Numerische Mathematik

, Volume 113, Issue 2, pp 299–324 | Cite as

Operator splitting methods for pricing American options under stochastic volatility

  • Samuli Ikonen
  • Jari ToivanenEmail author


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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  1. 1.
    Ball C.A., Roma A.: Stochastic volatility option pricing. J. Financ. Quant. Anal. 29, 589–607 (1994)CrossRefGoogle Scholar
  2. 2.
    Black F., Scholes M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)CrossRefGoogle Scholar
  3. 3.
    Brandt A., Cryer C.W.: Multigrid algorithms for the solution of linear complementarity problems arising from free boundary problems. SIAM Sci. Stat. Comput. 4, 655–684 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brennan M.J., Schwartz E.S.: The valuation of American put options. J. Finance 32, 449–462 (1977)CrossRefGoogle Scholar
  5. 5.
    Briggs W.L., Henson V.E., McCormick S.F.: A Multigrid Tutorial, 2nd edn. SIAM, Philadelphia (2000)zbMATHGoogle Scholar
  6. 6.
    Broadie M., Chernov M., Johannes M.: Model specification and risk premia: evidence from futures options. J. Finance 62, 1453–1490 (2007)CrossRefGoogle Scholar
  7. 7.
    Cash J.R.: Two new finite difference schemes for parabolic equations. SIAM J. Numer. Anal. 21, 433–446 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 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
  9. 9.
    Clarke N., Parrott K.: Multigrid for American option pricing with stochastic volatility. Appl. Math. Finance 6, 177–195 (1999)zbMATHCrossRefGoogle Scholar
  10. 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. 11.
    Duffie D.: Dynamic Asset Pricing Theory, 2nd edn. Princeton University Press, Princeton (1996)Google Scholar
  12. 12.
    Forsyth P.A., Vetzal K.R.: Quadratic convergence for valuing American options using a penalty method. SIAM J. Sci. Comput. 23, 2095–2122 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Fouque J.-P., Papanicolaou G., Sircar K.R.: Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  14. 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. 15.
    Glowinski R.: Finite element methods for incompressible viscous flow. Handbook of Numerical Analysis, vol. IX. North-Holland, Amsterdam (2003)Google Scholar
  16. 16.
    Hackbusch W.: Multigrid methods and applications. Springer Series in Computational Mathematics, vol. 4. Springer, Berlin (1985)Google Scholar
  17. 17.
    Heston S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Finance Stud. 6, 327–343 (1993)CrossRefGoogle Scholar
  18. 18.
    Huang J., Pang J.-S.: Option pricing and linear complementarity. J. Comput. Finance 2, 31–60 (1998)Google Scholar
  19. 19.
    Hull J., White A.: The pricing of options on assets with stochastic volatilities. J. Finance 42, 281–300 (1987)CrossRefGoogle Scholar
  20. 20.
    Hull J.C.: Options, Futures, and Other Derivatives, 3rd edn. Prentice Hall, Upper Saddle River (1997)Google Scholar
  21. 21.
    Ikonen S., Toivanen J.: Operator splitting methods for American option pricing. Appl. Math. Lett. 17, 809–814 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ikonen S., Toivanen J.: Efficient numerical methods for pricing American options under stochastic volatility. Numer. Methods Partial Differ. Equ. 24, 104–126 (2007)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Lions P.-L., Mercier B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Mitchell A.R., Griffiths D.F.: The Finite Difference Method in Partial Differential Equations. Wiley, Chichester (1980)zbMATHGoogle Scholar
  25. 25.
    Oosterlee C.W.: On multigrid for linear complementarity problems with application to American-style options. Electron. Trans. Numer. Anal. 15, 165–185 (2003)zbMATHMathSciNetGoogle Scholar
  26. 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
  27. 27.
    Trottenberg U., Oosterlee C.W., Schüller A.: Multigrid. Academic Press Inc., San Diego (2001)zbMATHGoogle Scholar
  28. 28.
    Wesseling P.: An Introduction to Multigrid Methods. Wiley, Chichester (1992)zbMATHGoogle Scholar
  29. 29.
    Wilmott P., Howison S., Dewynne J.: The Mathematics of Financial Derivatives. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  30. 30.
    Zvan R., Forsyth P.A., Vetzal K.R.: Penalty methods for American options with stochastic volatility. J. Comput. Appl. Math. 91, 199–218 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 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

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Nordea MarketsNordeaFinland
  2. 2.Department of Mathematical Information Technology, Agora40014 University of JyväskyläJyväskyläFinland

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