A comparison of numerical solutions of fractional diffusion models in finance. (English) Zbl 1180.91308

Summary: We compare the numerical solutions of three fractional partial differential equations that occur in finance. These fractional partial differential equations fall in the class of Lévy models. They are known as the FMLS (finite moment log stable), CGMY and KoBol models. Conditions for the convergence of each of these models is obtained.


91G60 Numerical methods (including Monte Carlo methods)
60G51 Processes with independent increments; Lévy processes
35R11 Fractional partial differential equations
Full Text: DOI


[1] Bachelier, L., Théorie de la spéculation, Ann. sci. ècole norm. sup., 3, 17, 21-86, (1900) · JFM 31.0241.02
[2] Mandelbrot, B., The variation of certain speculative prices, J. bus. univ. Chicago, 36, 394-419, (1963)
[3] Hull, J.C., Options futures and other derivatives, (2006), Pearson Prentice Hall, pp. 281-295
[4] E. Benhamou, Option pricing with levy process, London School of Economics, July 2000
[5] A. Cartea, Fractional diffusion models of option prices in markets with jumps, Birkbeck Working Papers in Economics and Finance, August 2006
[6] Gorenflo, R., Fractional calculus: integral and differential equations of fractional order, (1996), International Centre for Mechanical Sciences
[7] Meerschaert, M.M.; Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. numer. math., (2005) · Zbl 1086.65087
[8] Meerschaert, M.M.; Tadjeran, C.; Scheffler, H.P., A second-order accurate numerical approximation for the fractional diffusion equation, J. comput. phys., (2005)
[9] Miller, K.; Ross, B., An introduction to fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002
[10] Samko, S.; Kilbas, A.; Marichev, O., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach London · Zbl 0818.26003
[11] Barkai, E.; Metzler, R.; Klafter, J., From continuous time random walks to the fractional fokker – plank equation, Phys. rev. E, 61, 132-138, (2000)
[12] Bouchaud, J.P.; Georges, A., Anomalous diffusion in disordered media — statistical mechanisms, models and physical applications, Phys. rep., 195, 127-293, (1990)
[13] Klafter, J.; Blumen, A.; Shlesinger, M., Stochastic pathways to anomalous diffusion, Phys. rev. A, 35, 3081-3085, (1987)
[14] Lévy flights and related topics in physics, ()
[15] Raberto, M.; Scalas, E.; Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Physica A, 314, 749-755, (2002) · Zbl 1001.91033
[16] Sabatelli, L.; Keating, S.; Dudley, J.; Richmond, P., Waiting time distributions in financial markets, Eur. phys. J. B, 27, 273-275, (2002)
[17] Chen, W.; Lu, Z., An algorithm for Adomian decomposition method, Appl. math. comput., 159, 221-235, (2004) · Zbl 1062.65059
[18] Saha Ray, S.; Bera, R.K., An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl. math. comput., 167, 561-571, (2005) · Zbl 1082.65562
[19] Carr, P.; Geman, H.; Madan, D.B.; Yor, M., The fine structure of asset returns: an empirical investigation, J. bus., 75, 305-332, (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.