Skip to main content
SpringerLink
Log in

On the Modeling of Financial Time Series

  • Chapter
  • First Online:
Financial Econometrics and Empirical Market Microstructure

  • 2838 Accesses

Abstract

This paper discusses issues related to modeling of financial time series. We discuss so-called empirical “stylized facts” of real price time-series and the evolution of financial models from trivial random walk introduced by Louis Bachelier in 1900 to modern multifractal models, that nowadays are the most parsimonious and flexible models of stochastic volatility. We focus on a particular model of Multifractal Random Walk (MRW), which is the only continuous stochastic stationary causal process with exact multifractal properties and Gaussian infinitesimal increments. The paper presents a method of numerical simulation of realizations of MRW using the Circulant Embedding Method and discuss methods of its calibration.

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 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
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

Notes

  1. 1.

    We must notice that typically stochastic volatility models are defined not within the framework of Eq. (2), but as an extension of stochastic differential equation of the geometric Brownian motion. Strictly speaking, these equations do not always have solution in form of (2).

References

  • Aït-Sahalia, Y. & Hansen, L., (Eds.). (2009). Handbook of financial econometrics. Amsterdam: North Holland.

    Google Scholar 

  • Arneodo, A., Bacry, E., & Muzy, J.-F. (1998a). Random cascades on wavelet dyadic trees. Journal of Mathematical Physics, 39(8), 4142–4164.

    Article  Google Scholar 

  • Arneodo, A., Muzy, J.-F., & Sornette, D. (1998b). “Direct” causal cascade in the stock market. The European Physical Journal B - Condensed Matter and Complex Systems, 2(2), 277–282.

    Article  Google Scholar 

  • Bachelier, L. (1900). Théorie de la spéculation. Annales scientifiques de I’École normale supérieure, 3(17), 21–86.

    Google Scholar 

  • Bacry, E., Delour, J., & Muzy, J.-F. (2000). A multivariate multifractal model for return fluctuations. arXiv:cond-mat/0009260.

    Google Scholar 

  • Bacry, E., Delour, J., & Muzy, J.-F. (2001). Multifractal random walk. Physical Review E, 64(2), 456, 026103C.

    Google Scholar 

  • Bacry, E., Kozhemyak, A., & Muzy, J.-F. (2008). Log-Normal continuous cascades: aggregation properties and estimation. Application to financial time-series. arXiv:q-fin/0804.0185.

    Google Scholar 

  • Bacry, E. & Muzy, J.-F. (2003). Log-infinitely divisible multifractal processes. Communications in Mathematical Physics, 236(3), 449–475.

    Article  Google Scholar 

  • Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307–327.

    Article  Google Scholar 

  • Bollerslev, T. (2010). Glossary to ARCH (GARCH). Volatility and Time Series Econometrics, 28, 137–164.

    Google Scholar 

  • Bouchaud, J.-P., Matacz, A., & Potters, M. (2001). Leverage effect in financial markets: the retarded volatility model. Physical Review Letters, 87(22), 228701+.

    Google Scholar 

  • Bouchaud, J.-P. & Potters, M. (2000). Theory of financial risks: from statistical physics to risk management. Cambridge: Cambridge University Press.

    Google Scholar 

  • Breymann, W., Ghashghaie, S., & Talkner, P. (2000). A stochastic cascade model for FX dynamics. International Journal of Theoretical and Applied Finance, (3), 357–360.

    Google Scholar 

  • Calvet, L. E. & Fisher, A. J. (2002). Multifractality in asset returns: theory and evidence. Review of Economics and Statistics, 84(3), 381–406.

    Article  Google Scholar 

  • Calvet, L. E. & Fisher, A. J. (2004). How to forecast long-run volatility: regime switching and the estimation of multifractal processes. Journal of Financial Econometrics, 2(1), 49–83.

    Article  Google Scholar 

  • Calvet, L. E. & Fisher, Adlai J. (2008). Multifractal volatility theory, forecasting, and pricing. Burlington, MA: Academic Press. ISBN 9780080559964.

    Google Scholar 

  • Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1, 223–236.

    Article  Google Scholar 

  • Davis, M. (1987). Production of conditional simulations via the LU triangular decomposition of the covariance matrix. Mathematical Geology, 19(2), 91–98.

    Google Scholar 

  • Davis, R. A., & Mikosch, T. (2009). Extreme value theory for GARCH processes. In T. G. Andersen, R. A. Davis, J.-P. Kreiss, & T. Mikosch (Eds.), Handbook of financial time series (pp. 187–200). Springer: New York.

    Chapter  Google Scholar 

  • Dembo, A., Mallows, C. L., & Shepp, L. A. (1989). Embedding nonnegative definite Toeplitz matrices in nonnegative definite circulant matrices, with application to covariance estimation. IEEE Transactions on Information Theory, 35(6), 1206–1212.

    Article  Google Scholar 

  • Dietrich, C. R. & Newsam, G. N. (1997). Fast and exact simulation of stationary gaussian processes through circulant embedding of the covariance matrix. SIAM Journal on Scientific and Statistical Computing, 18(4), 1088–1107.

    Article  Google Scholar 

  • Engle, R. (2001). GARCH 101: The use of ARCH/GARCH models in applied econometrics. The Journal of Economic Perspectives, 15(4), 157–168.

    Article  Google Scholar 

  • Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the Econometric Society, 50(4), 987–1007.

    Article  Google Scholar 

  • Fama, E. F. (1970). Efficient capital markets: a review of theory and empirical work. The Journal of Finance, 25(2), 383–417.

    Article  Google Scholar 

  • Fama, E. F. (1991). Efficient capital markets: II. Journal of Finance, 46(5), 1575–1617.

    Article  Google Scholar 

  • Filimonov, V. (2010). Multifractal Models of Financial Time Series. HSE — Working Paper Series, P1/2010/06, (pp.4̃5).

    Google Scholar 

  • Filimonov, V. & Sornette, D. (2011). Self-excited multifractal dynamics. Europhysics Letters, 94(4), 46003.

    Article  Google Scholar 

  • Ghashghaie, S., Breymann, W., Peinke, J., Talkner, P., & Dodge, Y. (1996). Turbulent cascades in foreign exchange markets. Nature, 381, 767–770.

    Article  Google Scholar 

  • Gu, G.-F. & Zhou, W.-X. (2010). Detrending moving average algorithm for multifractals. Physical Review E, 82(1), 011136+.

    Google Scholar 

  • Kantelhardt, J. W., Zschiegner, S. A., Koscielny-Bunde, E., Havlin, S., Bunde, A., & Stanley, H. E. (2002). Multifractal detrended fluctuation analysis of nonstationary time series. Physica A: Statistical Mechanics and its Applications, 316(1-4), 87–114.

    Article  Google Scholar 

  • Kolmogorov, A. N. (1941). The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokladi Akademii Nauk SSSR, XXXI, 299–303.

    Google Scholar 

  • Kolmogorov, A. N. (1962). A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. Journal of Fluid Mechanics, 13(1), 82–85.

    Article  Google Scholar 

  • Liu, R., Matteo, T., & Lux, T. (2008). Multifractality and Long-Range Dependence of Asset Returns: The Scaling Behaviour of the Markov-Switching Multifractal Model with Lognormal Volatility Components. Kiel Working Papers, (pp. 1–15).

    Google Scholar 

  • Lux, T. (2009). Stochastic Behavioral Asset-Pricing Models and the Stylized Facts. In Handbook of Financial Markets: Dynamics and Evolution (pp. 161–215). Amsterdam: North Holland.

    Chapter  Google Scholar 

  • Mandelbrot, B. B. (1975). Les objets fractals: forme, hasard et dimension. Paris: Flammarion.

    Google Scholar 

  • Mandelbrot, B. B. (1982). The fractal geometry of nature. San Francisco: W. H. Freeman and Company.

    Google Scholar 

  • Mandelbrot, B. B. (1985). Self-affine fractals and fractal dimension. Physica Scripta, 32(4), 257–260.

    Article  Google Scholar 

  • Mandelbrot, B. B., Fisher, A. J., & Calvet, L. E. (1997). A Multifractal Model of Asset Returns. Cowles Foundation Discussion Paper #1164.

    Google Scholar 

  • Mandelbrot, B. B. & Van Ness, J. W. (1968). Fractional Brownian Motions, fractional noises and applications. SIAM Review, 10(4), 422–437.

    Article  Google Scholar 

  • Müller, U. A. (2000). Specially Weighted Moving Averages with Repeated Application of the EMA Operator. Technical report.

    Google Scholar 

  • Muzy, J.-F., Delour, J., & Bacry, E. (2000). Modelling fluctuations of financial time series: from cascade process to stochastic volatility model. The European Physical Journal B - Condensed Matter and Complex Systems, 17(3), 537–548.

    Article  Google Scholar 

  • Pochart, B. & Bouchaud, J.-P. (2002). The skewed multifractal random walk with applications to option smiles. Quantitative Finance, 2(4), 303–314.

    Article  Google Scholar 

  • Richardson, L. F. (1961). The problem of contiguity: an appendix of statistics of deadly quarrels. General Systems Yearbook, 6, 139–187.

    Google Scholar 

  • Saichev, A. & Filimonov, V. (2007). On the spectrum of multifractal diffusion process. Journal of Experimental and Theoretical Physics, 105(5), 1085–1093.

    Article  Google Scholar 

  • Saichev, A. & Filimonov, V. (2008). Numerical simulation of the realizations and spectra of a quasi-multifractal diffusion process. JETP Letters, 87(9), 506–510.

    Article  Google Scholar 

  • Saichev, A. & Sornette, D. (2006). Generic multifractality in exponentials of long memory processes. Physical Review E, 74(1), 011111+.

    Google Scholar 

  • Siven, J. V., & Lins, J. T. (2009). Gain/loss asymmetry in time series of individual stock prices and its relationship to the leverage effect. arXiv:0911.4679v2.

    Google Scholar 

  • Sornette, D. (2003). Why stock markets crash: critical events in complex financial systems. Princeton: Princeton University Press.

    Google Scholar 

  • Zakoian, J.-M. & Francq, C. (2010). GARCH models: structure, statistical inference and financial applications. Oxford: Wiley-Blackwell.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Prognoz Risk Lab, Perm, Russian Federation

    Aleksey Kutergin

  2. Department of Management, Technology and Economics ETH Zürich, Zürich, Switzerland

    Vladimir Filimonov

Authors
  1. Aleksey Kutergin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladimir Filimonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Economics, University of Illinois, Urbana, Illinois, USA

    Anil K. Bera

  2. Perm State University, Perm, Russia

    Sergey Ivliev

  3. Scuola Normale Superiore, Pisa, Italy

    Fabrizio Lillo

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kutergin, A., Filimonov, V. (2015). On the Modeling of Financial Time Series. In: Bera, A., Ivliev, S., Lillo, F. (eds) Financial Econometrics and Empirical Market Microstructure. Springer, Cham. https://doi.org/10.1007/978-3-319-09946-0_10

Download citation

Publish with us

Policies and ethics

Search

Navigation