Challet, Damien and Peirano, Pier Paolo (2008): The Ups and Downs of Modeling Financial Time Series with Wiener Process Mixtures.
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Abstract
Starting from inhomogeneous time scaling and linear decorrelation between successive price returns, Baldovin and Stella recently proposed a way to build a model describing the time evolution of a financial index. We first make it fully explicit by using Student distributions instead of power lawtruncated Lévy distributions; we also show that the analytic tractability of the model extends to the larger class of symmetric generalized hyperbolic distributions and provide a full computation of their multivariate characteristic functions; more generally, the stochastic processes arising in this framework are representable as mixtures of Wiener processes. The Baldovin and Stella model, while mimicking well volatility relaxation phenomena such as the Omori law, fails to reproduce other stylized facts such as the leverage effect or some time reversal asymmetries. We discuss how to modify the dynamics of this process in order to reproduce real data more accurately.
Item Type:  MPRA Paper 

Original Title:  The Ups and Downs of Modeling Financial Time Series with Wiener Process Mixtures 
Language:  English 
Keywords:  Stylized Facts; Student Processes; Hyperbolic Distributions; Wiener Process Mixtures 
Subjects:  C  Mathematical and Quantitative Methods > C4  Econometric and Statistical Methods: Special Topics > C40  General G  Financial Economics > G1  General Financial Markets > G10  General 
Item ID:  16358 
Depositing User:  Pier Paolo Peirano 
Date Deposited:  27 Jul 2009 07:10 
Last Modified:  04 Oct 2019 16:24 
References:  E. Bacry, J. Delour, and J. F. Muzy. Modelling ﬁnancial time series using multifractal random walks. Physica A, 299(12):84–92, 2001. E. Bacry, J. Delour, and J. F. Muzy. Multifractal random walk. Physical Review E, 64(2): 26103, 2001. E. Bacry, A. Kozhemyak, and J. F. Muzy. Are asset return tail estimations related to volatility longrange correlations? Physica A, 370(1):119–126, Oct 2006. F. Baldovin and A. L. Stella. Central limit theorem for anomalous scaling due to correlations.Physical Review E, 75(2):020101, 2007. F. Baldovin and A. L. Stella. Scaling and eﬃciency determine the irreversible evolution of a market. Proc. Natl. Acad. Sci. USA, 104(50):19741–4, 2007. F. Baldovin and A. L. Stella. Role of scaling in the statistical modeling of ﬁnance. Pramana Journal of Physics, 71:341, 2008. URL http://arxiv.org/abs/0804.0331. O.E. BarndorﬀNielsen and N. Shephard. NonGaussian OrnsteinUhlenbeckbased models and some of their uses in ﬁnancial economics. Journal of the Royal Statistical Society: B, 63 (2):167–241, 2001. N. H. Bingham and R. Kiesel. Modelling asset returns with hyperbolic distributions. In J. Knight and S. Satchell, editors, Return Distributions in Finance, chapter 1, pages 1–20. ButterworthHeinemann, 2001. N. H. Bingham and R. Kiesel. Semiparametric modelling in ﬁnance: theoretical foundations. Quantitative Finance, 1:1–10, 2001. T. Bollerslev, R. F. Engle, and D. B. Nelson. ARCH Models. In R. F. Engle and D. L. McFadden, editors, Handbook of Econometrics, pages 2959–3038. Elsevier, 1994. L. Borland. Option pricing formulas based on a nongaussian stock price model. Physical Review Letters, 89(9):98701, 2002. L. Borland and J. P. Bouchaud. On a multitimescale statistical feedback model for volatility ﬂuctuations. Science & Finance (CFM) working paper archive 500059, Science & Finance, Capital Fund Management, July 2005. L. Borland, J. P. Bouchaud, J. F. Muzy, and G. O. Zumbach. The Dynamics of Financial Markets – Mandelbrot’s multifractal cascades, and beyond. Science & Finance (CFM) working paper archive 500061, Science & Finance, Capital Fund Management, January 2005. J. P. Bouchaud. Elements for a theory of ﬁnancial risks. Physica A, 263:415–426, February 1999. J. P. Bouchaud and M. Potters. Theory of ﬁnancial risk and derivative pricing : from statistical physics to risk management. Cambridge Univ. Press, second edition, 2003. J. P. Bouchaud, M. Potters, and M. Meyer. Apparent multifractality in ﬁnancial time series. European Physical Journal B, 13:595–599, January 2000. J. P. Bouchaud, A. Matacz, and M. Potters. Leverage eﬀect in ﬁnancial markets: The retarded volatility model. Physical Review Letters, 87(22):228701, Nov 2001. P. Carr, H. Geman, D. Madan, and M. Yor. Selfdecomposability and option pricing. Mathematical ﬁnance, 17(1):31–57, 2007. G. Chang and J. Feigenbaum. A bayesian analysis of logperiodic precursors to ﬁnancial crashes. Quantitative Finance, 6:15–36, 2006. U. Cherubini, E. Luciano, and W. Vecchiato. Copula methods in ﬁnance. Wiley Finance. Wiley, 2004. R. Cont. Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1(2):223–236, February 2001. R. Cont and P. Tankov. Financial Modelling with Jump Processes, chapter 4. Financial Mathematics Series. CRC Press, 2004. R Development Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2008. URL http://www.rproject.org. I. Dreier and S. Kotz. A note on the characteristic function of the tdistribution. Statistics & Probability Letters, 57(3):221–224, 2002. S. Drozdz, M. Forczek, J. Kwapien, P. Oswiecimka, and R. Rak. Stock market return distributions: From past to present. Physica A, 383(1):59–64, Sep 2007. E. Eberlein and U. Keller. Hyperbolic distributions in ﬁnance. Bernoulli, 1(3):281–299, 1995. Z. Eisler and J. Kertész. Multifractal model of asset returns with leverage eﬀect. Physica A, 343:603–622, November 2004. A. Erdélyi. Higher Transcendental Functions (Vol. 2). McGraw–Hill Publisher, 1953. A. Helmstetter, D. Sornette, and J. R. Grasso. Mainshocks are aftershocks of conditional foreshocks: How do foreshock statistical properties emerge from aftershock laws. Journal of Geophysical Research, 108:2046, 2003. C. C. Heyde and N. N. Leonenko. Student processes. Advances in Applied Probability, 37: 342–365, 2005. S. Hurst. The characteristic function of the Student tdistribution. Technical Report SRR95044, Austrialian National University, Centre for Mathematics and its Applications, Canberra, September 1995. S. Jaﬀard. Multifractal Formalism for Functions Part I: Results Valid for All Functions. SIAM Journal on Mathematical Analysis, 28:944–970, 1997. S. Jaﬀard. Multifractal Formalism for Functions Part II: SelfSimilar Functions. SIAM Journal on Mathematical Analysis, 28:971–998, 1997. Z. Q. Jiang and W. X. Zhou. Multifractality in stock indexes: Fact or ﬁction? Physica A,387:3605–3614, June 2008. F. Lillo. Limit order placement as an utility maximization problem and the origin of power law distribution of limit order prices. European Physical Journal B, 55:453–459, February 2007. F. Lillo and R. N. Mantegna. Powerlaw relaxation in a complex system: Omori law after a ﬁnancial market crash. Physical Review E, 68(1):016119, Jul 2003. P. E. Lynch and G. O. Zumbach. Market heterogeneities and the causal structure of volatility. Quantitative Finance, 3(4):320–331, 2003. Y. Malevergne and D. Sornette. Extreme Financial Risks. Springer, 2006. B. Mandelbrot, A. Fisher, and L. Calvet. A multifractal model of asset returns. Cowles Foundation Discussion Papers 1164, Cowles Foundation, Yale University, September 1997. J. L. McCauley, K. E. Bassler, and G. H. Gunaratne. Martingales, the eﬃcient market hypothesis, and spurious stylized facts, October 2007. URL http://arxiv.org/abs/0710. M. Musiela and M. Rutkowski. Martingale Methods in Financial Modelling, chapter 7, pages 237–278. Springer Verlag, second edition, 2005. J. F. Muzy, E. Bacry, and A. Kozhemyak. Extreme values and fat tails of multifractal ﬂuctuations. Physical Review E, 73(6):066114, 2006. R. B. Nelsen. An introduction to copulas. Springer Series in Statistics. Springer, second edition, 2006. R. Osorio, L. Borland, and C. Tsallis. Distributions of highfrequency stock market observables. In M. GellMann and C. Tsallis, editors, Nonextensive entropy: interdisciplinary applications, page 321. Oxford University Press, 2004. R. H. Riedi. Multifractal processes. In P. Doukhan, G. Oppenheim, and M. S. Taqqu, editors, Longrange Dependence: Theory and Applications, pages 625–716. Birkhauser, 2002. I.J. Schoenberg. Positive deﬁnite functions on spheres. Duke Math. Journal, 9:96–108, 1942. I. M. Sokolov, A. V. Chechkin, and J. Klafter. Fractional diﬀusion equation for a powerlaw truncated Lévy process. Physica A, 336(34):245–251, May 2004. D. Sornette. Critical market crashes. Physics Reports, 378(1):1–98, 2003. D. Sornette, Y. Malevergne, and J. F. Muzy. Volatility ﬁngerprints of large shocks: Endogeneous versus exogeneous. arxiv:condmat/0204626. The Journal of Risk, 16(2):67–71, February 2003. R. S. Tsay. Analysis of Financial Time Series, chapter 3. John Wiley & Sons, 2002. M. Vellekoop and H. Nieuwenhuis. On option pricing models in the presence of heavy tails. Quantitative Finance, 7(5):563–573, Oct 2007. C. Vignat and A. Plastino. Scale invariance and related properties of qGaussian systems. Physics Letters A, 365:370–375, June 2007. P. Weber, F. Wang, I. VodenskaChitkushev, S. Havlin, and H. E. Stanley. Relation between volatility correlations in ﬁnancial markets and Omori processes occurring on all scales. Physical Review E, 76(1):016109, 2007. G. O. Zumbach. Volatility processes and volatility forecast with long memory. Quantitative Finance, 4(1):70–86, 2004. G. O. Zumbach. Time reversal invariance in ﬁnance, August 2007. URL http://arxiv.org/abs/0708.4022. G. O. Zumbach, M. M. Dacorogna, J. L. Olsen, and R. B. Olsen. Measuring shock in ﬁnancial markets. International Journal of Theoretical and Applied Finance, 3:347–355, 2000. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/16358 
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The ups and downs of the renormalization group applied to financial time series. (deposited 30 Jul 2008 10:30)
 The Ups and Downs of Modeling Financial Time Series with Wiener Process Mixtures. (deposited 27 Jul 2009 07:10) [Currently Displayed]