Finite difference methods for option pricing

Finite difference methods for option pricing are a numerical method 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]

Finite difference methods can solve derivative pricing problems that have, in general, the same level of complexity as those problems solved by tree approaches, ^[1] and are therefore 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.

The approach is based on the fact that the evolution of the option value can be modelled using a partial differential equation (see for example Black–Scholes PDE). Once in this form, a finite difference model can be derived, and the valuation obtained .^[2] Here, essentially, the PDE is expressed in a discretized form, using finite differences, and the evolution in the option price is then modelled recursively using a lattice with dimensions (of at least) {time; price of underlying}, where time runs from 0 to maturity and price runs from 0 to a "high" value, such that the option is deeply in- or out-the-money.

The option is valued as follows: ^[3]

• Maturity values are simply the difference between the exercise price of the option and the assumed ending value of the underlying instrument 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, 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 formula is discretized per the technique, 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.

• 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.

As above, these methods and tree-based methods are able to handle problems which are equivalent in complexity. In fact, when standard assumptions are applied it can be shown that the explicit technique encompasses the binomial and trinomial tree methods. ^[4] Tree based methods, then, suitably parameterized, are a special case of the explicit finite difference method. ^[5]

References

External links

• 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
• Numerical Methods for Option Pricing: Binomial and Finite-difference Approximations, Jouni Kerman Courant Institute of Mathematical Sciences, New York University
• The Finite Difference Method, Katia Rocha Instituto de Pesquisa Econômica Aplicada