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A penalty method for American options with jump diffusion processes

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Summary.

The fair price for an American option where the underlying asset follows a jump diffusion process can be formulated as a partial integral differential linear complementarity problem. We develop an implicit discretization method for pricing such American options. The jump diffusion correlation integral term is computed using an iterative method coupled with an FFT while the American constraint is imposed by using a penalty method. We derive sufficient conditions for global convergence of the discrete penalized equations at each timestep. Finally, we present numerical tests which illustrate such convergence.

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References

  1. Amadori, A.L.: The obstacle problem for nonlinear integro-differential equations arising in option pricing. Working paper, Istituto pre le Applicazione del Calcolo ‘‘M. Picone’’, Rome, www.iac.rm.cnr.it/∼amadori

  2. Amin, K.: Jump diffusion option valuation in discrete time. J. Finance 48, 1833–1863 (1993)

    Google Scholar 

  3. Andersen, L., Andreasen, J.: Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing. Rev. Derivatives Res. 4, 231–262 (2000)

    Article  Google Scholar 

  4. Ayache, E., Forsyth, P.A., Vetzal, K.R.: Next generation models for convertible bonds with credit risk. Wilmott Magazine, December 2002, pp. 68–77

  5. Barles, G.: Convergence of numerical schemes for degenerate parabolic equations arising in finance. In L.C.G. Rogers and D. Talay, (eds.), Numerical Methods in Finance, Cambridge University Press, Cambridge, 1997, pp. 1–21

  6. Briani, M., La Chioma, C., Natalini, R.: Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory. Working paper, Istituto pre le Applicazione del Calcolo ‘‘M. Picone’’, Rome, www.iac.rm.cnr.it/∼natalini, to appear in Numerische Mathematik

  7. Broadie, M., Yamamoto, Y.: Application of the Fast Gauss transform to option pricing. Working paper, Columbia School of Business, 2002

  8. Coleman, T.F., Li, Y., Verma, A.: Reconstructing the unknown local volatility function. J. Comput. Finance 2, 77–102 (1999)

    Google Scholar 

  9. Cottle, R.W., Pang, J.-S., Stone, R.E: The Linear Complementarity Problem. Academic Press, 1992

  10. Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bulletin of the Am. Math. Soc. 27, 1–67 July 1992

    Google Scholar 

  11. Cryer, C.W.: The efficient solution of linear complementarity problems for tridiagonal Minkowski matrices. ACM Trans. Math. Softw. 9, 199–214 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. d’Halluin, Y., Forsyth, P.A., Vetzal, K.R.: Robust numerical methods for contingent claims under jump diffusion processes. www.scicom.uwaterloo.ca/∼paforsyt/jump.pdf, submitted to IMA J. Numer. Anal.

  13. d’Halluin, Y., Forsyth, P.A., Vetzal, K.R., Labahn, G.: A numerical PDE approach for pricing callable bonds. Appl. Math. Finance 8, 49–77 (2001)

    Article  MATH  Google Scholar 

  14. Dutt, A., Rokhlin, V.: Fast Fourier transforms for nonequispaced data. SIAM J. Sci. Comput. 14, 1368–1393 November 1993

    Google Scholar 

  15. Elliot, E.M., Ockendon, J.R.: Weak and Variational Methods for Moving Boundary Problems. Pitman, 1982

  16. Forsyth, P.A., Vetzal, K.R.: Quadratic convergence of a penalty method for valuing American options. SIAM J. Sci. Comput. 23, 2096–2123 (2002)

    Google Scholar 

  17. Greengard, L., Strain, J.: The fast Gauss transform. SIAM J. Sci. Comput. 12, 79–94 (1991)

    Google Scholar 

  18. Hull, J.: Options, Futures, and Other Derivatives. Prentice Hall, Inc., Upper Saddle River, NJ, 3rd edition, 1997

  19. Johnson, C.: Numerical Solutions of Partial Differential Equations By the Finite Element Method. Cambridge University Press, Cambridge, 1987

  20. Kangro, R., Nicolaides, R.: Far field boundary conditions for Black-Scholes equations. SIAM J. Numer. Anal. 38(4), 1357–1368 (2000)

    Article  MATH  Google Scholar 

  21. Lewis, A.: Fear of jumps. Wilmott Magazine, December 2002, pp. 60–67

  22. Merton, R.C.: Option pricing when underlying stock returns are discontinuous. J. Financial Econ. 3, 125–144 (1976)

    Article  Google Scholar 

  23. Meyer, G.H.: The numerical valuation of options with underlying jumps. Acta Math. Univ. Comenianae 67, 69–82 (1998)

    MathSciNet  MATH  Google Scholar 

  24. Mulinacci, S.: An approximation of American option prices in a jump diffusion model. Stochastic Processes and their Applications 62, 1–17 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pham, H.: Optimal stopping of controlled jump diffusion processes: a viscosity solution approach. J. Math. Syst. Estimation and Control 8, 1–27 (1998)

    MATH  Google Scholar 

  26. Pooley, D.M., Forsyth, P.A., Vetzal, K.R.: Numerical convergence properties of option pricing PDEs with uncertain volatility. IMA J. Numer. Anal. 23, 241–267 (2003)

    MathSciNet  MATH  Google Scholar 

  27. Potts, D., Steidl, G., Tasche, M.: Fast Fourier transforms for nonequispaced data: A tutorial. In: Modern Sampling Theory: Mathematics and Application, J.J. Benedetto and P. Ferreira, (eds.), ch. 12, Birkhauser, 2000 , pp. 253–274

  28. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge (UK) and New York, 2nd edition, 1992

  29. Rannacher, R.: Finite element solution of diffusion problems with irregular data. Numer. Math. 43, 309–327 (1984)

    MathSciNet  MATH  Google Scholar 

  30. Tavella, D., Randall, C.: Pricing financial instruments: the finite difference method. John Wiley & Sons, Inc, 2000

  31. Vazquez, A.A., Oosterlee, C.W.: Numerical valuation of options with jumps in the underlying. Working paper, Delft University of Technology, 2003

  32. Ware, A.F.: Fast approximate Fourier transforms for irregularly spaced data. SIAM Rev. 40, 838–856 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wilmott, P.: Derivatives. John Wiley and Sons Ltd, Chichester, 1998

  34. Windcliff, H., Forsyth, P.A., Vetzal, K.R.: Shout options: a framework for pricing contracts which can be modified by the investor. J. Comput. Appl. Math. 134, 213–241 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  35. Windcliff, H., Forsyth, P.A., Vetzal, K.R.: Valuation of segregated funds: shout options with maturity extensions. Insurance: Math. Econ. 29, 1–21 (2001)

    Article  Google Scholar 

  36. Windcliff, H., Forsyth, P.A., Vetzal, K.R.: Analysis of the stability of the linear boundary condition for the Black-Scholes equation, 2003. Submitted to the J. of Comput. Finance

  37. Zhang, X.L.: Numerical analysis of American option pricing in a jump-diffusion model. Math. Oper. Res. 22, 668–690 (1997)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Zvan, R., Forsyth, P.A., Vetzal, K.R.: Discrete Asian barrier options. J. Comput. Finance 3(Fall), 41–67 (1999)

    Google Scholar 

  40. Zvan, R., Forsyth, P.A., Vetzal, K.R.: A finite element approach to the pricing of discrete lookbacks with stochastic volatility. Appl. Math. Finance 6, 87–106 (1999)

    Article  MATH  Google Scholar 

  41. Zvan, R., Forsyth, P.A., Vetzal, K.R.: A finite volume approach for contingent claims valuation. IMA J. Numer. Anal. 21, 703–731 (2001)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Computer Science, University of Waterloo, Waterloo ON, Canada, N2L 3G1J

    Y. d’Halluin, P.A. Forsyth & G. Labahn

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  1. Y. d’Halluin
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  2. P.A. Forsyth
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  3. G. Labahn
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Correspondence to Y. d’Halluin.

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Mathematics Subject Classification (1991): 65M12, 65M60, 91B28

Correspondence to: P.A. Forsyth

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d’Halluin, Y., Forsyth, P. & Labahn, G. A penalty method for American options with jump diffusion processes. Numer. Math. 97, 321–352 (2004). https://doi.org/10.1007/s00211-003-0511-8

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  • DOI: https://doi.org/10.1007/s00211-003-0511-8

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