Skip to main content
SpringerLink
Log in

Bayesian Maximum Likelihood Estimation in Fractional Stochastic Volatility Model

  • Chapter
  • First Online:
Parameter Estimation in Stochastic Volatility Models

  • 618 Accesses

Abstract

In mathematical finance, it is well accepted that volatility of a stock price is a stochastic process, not a constant. It is also known that volatility has long memory and clusters on high level. One way of modeling long memory is superposition of Ornstein–Uhlenbeck (supOU) processes as volatility models. The class of supOU processes can capture extremal clusters and long-range dependence. We consider volatility as a continuous model satisfying a stochastic differential equation driven by a persistent fractional Brownian motion. Long memory in volatility is a stylized fact in finance due to volatility clustering and persistence.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Alos, E., Mazet, O. and Nualart, D. (2000): Stochastic calculus with respect to fractional Brownian motion with Hurst parameter less than \(\frac {1}{2}\), Stoch. Proc. Appl. 86 (1), 121–139.

    MATH  Google Scholar 

  • Alos, E., Mazet, O. and Nualart, D. (2001): Stochastic calculus with respect to Gaussian processes, Annals of Probability 29 (2), 766–801.

    MathSciNet  MATH  Google Scholar 

  • Beran, R. (1994): Statistics for Long-Memory Processes, Chapman & Hall: New York.

    MATH  Google Scholar 

  • Bishwal, J.P.N. (2008a): Parameter Estimation in Stochastic Differential Equations, Lecture Notes in Mathematics, 1923, Springer-Verlag.

    Google Scholar 

  • Bishwal, J.P.N. (2011b): Minimum contrast estimation in fractional Ornstein-Uhlenbeck process: continuous and discrete sampling, Fractional Calculus and Applied Analysis 14 (3), 375–410.

    MathSciNet  MATH  Google Scholar 

  • Bishwal, J.P.N. (2011c): Sieve estimator for fractional stochastic partial differential equation, Annals of Constantin Brancusi-Engineering Series 5 (1), 9–18.

    MathSciNet  Google Scholar 

  • Bishwal, J.P.N. (2011i): Maximum quasi-likelihood estimation in fractional Levy stochastic volatility model, Journal of Mathematical Finance 1 (3), 12–15.

    MathSciNet  Google Scholar 

  • Bishwal, J.P.N. (2017b): Valuation of real options under persistent shocks, Journal of Statistics and Management Systems 20 (5), 801–815.

    Google Scholar 

  • Bishwal, J.P.N. (2017c): Hypothesis testing for fractional stochastic partial differential equations with applications to neurophysiology and finance, Asian Research Journal of Mathematics 4 (1), 1–24.

    MathSciNet  Google Scholar 

  • Campillo, F. and Le Gland, F. (1989): MLE for partially observed diffusions: direct maximization vs. the EM algorithm, Stoch. Process. Appl. 33, 245–274.

    MathSciNet  MATH  Google Scholar 

  • Coutin, L. and Decreusefond, L. (1997): Stochastic differential equations driven by a fractional Brownian motion, Tech. Report 97C004, ENST Paris.

    Google Scholar 

  • Coutin, L. and Decreusefond, L. (1999): Abstract nonlinear filtering in the presence of fractional Brownian motion, Ann. Applied Probab. 9 (4), 1058–1090.

    MathSciNet  MATH  Google Scholar 

  • Dai, W. and Heyde, C.C. (1996): ItĂ´ formula with respect to fractional Brownian motion and its applications, J. Appl. Math. Stoch. Anal., 9, 439–448.

    MathSciNet  MATH  Google Scholar 

  • Decreusefond, L. and Ustunel, A.S. (1998): Fractional Brownian motion: theory and applications, In: Fractional Differential Systems: Models, Methods and Applications, ESAIM Proceedings 5, 75–86.

    MathSciNet  MATH  Google Scholar 

  • Decreusefond, L. and Ustunel, A.S. (1999): Stochastic analysis of the fractional Brownian motion, Potential Analysis 10, 177–214.

    MathSciNet  MATH  Google Scholar 

  • Duncan, T.E., Hu, Y. and Pasik-Duncan, B. (2000): Stochastic calculus for fractional Brownian motion I, SIAM J. Control. Optim. 38, 582–612.

    MathSciNet  MATH  Google Scholar 

  • James, M. and Le Gland, F. (1995): Consistent parameter estimation for partially observed diffusions with small noise, Applied Math. Optim. 32, 47–72.

    MathSciNet  MATH  Google Scholar 

  • Kallianpur, G. (1980): Stochastic Filtering Theory, Springer-Verlag: New York.

    MATH  Google Scholar 

  • Kallianpur, G. and Selukar, R.S. (1991): Parameter estimation in linear filtering, J. Multivariate Anal. 39, 284–304.

    MathSciNet  MATH  Google Scholar 

  • Kitagawa, G. (1987): Non-Gaussian state space modelling of nonstationary time series (with discussion), Journal of the American Statistical Association 82, 1032–1063.

    MathSciNet  MATH  Google Scholar 

  • Kleptsyna, M.L. Kloeden, P.E. and Anh, V.V. (1998a): Linear filtering with fractional Brownian motion, Stoch. Anal. Appl. 16, 907–914.

    MathSciNet  MATH  Google Scholar 

  • Kleptsyna, M.L., Kloeden, P.E. and Anh, V.V. (1998b): Nonlinear filtering with fractional Brownian motion, Problems of Information Transmission 34 (2) 65–76.

    MathSciNet  MATH  Google Scholar 

  • Kleptsyna, M.L. , Le Breton, A. and Roubaud, M.C. (1998): An elementary approach to filtering in systems with fractional Brownian motion observation noise, In: Probability Theory and Mathematical Statistics, B. Grigelionis (Eds.), VSP/TEV, 373–392.

    Google Scholar 

  • Kleptsyna, M.L. , Le Breton, A. , Roubaud, M.C. (2000): General approach to filtering with fractional Brownian noises- application to linear systems, Stochastics and Stochastics Reports 71, 119–140.

    MathSciNet  MATH  Google Scholar 

  • Kolmogorov, A.N. (1940): Wiener skewline and other interesting curves in Hilbert space, Doklady Akad. Nauk 26, 115–118.

    Google Scholar 

  • Konecky, F. (1990): Maximum likelihood estimator from a partially observed diffusions in the case of small measurement noise, Statist. Decisions 8, 115–130.

    MathSciNet  Google Scholar 

  • Kutoyants, Yu. A. (1984a): Parameter Estimation for Stochastic Processes (Translated and edited by B.L.S. Prakasa Rao), Research and Expositions in Mathematics, Vol. 6, Heldermann-Verlag, Berlin.

    Google Scholar 

  • Kutoyants, Yu. A. (1984b): Parameter estimation for diffusion type processes of observations, Math. Operationsforch. U. Statist., Ser. Statist. 15, 541–551.

    MathSciNet  MATH  Google Scholar 

  • Kutoyants, Yu. A. (1994a): Identification of Dynamical Systems with Small Noise, Kluwer, Dordrecht.

    Book  MATH  Google Scholar 

  • Kutoyants, Yu. A. (1994b): Nonconsistent estimation by diffusion type observations, Statist. Probab. Letters. 20, 1–7.

    Article  MathSciNet  MATH  Google Scholar 

  • Kutoyants, Yu. A. and Pohlman, H. (1994): Parameter estimation for Kalman-Bucy filter with small noise, Statistics 25, 307–323.

    MathSciNet  MATH  Google Scholar 

  • Le Breton, A. (1998a): Filtering and parameter estimation in a simple linear system driven by fractional Brownian motion, Statist. Probab. Letters 38, 263–274.

    MathSciNet  MATH  Google Scholar 

  • Le Breton, A. (1998b): A Girsanov type approach to filtering in a simple linear system driven by fractional Brownian motion, C. R. Acad. Sci. Paris, Series I, 326, 997–1002.

    MATH  Google Scholar 

  • Levy, P. (1948): Processus stochastiques et movement Brownien, Paris.

    MATH  Google Scholar 

  • Lin, S. J. (1995): Stochastic analysis of fractional Brownian motions, Stochastics and Stochastics Reports 55, 121–140.

    MathSciNet  MATH  Google Scholar 

  • Liptser, R.S. and Shiryayev, A.N. (1989): Theory of Martingales, Kluwer, Dordrecht.

    MATH  Google Scholar 

  • Lyons, T.J. and Zhang, T.S. (1994): Decomposition of Dirichlet processes and its application, Ann. Probab. 22, 494–524.

    MathSciNet  MATH  Google Scholar 

  • Mandelbrot, B. and Van Ness, J.W. (1968): Fractional Brownian motions, fractional noises and applications, SIAM Review 10, 422–437.

    Article  MathSciNet  MATH  Google Scholar 

  • Norros, I., Valkeila, E. and Virtamo, J. (1999): An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motion, Bernoulli 5, 571–587.

    Article  MathSciNet  MATH  Google Scholar 

  • Peltier, R.F., Levy Vehel, J. (1994): A new method for estimating the parameter of fractional Brownian motion, INRIA Research Report No. 2396.

    Google Scholar 

  • Revuz, D. and Yor, M. (1991): Continuous Martingales and Brownian Motion, Springer-Verlag.

    Book  MATH  Google Scholar 

  • Ruzmaikina, A.A. (2000): Stochastic calculus for fractional Brownian motion, J. Statistical Physics, 100, 1049–1069.

    MathSciNet  MATH  Google Scholar 

  • Zahle, M. (1998): Integration with respect to fractal functions and stochastic calculus, Prob. Theory. Rel. Fields 111, 333–374.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC, USA

    Jaya P. N. Bishwal

Authors
  1. Jaya P. N. Bishwal
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bishwal, J.P.N. (2022). Bayesian Maximum Likelihood Estimation in Fractional Stochastic Volatility Model. In: Parameter Estimation in Stochastic Volatility Models. Springer, Cham. https://doi.org/10.1007/978-3-031-03861-7_12

Download citation

Publish with us

Policies and ethics

Search

Navigation