Option Pricing in Subdiffusive Bachelier Model

Journal of Statistical Physics

Volumn 145, Issue 1, 2011, Pages 187-203

  Magdziarz, Marcin ^a   Orzeł, Sebastian ^a   Weron, Aleksander ^a  

^a WROCLAW UNIVERSITY OF TECHNOLOGY   (Poland)

Author keywords

Bachelier model; Fractional Fokker Planck equation; Infinitely divisible distribution; Model of financial markets; Option pricing; Subdiffusion; Tempered stable distribution

Indexed keywords

EID: 80053119283     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-011-0310-z     Document Type: Article

References (42)

1. Bachelier, L.: Théorie de la spéculation. Ann. Éc. Norm. Super. 17, 21-86 (1900).
2. Baeumer, B., Meerschaert, M. M.: Tempered stable Lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233, 2438-2448 (2010).
3. Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996).
4. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637-654 (1973).
5. Cartea, Á., del-Castillo-Negrete, D.: Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. Phys. Rev. E 76, 041105 (2007).
6. Carr, P., Geman, H., Madan, D. B., Yor, M.: Self-decomposability and option pricing. Math. Finance 17, 31-51 (2007).
7. Cont, R., Tankov, P.: Financial Modeling with Jump Processes. Chapman & Hall/CRC, London (2004).
8. Courtault, J. M., et al.: Louis Bachelier's on the centenary of Théorie de la Spéculation. Math. Finance 10, 341-353 (2000).
9. Deng, W.: Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal. 47, 204-226 (2008).
10. Dybiec, B.: Anomalous diffusion on finite intervals. J. Stat. Mech., P01011 (2010).
11. Dybiec, B.: Approaching stationarity: competition between long jumps and long waiting times. J. Stat. Mech., P03019 (2010).
12. Dybiec, B., Gudowska-Nowak, E.: Discriminating between normal and anomalous random walks. Phys. Rev. E 80, 061122 (2009).
13. Dybiec, B., Sokolov, I. M., Checkin, A. V.: Stationary states in single-well potentials under symmetric Lévy noises. J. Stat. Mech., P07008 (2010).
14. Einstein, A.: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. (Leipz.) 17, 549-560 (1905).
15. Galloway, M. L., Nolder, C. A.: Subordination, self-similarity, and option pricing. J. Appl. Math. Decis. Sci., 397028 (2008).
16. Itô, K.: On stochastic processes. Jpn. J. Math. 18, 261-301 (1942).
17. Janczura, J., Wyłomańska, A.: Subdynamics of financial data from fractional Fokker-Planck equation. Acta Phys. Pol. B 40(5), 1341-1351 (2009).
18. Janczura, J., Orzeł, S., Wyłomańska, A.: Subordinated Ornstein-Uhlenbeck process as a tool of financial data description. Preprint (2011).
19. Janicki, A., Weron, A.: A Simulation and Chaotic Behavior of α-Stable Stochatic Processes. Dekker, New York (1994).
20. Kullback, S., Leibler, R. A.: On information and sufficiency Ann. Math. Stat. 22(1), 79-86 (1951).
21. Lévy, P.: Processus stochastiques at mouvement Brownian. Gauthier-Villars, Paris (1948).
22. Magdziarz, M.: Langevin picture of subdiffusion with infinitely divisible waiting times. J. Stat. Phys. 135, 763-772 (2009).
23. Magdziarz, M.: Black-Scholes formula in subdiffusive regime. J. Stat. Phys. 136, 553-564 (2009).
24. Magdziarz, M.: Path properties of subdiffusion-a martingale approach. Stoch. Models 26, 256-271 (2010).
25. Magdziarz, M., Weron, A., Weron, K.: Fractional Fokker-Planck dynamics: stochastic representation and computer simulation. Phys. Rev. E 75, 016708 (2007).
26. Merton, R. C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, 141-183 (1973).
27. Metzler, R., Klafter, J.: The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1 (2000).
28. Musiela, M., Rutkowski, M.: Martingale Methods in Financial Modelling. Springer, Berlin (2005).
29. Orzeł, S., Weron, A.: Calibration of the subdiffusive Black-Scholes model. Acta Phys. Pol. B 41(5), 1051-1059 (2010).
30. Orzeł, S., Wyłomańska, A.: Calibration of the subdiffusive arithmetic Brownian motion with tempered stable waiting-times. J. Stat. Phys. 143(3), 447-454 (2011). doi: 10. 1007/s10955-011-0191-1.
31. Protter, P.: Stochastic Integration and Differential Equations. A New Approach. Springer, Berlin/Heidelberg (1990).
32. Rosinski, J.: Tempering stable processes. Stoch. Process. Appl. 117, 677-707 (2007).
33. Rosinski, J.: Simulation of Lévy processes. In: Encyclopedia of Statistics in Quality and Reliability: Computationally Intensive Methods and Simulation. Wiley, New York (2008).
34. Samko, S. G., Kilbas, A. A., Maritchev, D. I.: Integrals and Derivatives of the Fractional Order and Some of Their Applications. Gordon & Breach, Amsterdam (1993).
35. Samorodnitsky, G., Taqqu, M. S.: Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York (1994).
36. Samuelson, P. A.: Rational theory of warrant pricing. Ind. Manage. Rev. 6, 13-31 (1965).
37. Schachermayer, W., Teichmann, J.: How close are the option pricing formulas of Bachelier and Black-Merton-Scholes? Math. Finance 18, 155-170 (2008).
38. Smoluchowski, M.: Sur le chemin moyen parcouru par les molécules dún gaz et sur son rapport avec la théorie de la diffusion. Bull. Int. Acad. Sci. Cracovie, 202-213 (1906).
39. Stanislavsky, A. A.: Black-Scholes model under subordination. Physica A 318, 469-474 (2003).
40. Weron, R.: On the Chambers-Mallows-Stuck method for simulating skewed stable random variables. Stat. Probab. Lett. 28, 165-171 (1996).
41. Weron, A., Magdziarz, M.: Anomalous diffusion and semimartingales. Europhys. Lett. 86, 60010 (2009).
42. Wiener, N.: The average of an analytic functional and the Brownian movement. Proc. Natl. Acad. Sci. USA 7, 294-298 (1921).