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A Robust Spectral Method for Solving Heston’s Model

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Abstract

In this paper, we consider the Heston’s volatility model (Heston in Rev. Financ. Stud. 6: 327–343, 1993]. We simulate this model using a combination of the spectral collocation method and the Laplace transforms method. To approximate the two dimensional PDE, we construct a grid which is the tensor product of the two grids, each of which is based on the Chebyshev points in the two spacial directions. The resulting semi-discrete problem is then solved by applying the Laplace transform method based on Talbot’s idea of deformation of the contour integral (Talbot in IMA J. Appl. Math. 23(1): 97–120, 1979).

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References

  1. Hull, J.C., White, A.: The pricing of options on assets with stochastic volatility. J. Finance 42(2), 281–300 (1987)

    Article  Google Scholar 

  2. Merton, R.C.: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125–144 (1975)

    Article  Google Scholar 

  3. Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327–343 (1993)

    Article  Google Scholar 

  4. In’t Hout, K.J., Foulon, S.: ADI finite difference schemes for option pricing in the Heston model with correlation. Int. J. Numer. Anal. Model. 7(2), 303–320 (2010)

    MathSciNet  Google Scholar 

  5. Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  6. McKee, S., Wall, D.P., Wilson, S.K.: An alternating direction implicit scheme for parabolic equations with mixed derivative and convective terms. J. Comp. Physiol. 126(1), 64–76 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Craig, I.J.D., Sneyd, A.D.: An alternating-direction implicit scheme for parabolic equations with mixed derivatives. Comput. Math. Appl. 16, 341–350 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  8. In’t Hout, K.J., Welfert, B.D.: Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative. Appl. Numer. Math. 59(3–4), 677–692 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hundsdorfer, W.: Accuracy and stability of splitting with stabilizing corrections. Appl. Numer. Math. 42, 213–233 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hundsdorfer, W., Verwer, J.G.: Numerical Solution of Time-Dependent Advection-Diffusion–Reaction Equations. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  11. Haentjens, T.: ADI finite difference discretization of the Heston–Hull–White PDE. In: AIP Conference Proceedings, ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010, vol. 1281, pp. 1995–1999 (2010)

    Google Scholar 

  12. Ikonen, S., Toivanen, J.: Operator splitting methods for pricing American options under stochastic volatility. Numer. Math. 113(2), 299–324 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Oosterlee, C.W.: On multigrid for linear complementarity problems with application to American-style options. Electron. Trans. Numer. Anal. 15, 165–185 (2003)

    MATH  MathSciNet  Google Scholar 

  14. Zvan, R., Forsyth, P.A., Vetzal, K.R.: Penalty methods for American option with stochastic volatility. J. Comput. Appl. Math. 91(2), 199–218 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Zhu, S.P., Chen, W.T.: A new analytical approximation for European puts with stochastic volatility. Appl. Math. Lett. 23, 687–692 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. In’t Hout, K.J., Weideman, J.A.C.: Appraisal of a contour integral method for the Black-Scholes and Heston equations. SIAM J. Sci. Comput. 33, 763–785 (2010)

    Article  MathSciNet  Google Scholar 

  17. Albrecher, H., Mayer, P., Schoutens, W., Tistaert, J.: The little Heston trap. Wilmott Mag. January, 83–92 (2007)

    Google Scholar 

  18. Carr, P., Madan, D.: Option valuation using the Fast Fourier Transform. J. Comput. Finance 2(4), 61–73 (1999)

    Google Scholar 

  19. Clarke, N., Parrott, K.: Multigrid for American option pricing with stochastic volatility. Appl. Math. Finance 6, 177–195 (1999)

    Article  MATH  Google Scholar 

  20. Gatheral, J.: The Volatility Surface: A Practioner’s Guide. Wiley Finance, New York (2006)

    Google Scholar 

  21. Grzelak, L.A., Oosterlee, C.W.: On the Heston model with stochastic interest rates. SIAM J. Financ. Math. 2, 255–286 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kahl, C., Jäckel, P.: Not-so-complex logarithms in the Heston model. Wilmott Mag. September, 94–103 (2005)

    Google Scholar 

  23. Kim, J., Kim, B., Moon, K.-S., Wee, I.-S.: Valuation of power options under Heston’s stochastic volatility model. J. Econ. Dyn. Control 36(11), 1796–1813 (2012)

    Article  MathSciNet  Google Scholar 

  24. Langrock, R., MacDonald, I.L., Zucchini, W.: Some nonstandard stochastic volatility models and their estimation using structured hidden Markov models. J. Empir. Finance 19(1), 147–161 (2012)

    Article  Google Scholar 

  25. Laurent, S., Rombouts, J.V.K., Violante, F.: On loss functions and ranking forecasting performances of multivariate volatility models. J. Economet. 173(1), 1–10 (2013)

    Article  MathSciNet  Google Scholar 

  26. Schneider, C., Werner, W.: Some new aspect of rational interpolation. Math. Comput. 47(175), 285–299 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  27. Tee, W.: A rational collocation with adaptively transformed Chebyshev grid points. SIAM J. Sci. Comput. 28(5), 8–1811 (2006)

    Article  MathSciNet  Google Scholar 

  28. Heinrichs, W.: Improved condition number of spectral methods. Math. Comput. 53(187), 103–119 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  29. Julien, K., Watson, M.: Efficient multi-dimensional solution of PDEs using Chebyshev spectral methods. J. Comp. Physiol. 228(5), 1480–1503 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  30. Trefethen, L.N.: Spectral Method in MATLAB. SIAM, Philadelphia (2000)

    Book  Google Scholar 

  31. Talbot, A.: The accurate numerical inversion of Laplace transforms. IMA J. Appl. Math. 23(1), 97–120 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  32. Spiegel, M.R.: Theory and Problems of Laplace Transforms. McGraw-Hill, New York (1965)

    Google Scholar 

  33. Weideman, J.A.C., Trefethen, L.N.: Parabolic and hyperbolic contours for computing the Bromwich integral. Math. Comput. 76(259), 1341–1356 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  34. López-Fernández, M., Palencia, C.: On the numerical inversion of the Laplace transform of certain holomorphic mappings. Appl. Numer. Math. 51(2–3), 289–303 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  35. Martensen, V.E.: Zur numerischen Auswertung uneigentlicher Integrale. Z. Angew. Math. Mech. 48, T83–T85 (1968)

    MATH  MathSciNet  Google Scholar 

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Acknowledgements

The two authors, Ngounda and Pindza, acknowledge the Agence National des Bourses du Gabon for the financial support. Patidar’s research was supported by the South African National Research Foundation. Furthermore, we acknowledge the anonyms referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville, 7535, South Africa

    E. Ngounda, K. C. Patidar & E. Pindza

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  1. E. Ngounda
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  2. K. C. Patidar
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  3. E. Pindza
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Correspondence to K. C. Patidar.

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Ngounda, E., Patidar, K.C. & Pindza, E. A Robust Spectral Method for Solving Heston’s Model. J Optim Theory Appl 161, 164–178 (2014). https://doi.org/10.1007/s10957-013-0284-x

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  • DOI: https://doi.org/10.1007/s10957-013-0284-x

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