info/Courses/Spring-98/CS522/content/lab4.pdf Finite Difference Approach to Option Pricing] (includes Matlab Code); [https://www.cs.cornell.edu/Info/Courses/Spring-98/CS522/content/lecture2.math.pdf Numerical Solution of Black–Scholes Equation], Tom Coleman, Cornell University * [http://www.goddardconsulting.ca/option-pricing-finite-diff-index.html Option Pricing – Finite Difference Methods], Dr. Phil Goddard * [https://www.sfu.ca/~rjones/bus864/notes/notes2.pdf Numerically Solving PDE’s: Crank-Nicolson Algorithm], Prof. R. Jones, Simon Fraser University * [http://www.math.ust.hk/~maykwok/courses/ma571/06_07/Kwok_Chap_6.pdf Numerical Schemes for Pricing Options], Prof. Yue Kuen Kwok, Hong Kong University of Science and Technology * [https://web.archive.org/web/20110719120702/http://mit.econ.au.dk/vip_htm/cmunk/noter/PDENOTE.pdf Introduction to the Numerical Solution of Partial Differential Equations in Finance], Claus Munk, University of Aarhus * [http://users.aims.ac.za/~bundi/Thesis.pdf Numerical Methods for the Valuation of Financial Derivatives] {{Webarchive|url=https://web.archive.org/web/20111005044000/http://users.aims.ac.za/~bundi/Thesis.pdf |date=2011-10-05 }}, D.B. Ntwiga, University of the Western Cape * [https://web.archive.org/web/20060821114429/http://www.puc-rio.br/marco.ind/katia-num.html#finite-differences The Finite Difference Method], Katia Rocha, Instituto de Pesquisa Econômica Aplicada * [https://web.archive.org/web/20100813163124/http://janroman.dhis.org/doc/lnotes/AFI/Finite%20Difference%20and%20MC.pdf Analytical Finance: Finite difference methods], Jan Röman, Mälardalen University
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Category:Mathematical finance Category:Options (finance) Category:Numerical differential equations