Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

6 Issues per year


The journal celebrates now its 20 years!

IMPACT FACTOR 2016: 2.034
5-year IMPACT FACTOR: 2.359

CiteScore 2016: 2.18

SCImago Journal Rank (SJR) 2016: 1.372
Source Normalized Impact per Paper (SNIP) 2016: 1.492

Mathematical Citation Quotient (MCQ) 2016: 0.61

Online
ISSN
1314-2224
See all formats and pricing
More options …

Almost sure and moment stability properties of fractional order Black-Scholes model

Caibin Zeng
  • School of Sciences, South China University of Technology, Guangzhou, 510640, China
  • Mechatronics, Embedded Systems and Automation (MESA) Lab School of Engineering, University of California, Merced 5200 North Lake Road, Merced, CA, 95343, USA
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ YangQuan Chen
  • Mechatronics, Embedded Systems and Automation (MESA) Lab School of Engineering, University of California, Merced 5200 North Lake Road, Merced, CA, 95343, USA
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Qigui Yang
Published Online: 2013-03-19 | DOI: https://doi.org/10.2478/s13540-013-0020-0

Abstract

We deal with the stability problem of the fractional order Black-Scholes model driven by fractional Brownian motion (fBm). First, necessary and sufficient conditions are established for almost sure asymptotic stability and pth moment asymptotic stability by means of the largest Lyapunov exponent and the pth moment Lyapunov exponent, respectively. Moreover, we are able to present large deviations results for this fractional process. In particular, for the first time it is found that the Hurst parameter affects both stability conclusions and large deviations. Interestingly, large deviations always happen for the considered system when 1/2 < H < 1. This fact is due to the long-range dependence (LRD) property of the fBm. Numerical simulation results are presented to illustrate the above findings.

MSC: Primary 60G22; Secondary 93E15, 60F10, 60H10, 60H35

Keywords: fractional Brownian motion; stochastic stability; Black-Scholes model; large deviations; Hurst parameter

  • [1] E. Alòs, O. Mazet, D. Nualart, Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29, No 2 (2001), 766–801. http://dx.doi.org/10.1214/aop/1008956692CrossrefGoogle Scholar

  • [2] T. G. Andersen, T. Bollerslev, Heterogeneous information arrivals and return volatility dynamics: uncovering the long-run in high frequency returns. J. Finance 52, No 3 (1997), 975–1006. http://dx.doi.org/10.1111/j.1540-6261.1997.tb02722.xCrossrefGoogle Scholar

  • [3] M. A. Arcones, On the law of the iterated logarithm for Gaussian processes. J. Theor. Probab. 8, No 4 (1995), 877–890. http://dx.doi.org/10.1007/BF02410116CrossrefGoogle Scholar

  • [4] L. Arnold, E. Oeljeklaus, E. Pardoux, Almost sure and moment stability for linear Itô equations. In: L. Arnold, V. Wihstutz (Eds.), Lyapunov Exponents, Springer-Verlag (1986), 129–159. http://dx.doi.org/10.1007/BFb0076837Google Scholar

  • [5] C. Bender, An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter. Stoch. Proc. Appl. 104, No 1 (2003), 81–106. http://dx.doi.org/10.1016/S0304-4149(02)00212-0CrossrefGoogle Scholar

  • [6] J. Beran, Statistics for Long-Memory Processes. Chapman & Hall/CRC, New-York (1994). Google Scholar

  • [7] F. Biagini, Y. Hu, B. Oksendal, T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications. Springer-Verlag (2008). http://dx.doi.org/10.1007/978-1-84628-797-8CrossrefGoogle Scholar

  • [8] F. Biagini, B. Øksendal, A. Sulem, N. Wallner, An introduction to white noise theory and Malliavin calculus for fractional Brownian motion. Proc. R. Soc. 460, No 2041 (2004), 347–372. http://dx.doi.org/10.1098/rspa.2003.1246CrossrefGoogle Scholar

  • [9] Jaya P. N. Bishwal, Minimum contrast estimation in fractional Ornstein-Uhlenbeck process: Continuous and discrete sampling. Fract. Calc. Appl. Anal. 14, No 3 (2011), 375–410; DOI:10.2478/s13540-011-0024-6; http://link.springer.com/journal/13540/14/3/ CrossrefWeb of ScienceGoogle Scholar

  • [10] F. Black, M. Scholes, The pricing of options and corporate liabilities. J. Polit. Econ. 81, No 3 (1973), 637–654. http://dx.doi.org/10.1086/260062CrossrefGoogle Scholar

  • [11] P. Carmona, L. Coutin, G. Montseny, Stochastic integration with respect to fractional Brownian motion. Ann. I. H. Poincaré Probab. Stat. 39, No 1 (2003), 27–68. http://dx.doi.org/10.1016/S0246-0203(02)01111-1CrossrefGoogle Scholar

  • [12] L. Decreusefond, A. Üstünel, Stochastic analysis of the fractional Brownian motion. Potential Anal. 10, No 2 (1999), 177–214 http://dx.doi.org/10.1023/A:1008634027843CrossrefGoogle Scholar

  • [13] C. Dellacherie, P. Meyer, Probability and Potentials B. Theory of Martingales. North-Holland, Amsterdam (1982). Google Scholar

  • [14] E. Derman, I. Kani, The volatility smile and its implied tree. Available at: http://www.ederman.com/new/docs/gs-volatility smile.pdf Google Scholar

  • [15] T. E. Duncan, Y. Hu, B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion I, Theory. SIAM J. Control Optim. 38, No 2 (2000), 582–612. http://dx.doi.org/10.1137/S036301299834171XCrossrefGoogle Scholar

  • [16] T. E. Duncan, B. Maslowski, B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise. Stoch. Proc. Appl. 115, No 8 (2005), 1357–1383. http://dx.doi.org/10.1016/j.spa.2005.03.011CrossrefGoogle Scholar

  • [17] R. Elliott, J. Van Der Hoek, A general fractional white noise theory and applications to finance. Math. Finan. 13, No 2 (2003), 301–330. http://dx.doi.org/10.1111/1467-9965.00018CrossrefGoogle Scholar

  • [18] C. L. E. Franzke, T. Graves, N. W. Watkins, R. B. Gramacy, C. Hughes, Robustness of estimators of long-range dependence and self-similarity under non-Gaussianity. Phil. Trans. Math. Phys. Eng. Sci. 370, No 1962 (2012), 1250–1267. http://dx.doi.org/10.1098/rsta.2011.0349CrossrefGoogle Scholar

  • [19] Y. Hu, B. Øksendal, Fractional white noise calculus and applications to finance. Infin. Dimens. Anal. Qu. 6, No 1 (2003), 1–32. http://dx.doi.org/10.1142/S0219025703001110CrossrefGoogle Scholar

  • [20] G. A. Hunt, Random Fourier transforms. Trans. Am. Math. Soc. 71, No 1 (1951), 38–69. http://dx.doi.org/10.1090/S0002-9947-1951-0051340-3CrossrefGoogle Scholar

  • [21] H. E. Hurst, Long-term storage capacity in reservoirs. Trans. Amer. Soc. Civil Eng. 116, (1951), 400–410. Google Scholar

  • [22] M. Jolis, On the wiener integral with respect to the fractional Brownian motion on an interval. J. Math. Anal. Appl. 330, No 2 (2007), 1115–1127. http://dx.doi.org/10.1016/j.jmaa.2006.07.100CrossrefGoogle Scholar

  • [23] R. Khasminskii, Stochastic Stability of Differential Equations. Springer (2011). Google Scholar

  • [24] A. N. Kolmogorov, The Wiener spiral and some other interesting curves in Hilbert space. Dokl. Akad. Nauk SSSR 26, (1940), 115–118. Google Scholar

  • [25] S. Lin, Stochastic analysis of fractional Brownian motions. Stoch. Stoch. Rep. 55, No 1–2 (1995), 121–140. Google Scholar

  • [26] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010). http://dx.doi.org/10.1142/p614CrossrefGoogle Scholar

  • [27] R. C. Merton, Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, No 1 (1973), 141–183. http://dx.doi.org/10.2307/3003143CrossrefGoogle Scholar

  • [28] Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer (2008). http://dx.doi.org/10.1007/978-3-540-75873-0CrossrefGoogle Scholar

  • [29] W. A. Müller, M. A. Dacorogna, O. V. Pictet, Heavy tails in high frequency financial data. In: R. A. Adler, R. E. Feldman, M. S. Taqqu (Eds.), A Practical Guide to Heavy Tails: Statistical Techniques and Applications, Birkhäuser, Boston (1998), 55–77. Google Scholar

  • [30] L. Rogers, Arbitrage with fractional Brownian motion. Math. Finan. 7, No 1 (1997), 95–105. http://dx.doi.org/10.1111/1467-9965.00025CrossrefGoogle Scholar

  • [31] H. Sheng, Y. Q. Chen, T. Qiu, On the robustness of Hurst estimators. IET Signal Processing 5, No 2 (2011), 209–225. http://dx.doi.org/10.1049/iet-spr.2009.0241Web of ScienceCrossrefGoogle Scholar

  • [32] H. Sheng, Y. Q. Chen, T. Qiu, Fractional Processes and Fractional-Order Signal Processing. Springer-Verlag, London (2012). http://dx.doi.org/10.1007/978-1-4471-2233-3CrossrefGoogle Scholar

  • [33] H. S. Shu, C. L. Chen, G. L. Wei, Stability of linear stochastic differential equations with respect to fractional Brownian motion. J. Donghua Univ. 26, No 2 (2009), 119–125. Google Scholar

  • [34] O. Vivero, W. P. Heath, A regularised estimator for long-range dependent processes. Automatica 48, No 2 (2012), 287–296. http://dx.doi.org/10.1016/j.automatica.2011.07.012CrossrefWeb of ScienceGoogle Scholar

  • [35] W. Wyss, The fractional Black-Scholes equation. Fract. Calc. Appl. Anal. 3, No 1 (2000), 51–61. Google Scholar

  • [36] C. Zeng, Q. Yang, Y. Q. Chen, Solving nonlinear stochastic differential equations with fractional Brownian motion using reducibility approach. Nonlinear Dyn. 67, No 4 (2012), 2719–2726. http://dx.doi.org/10.1007/s11071-011-0183-3CrossrefGoogle Scholar

  • [37] C. Zeng, Y. Q. Chen, Q. Yang, The fBm-driven Ornstein-Uhlenbeck process: Probability density function and anomalous diffusion. Fract. Calc. Appl. Anal. 15, No 3 (2012), 479–492; DOI:10.2478/s13540-012-0034-z; http://link.springer.com/journal/13540/15/3/ Web of ScienceCrossrefGoogle Scholar

  • [38] C. Zeng, Y. Q. Chen, Q. Yang, Almost sure and moment stability properties of LTI stochastic dynamic systems driven by fractional Brownian motion. In: 51st IEEE Conference on Decision and Control, Maui, Hawaii (2012), Accepted. Google Scholar

About the article

Published Online: 2013-03-19

Published in Print: 2013-06-01


Citation Information: Fractional Calculus and Applied Analysis, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-013-0020-0.

Export Citation

© 2013 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Robert G Wallace, Marius Gilbert, Rodrick Wallace, Claudia Pittiglio, Raffaele Mattioli, and Richard Kock
Environment and Planning A, 2014, Volume 46, Number 11, Page 2533
[2]
HongGuang Sun, Xiaoxiao Hao, Yong Zhang, and Dumitru Baleanu
Physica A: Statistical Mechanics and its Applications, 2017, Volume 468, Page 590
[3]
Rodrick Wallace
Physics Letters A, 2016, Volume 380, Number 5-6, Page 726
[5]
Caibin Zeng, Qigui Yang, and YangQuan Chen
Abstract and Applied Analysis, 2014, Volume 2014, Page 1
[6]
Qigui Yang, Caibin Zeng, and Cong Wang
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2013, Volume 23, Number 4, Page 043120

Comments (0)

Please log in or register to comment.
Log in