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

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