Electronic Communications in Probability

Fractional Brownian motion with zero Hurst parameter: a rough volatility viewpoint

Eyal Neuman and Mathieu Rosenbaum

Full-text: Open access

Abstract

Rough volatility models are becoming increasingly popular in quantitative finance. In this framework, one considers that the behavior of the log-volatility process of a financial asset is close to that of a fractional Brownian motion with Hurst parameter around 0.1. Motivated by this, we wish to define a natural and relevant limit for the fractional Brownian motion when $H$ goes to zero. We show that once properly normalized, the fractional Brownian motion converges to a Gaussian random distribution which is very close to a log-correlated random field.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 61, 12 pp.

Dates
Received: 1 November 2017
Accepted: 26 July 2018
First available in Project Euclid: 12 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1536718014

Digital Object Identifier
doi:10.1214/18-ECP158

Mathematical Reviews number (MathSciNet)
MR3863917

Zentralblatt MATH identifier
1401.60064

Subjects
Primary: 60G22: Fractional processes, including fractional Brownian motion 60G15: Gaussian processes 60G57: Random measures
Secondary: 60G18: Self-similar processes 28A80: Fractals [See also 37Fxx]

Keywords
fractional Brownian motion log-correlated random field rough volatility multifractal processes

Rights
Creative Commons Attribution 4.0 International License.

Citation

Neuman, Eyal; Rosenbaum, Mathieu. Fractional Brownian motion with zero Hurst parameter: a rough volatility viewpoint. Electron. Commun. Probab. 23 (2018), paper no. 61, 12 pp. doi:10.1214/18-ECP158. https://projecteuclid.org/euclid.ecp/1536718014


Export citation

References

  • [1] E. Bacry, J. Delour, and J.F. Muzy. Multifractal random walk. Physical Review E, 64(2):026103, 2001.
  • [2] E. Bacry and J.F. Muzy. Log-infinitely divisible multifractal processes. Communications in Mathematical Physics, 236(3):449–475, Jun 2003.
  • [3] J. Barral and B.B. Mandelbrot. Multifractal products of cylindrical pulses. Probability Theory and Related Fields, 124(3):409–430, Nov 2002.
  • [4] C. Bayer, P.K. Friz, P. Gassiat, J. Martin, and B. Stemper. A regularity structure for rough volatility. arXiv preprint arXiv:1710.07481, 2017.
  • [5] C. Bayer, P.K. Friz, and J. Gatheral. Pricing under rough volatility. Quantitative Finance, 16(6):887–904, 2016.
  • [6] C. Bayer, P.K. Friz, A. Gulisashvili, B. Horvath, and B. Stemper. Short-time near-the-money skew in rough fractional volatility models. arXiv preprint arXiv:1703.05132, 2017.
  • [7] M. Bennedsen, A. Lunde, and M.S. Pakkanen. Decoupling the short-and long-term behavior of stochastic volatility. arXiv preprint arXiv:1610.00332, 2016.
  • [8] M. Bennedsen, A. Lunde, and M.S. Pakkanen. Hybrid scheme for Brownian semistationary processes. Finance and Stochastics, 21(4):931–965, 2017.
  • [9] L. E. Calvet and A.J. Fisher. How to forecast long-run volatility: Regime switching and the estimation of multifractal processes. Journal of Financial Econometrics, 2(1):49–83, 2004.
  • [10] L. Chevillard, R. Robert, and V. Vargas. A stochastic representation of the local structure of turbulence. EPL (Europhysics Letters), 89(5):54002, 2010.
  • [11] F. Comte and E. Renault. Long memory in continuous-time stochastic volatility models. Mathematical Finance, 8(4):291–323, 1998.
  • [12] B. Duplantier, R. Rhodes, S. Sheffield, and V. Vargas. Log-correlated Gaussian fields: an overview. arXiv preprint arXiv:1407.5605, 2014.
  • [13] O. El Euch and M. Rosenbaum. The characteristic function of rough Heston models. Mathematical Finance, to appear, 2016.
  • [14] M. Forde and H. Zhang. Asymptotics for rough stochastic volatility and Lévy models. preprint available at http://www.mth.kcl.ac.uk/fordem, 2015.
  • [15] U. Frisch and G. Parisi. On the singularity structure of fully developed turbulence. Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, pages 84–88, 1985.
  • [16] M. Fukasawa. Short-time at-the-money skew and rough fractional volatility. Quantitative Finance, 17(2):189–198, 2017.
  • [17] Y. V. Fyodorov and J.P. Bouchaud. Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential. Journal of Physics A: Mathematical and Theoretical, 41(37):372001, 2008.
  • [18] Y. V. Fyodorov, P. Le Doussal, and A. Rosso. Freezing transition in decaying burgers turbulence and random matrix dualities. EPL (Europhysics Letters), 90(6):60004, 2010.
  • [19] Y. V. Fyodorov, B. A. Khoruzhenko, and N. J. Simm. Fractional Brownian motion with hurst index ${H}=0$ and the Gaussian unitary ensemble. The Annals of Probability, 44(4):2980–3031, 2016.
  • [20] J. Gatheral, T. Jaisson, and M. Rosenbaum. Volatility is rough. arXiv preprint arXiv:1410.3394, 2014.
  • [21] A. Jacquier, M.S. Pakkanen, and H. Stone. Pathwise large deviations for the rough Bergomi model. arXiv preprint arXiv:1706.05291, 2017.
  • [22] J.P. Kahane. Sur le chaos multiplicatif. Prépublications mathématiques d’Orsay. Département de mathématique, 1985.
  • [23] W.E. Leland, M.S. Taqqu, W. Willinger, and D.V. Wilson. On the self-similar nature of ethernet traffic (extended version). IEEE/ACM Transactions on networking, 2(1):1–15, 1994.
  • [24] A. Lodhia, S. Sheffield, X. Sun, and S. S. Watson. Fractional Gaussian fields: a survey. arXiv preprint arXiv:1407.5598v2, 2016.
  • [25] T. Madaule, R. Rhodes, and V. Vargas. Glassy phase and freezing of log-correlated Gaussian potentials. The Annals of Applied Probability, 26(2):643–690, 2016.
  • [26] B.B. Mandelbrot, A.J. Fisher, and L. Calvet. A multifractal model of asset returns. Cowles Foundation Discussion Papers 1164, Cowles Foundation for Research in Economics, Yale University, September 1997.
  • [27] B.B. Mandelbrot, J.W. Van Ness. Fractional Brownian Motions, Fractional Noises and Applications. SIAM Rev., 10(4):422–437, 1968.
  • [28] T. Mikosch, S. Resnick, H. Rootzén, and A. Stegeman. Is network traffic approximated by stable Lévy motion or fractional Brownian motion? The Annals of Applied Probability, 12(1):23–68, 2002.
  • [29] F.J. Molz, H.H. Liu, and J. Szulga. Fractional Brownian motion and fractional Gaussian noise in subsurface hydrology: A review, presentation of fundamental properties, and extensions. Water Resources Research, 33(10):2273–2286, 1997.
  • [30] R. Rhodes and V. Vargas. Gaussian multiplicative chaos and applications: a review. Probability Surveys, 11:315–392, 2014.
  • [31] R. Rhodes and V. Vargas. Lecture notes on Gaussian multiplicative chaos and Liouville quantum gravity. arXiv prepint arXiv:1602.07323, 2016.
  • [32] R. Robert and V. Vargas. Gaussian multiplicative chaos revisited. The Annals of Probability, 38(2):605–631, 2010.
  • [33] A. Shamov. On Gaussian multiplicative chaos. Journal of Functional Analysis, 270(9):3224–3261, 2016.