Operator splitting methods for pricing American options under stochastic volatility
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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.
Mathematics Subject Classification (2000)35K85 65M06 65M55 65Y20 91B28
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- 8.Clarke, N., Parrott, K.: The multigrid solution of two-factor American put options. Technical Report 96-16, Oxford Comp. Lab, Oxford (1996)Google Scholar
- 10.Coleman T.F., Li Y., Verma A.: A Newton method for American option pricing. J. Comput. Finance 5, 51–78 (2002)Google Scholar
- 11.Duffie D.: Dynamic Asset Pricing Theory, 2nd edn. Princeton University Press, Princeton (1996)Google Scholar
- 14.Giles M.B., Carter R.: Convergence analysis of Crank–Nicolson and Rannacher time-marching. J. Comput. Finance 9, 89–112 (2006)Google Scholar
- 15.Glowinski R.: Finite element methods for incompressible viscous flow. Handbook of Numerical Analysis, vol. IX. North-Holland, Amsterdam (2003)Google Scholar
- 16.Hackbusch W.: Multigrid methods and applications. Springer Series in Computational Mathematics, vol. 4. Springer, Berlin (1985)Google Scholar
- 18.Huang J., Pang J.-S.: Option pricing and linear complementarity. J. Comput. Finance 2, 31–60 (1998)Google Scholar
- 20.Hull J.C.: Options, Futures, and Other Derivatives, 3rd edn. Prentice Hall, Upper Saddle River (1997)Google Scholar
- 26.Pooley D.M., Vetzal K.R., Forsyth P.A.: Convergence remedies for non-smooth payoffs in option pricing. J. Comput. Finance 6, 25–40 (2003)Google Scholar
- 31.Zvan R., Forsyth P.A., Vetzal K.R.: Negative coefficients in two factor option pricing models. J. Comput. Finance 7, 37–73 (2003)Google Scholar