# 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

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 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:
1. 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);
2. the value at each point is then found using the technique in question.

## 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,[1] but, given their relative complexity, are 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. [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

1. ^ a b c Hull, John C. (2002). Options, Futures and Other Derivatives (5th ed.). Prentice Hall. ISBN 0-13-009056-5.
2. ^ 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.
3. ^ a b Boyle, Phelim; Feidhlim Boyle (2001). Derivatives: The Tools That Changed Finance. Risk Publications. ISBN 189933288X.
4. ^ Phil Goddard (N.D.). Option Pricing - Finite Difference Methods
5. ^ Wilmott, P.; Howison, S.; Dewynne, J. (1995). The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press. ISBN 0-521-49789-2.
6. ^ 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 (University of Washington School of Business Administration) 13 (3): 461–474. JSTOR 2330152. doi:10.2307/2330152.
7. ^ Rubinstein, M. (2000). "On the Relation Between Binomial and Trinomial Option Pricing Models". Journal of Derivatives 8 (2): 47–50. doi:10.3905/jod.2000.319149.