Finite difference methods for option pricing
Finite difference methods for option pricing are numerical methods used in mathematical finance for the valuation of options.^{[1]} Finite difference methods were first applied to option pricing by Eduardo Schwartz in 1977.^{[2]}^{[3]}^{: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.^{[4]} 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.^{[2]}
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.^{[1]}
Method[edit]
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:^{[5]}
- 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 moneyness or arbitrage bounds on option prices.
- 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.
- 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[edit]
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,^{[1]} 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. ^{[3]}^{:182} Note that, when standard assumptions are applied, the explicit technique encompasses the binomial- and trinomial tree methods.^{[6]} Tree based methods, then, suitably parameterized, are a special case of the explicit finite difference method.^{[7]}
References[edit]
- ^ ^{a} ^{b} ^{c} Hull, John C. (2002). Options, Futures and Other Derivatives (5th ed.). Prentice Hall. ISBN 978-0-13-009056-0.
- ^ ^{a} ^{b} Schwartz, E. (January 1977). "The Valuation of Warrants: Implementing a New Approach". Journal of Financial Economics. 4: 79–94. doi:10.1016/0304-405X(77)90037-X.
- ^ ^{a} ^{b} Boyle, Phelim; Feidhlim Boyle (2001). Derivatives: The Tools That Changed Finance. Risk Publications. ISBN 978-1899332885.
- ^ Phil Goddard (N.D.). Option Pricing – Finite Difference Methods
- ^ Wilmott, P.; Howison, S.; Dewynne, J. (1995). The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press. ISBN 978-0-521-49789-3.
- ^ Brennan, M.; Schwartz, E. (September 1978). "Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis". Journal of Financial and Quantitative Analysis. 13 (3): 461–474. doi:10.2307/2330152. JSTOR 2330152.
- ^ Rubinstein, M. (2000). "On the Relation Between Binomial and Trinomial Option Pricing Models". Journal of Derivatives. 8 (2): 47–50. CiteSeerX 10.1.1.43.5394. doi:10.3905/jod.2000.319149. Archived from the original on June 22, 2007.
External links[edit]
- Option Pricing Using Finite Difference Methods, Prof. Don M. Chance, Louisiana State University
- Finite Difference Approach to Option Pricing (includes Matlab Code); Numerical Solution of Black–Scholes Equation, Tom Coleman, Cornell University
- Option Pricing – Finite Difference Methods, Dr. Phil Goddard
- Numerically Solving PDE’s: Crank-Nicolson Algorithm, Prof. R. Jones, Simon Fraser University
- Numerical Schemes for Pricing Options, Prof. Yue Kuen Kwok, Hong Kong University of Science and Technology
- Introduction to the Numerical Solution of Partial Differential Equations in Finance, Claus Munk, University of Aarhus
- Numerical Methods for the Valuation of Financial Derivatives, D.B. Ntwiga, University of the Western Cape
- The Finite Difference Method, Katia Rocha, Instituto de Pesquisa Econômica Aplicada
- Analytical Finance: Finite difference methods, Jan Röman, Mälardalen University