Challet, Damien and Peirano, Pier Paolo (2008): The ups and downs of the renormalization group applied to financial time series.
Download (786kB) | Preview
Starting from inhomogeneous time scaling and linear decorrelation between successive price returns, Baldovin and Stella recently devised a model describing the time evolution of a financial index. We first make it fully explicit by using Student distributions instead of power law-truncated Levy 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. 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 the renormalization group applied to financial time series|
|Keywords:||Stylized Facts; Student Processes; Hyperbolic Distributions; Renormalization Group|
|Subjects:||C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C40 - General
G - Financial Economics > G1 - General Financial Markets > G10 - General
|Depositing User:||Pier Paolo Peirano|
|Date Deposited:||30. Jul 2008 10:30|
|Last Modified:||22. Feb 2013 09:05|
E. Bacry, J. Delour, and J. F. Muzy. Modelling ﬁnancial time series using multifractal random walks. Physica A, 299(1-2):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 long-range 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, 2008. URL http://arxiv.org/abs/0804.0331. Based on the Key Note lecture by A.L. Stella at the Conference on “Statistical Physics Approaches to Multi-Disciplinary Problems”, IIT Guwahati, India, 7-13 January 2008.
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. Butterworth-Heinemann, 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 non-gaussian stock price model. Physical Review Letters, 89(9):98701, 2002.
L. Borland and J. P. Bouchaud. On a multi-timescale 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.
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.
I. Dreier and S. Kotz. A note on the characteristic function of the t-distribution. 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.
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 t distribution. Technical Report SRR95-044, 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: Self-Similar 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. Power-law 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. 2583.
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 high-frequency stock market observables. In M. Gell-Mann and C. Tsallis, editors, Nonextensive entropy: interdisciplinary applications, page 321. Oxford University 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.r-project. org.
R. H. Riedi. Multifractal processes. In P. Doukhan, G. Oppenheim, and M. S. Taqqu, editors, Long-range Dependence: Theory and Applications, pages 625–716. Birkhauser, 2002.
I. M. Sokolov, A. V. Chechkin, and J. Klafter. Fractional diﬀusion equation for a power-law-truncated Lévy process. Physica A, 336(3-4):245–251, May 2004.
R. S. Tsay. Analysis of Financial Time Series, chapter 3. John Wiley & Sons, 2002.
C. Vignat and A. Plastino. Scale invariance and related properties of q-Gaussian systems. Physics Letters A, 365:370–375, June 2007.
P. Weber, F. Wang, I. Vodenska-Chitkushev, 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.
Available Versions of this Item
- The ups and downs of the renormalization group applied to financial time series. (deposited 30. Jul 2008 10:30) [Currently Displayed]