[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.
[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; an example here, being changing interest rates and / or time linked dividend policy.
[3]: 182 Note that, when standard assumptions are applied, the explicit technique encompasses the binomial- and trinomial tree methods.