Fractional diffusion models of option prices in markets with jumps

Physica A: Statistical Mechanics and its Applications

Volumn 374, Issue 2, 2007, Pages 749-763

  Cartea, Álvaro ^a   del Castillo Negrete, Diego ^b  

^a UNIVERSITY OF LONDON   (United Kingdom)

^b OAK RIDGE NATIONAL LABORATORY   (United States)

Author keywords

Barrier options; CGMY; Double knock out; Down and out; FMLS; Fractional calculus; Fractional Black Scholes; KoBoL; L vy stable processes; Riemann Liouville fractional derivative; Up and out

Indexed keywords

MARKETING; MATHEMATICAL MODELS; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS;

BARRIER OPTIONS; DOUBLE KNOCK-OUT; FRACTIONAL CALCULUS; FRACTIONAL DERIVATIVES; LÉVY-STABLE PROCESSES;

DIFFUSION;

EID: 33751093078     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physa.2006.08.071     Document Type: Article

References (34)

1. Merton R. Continuous-Time Finance. first ed. (1990), Blackwell, Oxford
2. Wilmott P., Howison S., and Dewynne J. The Mathematics of Financial Derivatives, a Student Introduction. first ed. (1995), Cambridge University Press, Cambridge
3. Schoutens W. Lévy Processes in Finance, Wiley Series in Probability and Statistics. first ed. (2003), Wiley, England
4. Cont R., and Voltchkova E. Finite difference methods for option pricing in jump-diffusion and exponential Lévy models. SIAM J. Numer. Anal. 43 4 (2005) 1596-1626
5. Carr P., and Madan D. Option valuation using the fast Fourier transform. J. Comput. Finance 2 (1999) 61-73
6. D. del-Castillo-Negrete, Fractional diffusion models on nonlocal transport, Phys. Plasmas 3 (2006) 082308-1-082308-16.
7. D. del-Castillo Negrete, Finite difference methods for fractional diffusion equations, Working paper, 2006.
8. Boyarchenko S., and Levendorskiǐ S. Barrier options and touch-and-out options under regular Lévy processes of exponential type. Ann. Appl. Probab. 12 4 (2002) 1261-1298
9. Á. Cartea, Dynamic hedging of financial instruments when the underlying follows a non-Gaussian process, Working paper, Birkbeck College, University of London, 2005.
10. Björk T. Arbitrage Theory in Continuous Time. first ed. (1998), Oxford University Press, Oxford
11. Scalas E., Gorenflo R., and Mainardi F. Fractional calculus and continuous-time finance. Physica A 284 (2000) 376-384
12. Mainardi F., Raberto M., Gorenflo R., and Scalas E. Fractional calculus and continuous-time finance II: the waiting-time distribution. Physica A 287 (2000) 469-481
13. Press S.J. A compound events model for security prices. J. Business 40 (1967) 317-335
14. Mantegna R.M., and Stanley H.E. Stochastic processes with ultraslow convergence to a Gaussian: the truncated Levy flight. Phys. Rev. Lett. 73 (1994) 2946-2949
15. R.N. Mantegna, H.E. Stanley, Scaling behavior in the dynamics of an economic index, Nature (376) (1995) 46-49.
16. R.N. Mantegna, H.E Stanley, An Introduction to Econophysics, Correlations and Complexity in Finance, first ed., Cambridge, 2000.
17. Mandelbrot B. Fractals and Scaling in Finance. first ed. (1997), Springer, Berlin
18. Koponen I. Analytic approach to the problem of convergence of truncated Levy flights towards the Gaussian stochastic process. Phys. Rev. E 52 (1995) 1197-1199
19. S. Boyarchenko, S. Levendorskiǐ, Non-Gaussian Merton-Black-Scholes Theory, vol. 9, World Scientific, Singapore, 2002.
20. Carr P., Geman H., Madan D., and Yor M. The fine structure of asset returns: an empirical investigation. J. Bus. 75 2 (2002) 305-332
21. W. Feller, An Introduction to Probability Theory and its Applications, vol. II, Wiley, New York, 1966.
22. S. Boyarchenko, S. Levendorskiǐ, Generalization of the Black-Scholes equation for truncated Lévy processes, Working paper, 2006.
23. Cont R., and Tankov P. Financial Modelling with Jump Processes (2004), Chapman & Hall, London
24. Carr P., and Wu L. The finite moment logstable process and option pricing. J. Finance LVIII 2 (2003) 753-777
25. I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198, first ed., Academic Press, San Diego, CA, 1999.
26. Metzler R., and Klafter J. The random walk guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. (2000) 339
27. Sokolov I.M., Chechkin A.V., and Klafter J. Fractional diffusion equation for a power-law-truncated Lévy process. Physica A (2004) 336
28. Samorodnitsky G., and Taqqu M. Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance, Stochastic Modelling. first ed. (1994), Chapman & Hall, London
29. Samko S.G., Kilbas A.A., and Marichev O.I. Fractional Integrals and Derivatives, Theory and Applications (1993), Gordon and Breach Science Publishers, London
30. Lynch V., Carreras B., and del-Castillo-Negrete D. Numerical methods for the solution of partial differential equations of fractional order. J. Comput. Phys. (2003) 406-421
31. Morton K.W., and Mayers D.F. Numerical Solutions of Partial Differential Equations (1996), Cambridge University Press, Cambridge
32. Boyarchenko S., and Levendorskiǐ S. Perpetual American options under Lévy processes. SIAM J. Control Optim. 40 (2002) 1663-1696
33. Mordecki E. Optimal stopping and perpetual options for Lévy processes. Finance and Stochastics VI 4 (2002) 473-493
34. A. Lewis, A simple option formula for general jump-diffusion and other exponential Lévy processes, Working paper, September 2001.