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Operator splitting methods for pricing American options under stochastic volatility. (English) Zbl 1204.91126

Summary: We consider the numerical pricing of American options under Heston’s stochastic volatility model. The price is given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. We propose operator splitting methods for performing time stepping after a finite difference space discretization. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank-Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting methods have comparable discretization errors. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
91G60 Numerical methods (including Monte Carlo methods)
65K15 Numerical methods for variational inequalities and related problems
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators

Software:

Wesseling
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References:

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