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Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory. (English) Zbl 1065.65145
The authors study the numerical approximation of a class of semilinear strongly degenerate parabolic integro-differential Cauchy problems. Convergence is shown for monotone schemes for viscosity solutions to problems arising in financial theory. Similar models arise in option pricing. Moreover, numerical tests are presented and analyzed.
MSC:
65R20Integral equations (numerical methods)
45K05Integro-partial differential equations
45G10Nonsingular nonlinear integral equations
49L25Viscosity solutions (infinite-dimensional problems)
91B28Finance etc. (MSC2000)
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)
[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)
[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) · Zbl 1274.91398 · doi:10.1023/A:1011354913068
[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)
[10]Bates, D.: Jumps and stochastic volatility: exchange rate processes implicit in Deutsch mark options. Review of financial studies 9, 69-107 (1996) · doi:10.1093/rfs/9.1.69
[11]Black, F., Scholes, M.: The pricing of option and corporate liabilities. J. Political Econom. 72, 637-659 (1973)
[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) · Zbl 0884.90014 · doi:10.1111/1467-9965.00031
[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) · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5
[16]Crandall, M.G., Lions, P.L.: Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput. 43(167), 1-19 (1994)
[17]Das, S., Foresi, S.: Exact solutions for bond and option prices with systematic jump risk. J. Financ. Quantitative Anal. 34, 211-240 (1999) · doi:10.2307/2676279
[18]Das, S.R.: The surprise element: jumps in interest rates. J. Econometrics 106(1), 27-65 (2002) · Zbl 1051.62106 · doi:10.1016/S0304-4076(01)00085-9
[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) · Zbl 1055.91524 · doi:10.1111/1468-0262.00164
[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)
[23]Garroni, M.G., Menaldi, J.L.: Maximum principle for integro-differential parabolic operators. Differ. Integral Equ. 8(1), 161-182 (1995)
[24]Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89-112 (2001) · Zbl 0967.65098 · doi:10.1137/S003614450036757X
[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) · Zbl 0866.90018 · doi:10.1111/j.1467-9965.1995.tb00070.x
[26]Karlsen, K.H., Risebro, N.H.: An operator splitting method for nonlinear convection-diffusion equations. Numer. Math. 77(3), 365-382 (1997) · Zbl 0882.35074 · doi:10.1007/s002110050291
[27]Kou, S.G.: A jump diffusion model for option pricing. Management Science 48, 1086-1101 (2002) · Zbl 1216.91039 · doi:10.1287/mnsc.48.8.1086.166
[28]Kurganov, A., Tadmor, E.: New High-Resolution Semi-Discrete Central Schemes for Hamilton-Jacobi Equations. J. Comput. Phys. 160(2), 720-742 (2000) · Zbl 0961.65077 · doi:10.1006/jcph.2000.6485
[29]Lin, C.-T., Tadmor, E.: High-resolution non-oscillatory central scheme for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2163-2186 (2000) · Zbl 0964.65097 · doi:10.1137/S1064827598344856
[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) · Zbl 1131.91344 · doi:10.1016/0304-405X(76)90022-2
[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) · Zbl 0736.65066 · doi:10.1137/0728049
[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) · doi:10.2307/2330691
[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