Skip to main content
Log in

Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary.

We study the numerical approximation of viscosity solutions for integro-differential, possibly degenerate, parabolic problems. Similar models arise in option pricing, to generalize the celebrated Black–Scholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Convergence is proven for monotone schemes and numerical tests are presented and discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ait-Sahalia, Y., Wang, Y., Yared, F.: Do option markets correctly price the probabilities of movements of the underlying asset?. Working paper, Univ. of Chicago, 1998

  2. Alvarez, O., Tourin, A.: Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 13(3), 293–317 (1996)

    MATH  Google Scholar 

  3. Amadori, A.L.: Differential And Integro–Differential Nonlinear Equations of Degenerate Parabolic Type Arising in the Pricing of Derivatives in Incomplete Markets. Ph.D. Thesis, Università di Roma “La Sapienza”, 2000

  4. Amadori, A.L.: Nonlinear integro-differential evolution problems arising in option pricing: a viscosity solutions approach. J. Differential and Integral Equations 16(7), 787–811 (2003)

    MATH  Google Scholar 

  5. Amadori, A.L.: The obstacle problem for nonlinear integro–differential operators arising in option pricing Quaderno IAC, Q21-000, (2000)

  6. Andersen, L., Andreasen, J.: Jump-Diffusion Processes: volatility smile fitting and numerical methods for pricing. Rev. Derivative Res. 4, 231–262 (2000)

    Article  Google Scholar 

  7. Andersen, L., Benzoni, L., Lund, J.: Estimating jump–diffusions for equity returns. Working paper, Northweterns Univ. and Aarhus School of Business, 1999

  8. Attari, M.: Discontinuous interest rate processes: an equilibrium model of bond option prices. Working paper, Univ. Madison (Winsconsin), 1997

  9. Barles, G., Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal 4(3), 271–283 (1991)

    MATH  Google Scholar 

  10. Bates, D.: Jumps and stochastic volatility: exchange rate processes implicit in Deutsch mark options. Review of financial studies 9, 69–107 (1996)

    Article  MATH  Google Scholar 

  11. Black, F., Scholes, M.: The pricing of option and corporate liabilities. J. Political Econom. 72, 637–659 (1973)

    Article  Google Scholar 

  12. Björk, T.: Arbitrage Theory in Continuous Time. Oxford: Oxford University Press, 1998

  13. Bjork, T., Kabanov, Y., Runggaldier, W.: Bond market structure in the presence of marked point processes. Math. Finance 7(2), 211–239 (1997)

    Article  Google Scholar 

  14. Burneta, A.N., Ritchken, P.: On rational jump-diffusion models in the Flesaker-Hughston paradigm, Working paper, Case western reserves University, 1996

  15. Crandall, M.G., Ishi, H., Lions, P.L.: Users’ guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc., (New Ser.) 27(1), 1–67 (1992)

    Google Scholar 

  16. Crandall, M.G., Lions, P.L.: Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput. 43(167), 1–19 (1994)

    MATH  Google Scholar 

  17. Das, S., Foresi, S.: Exact solutions for bond and option prices with systematic jump risk. J. Financ. Quantitative Anal. 34, 211–240 (1999)

    Google Scholar 

  18. Das, S.R.: The surprise element: jumps in interest rates. J. Econometrics 106(1), 27–65 (2002)

    Article  MATH  Google Scholar 

  19. Davis, P.J., Rabinowitz, P.: Methods of numerical integration. Academic Press, Inc. (1975)

  20. Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6), 1343–1376 (2000)

    Article  MATH  Google Scholar 

  21. Garroni, M.G., Menaldi, J.L.: Green functions for second order parabolic integro-differential problems. Pitman Res. Notes Math. Ser. (1992)

  22. Garroni, M.G., Menaldi, J.L.: Regularizing effect for integro-differential parabolic equations. Commun. Partial Differential Equations 18(12), 2023–2025 (1993)

    MATH  Google Scholar 

  23. Garroni, M.G., Menaldi, J.L.: Maximum principle for integro-differential parabolic operators. Differ. Integral Equ. 8(1), 161–182 (1995)

    MATH  Google Scholar 

  24. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jarrow, R., Madan, D.: Option pricing using the term structure of interest rates to hedge systematic discontinuities in asset returns. Math. Finance 5(4), 311–336 (1995)

    MATH  Google Scholar 

  26. Karlsen, K.H., Risebro, N.H.: An operator splitting method for nonlinear convection-diffusion equations. Numer. Math. 77(3), 365–382 (1997)

    Article  MATH  Google Scholar 

  27. Kou, S.G.: A jump diffusion model for option pricing. Management Science 48, 1086–1101 (2002)

    Article  Google Scholar 

  28. Kurganov, A., Tadmor, E.: New High-Resolution Semi-Discrete Central Schemes for Hamilton-Jacobi Equations. J. Comput. Phys. 160(2), 720–742 (2000)

    Article  MATH  Google Scholar 

  29. Lin, C.-T., Tadmor, E.: High-resolution non-oscillatory central scheme for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2163–2186 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. Magill, M., Quinzii, M.: Theory of incomplete markets. vol. 1 – MIT Press, 1996

  31. Merton, R.C.: Option pricing when the underlying stocks returns are discontinuous. J. Financ. Econ. 5, 125–144 (1976)

    Article  Google Scholar 

  32. Osher, S., Shu, C.-W.: High-order essentially non-oscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28(4), 907–922 (1991)

    MATH  Google Scholar 

  33. Page, F.H., Sanders, A.B.: A general derivation of the jump process option pricing formula. J. Financial and Quantitative Anal. 21, 437–446 (1986)

    Google Scholar 

  34. Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations. Chapman & Hall, New York

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

  36. Yong, J., Zhou, X.Y.: Stochastic controls. Hamiltonian systems and HJB equations. Applications of Mathematics, 43. New York: Springer-Verlag, 1999

Download references

Acknowledgments.

The first two authors would like to thank the whole staff of IAC-CNR for their kind hospitality during the development of this work.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Roberto Natalini.

Additional information

Mathematics Subject Classification (1991): 65M12, 35K55, 49L25

Revised version received February 13, 2003

Rights and permissions

Reprints and permissions

About this article

Cite this article

Briani, M., Chioma, C. & Natalini, R. Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory. Numer. Math. 98, 607–646 (2004). https://doi.org/10.1007/s00211-004-0530-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-004-0530-0

Keywords

Navigation