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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]}
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]}
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]}
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