Log-normal continuous cascade model of asset returns: Aggregation properties and estimation

Quantitative Finance

Volumn 13, Issue 5, 2013, Pages 795-818

  Bacry, E ^a   Kozhemyak, A ^b   Muzy, J F ^c  

^a CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

^b Calyon RPC DRM MCRA   (France)

^c UNIVERSITÉ DE CORSE   (France)

Author keywords

Econophysics; Empirical time series analysis; Fractals; Heteroskedasticity; Multifractal model of asset returns; Statistical methods; Stochastic volatility; Volatility modelling

Indexed keywords

EID: 84877309885     PISSN: 14697688     EISSN: 14697696     Source Type: Journal    
DOI: 10.1080/14697688.2011.647411     Document Type: Article

References (41)

1. Andersen, TG, Bollerslev, T, Christoffersen, P and Diebold, FX. 2005. " Volatility and correlation forecasting ". In Handbook of Economic Forecasting, Edited by: Elliott, G, Granger, CWJ and Timmermann, A. Amsterdam: North-Holland.
2. Bacry, E, Delour, J and Muzy, JF. 2001. Multifractal random walks. Phys. Rev. E, 64: 026103 - 026106.
3. Bacry, E, Gloter, A, Hoffmann, M and Muzy, JF. Preprint, 2008a. Estimation of the multifractal spectrum in a mixed asymptotic framework,
4. Bacry, E, Kozhemyak, A and Muzy, JF. 2008b. Continuous cascade models for asset returns. J. Econ. Dynam. Control, 32: 156 - 199.
5. Bacry, E and Muzy, JF. 2003. Log-infinitely divisible multifractal process. Commun. Math. Phys., 236: 449 - 475.
6. Barral, J. 2004. Non-degeneracy, Moments, Dimension, and Multifractal Analysis for Random Multiplicative Measures (Random Multiplicative Multifractal Measures, Part III), Vol. 72, 53 - 90. Providence: AMS.
7. Barral, J and Mandelbrot, BB. 2002. Multifractal products of cylindrical pulses. Prob. Theory Relat. Fields, 124: 409 - 430.
8. Barral, J and Mandelbrot, BB. 2004. Non-degeneracy, moments, dimension, and multifractal analysis for random multiplicative measures (random multiplicative multifractal measures, Part II). Proc. Symp. Pures Math., 72: 17 - 52.
9. Bollerslev, T. 1986. A generalized autoregressive conditional heteroskedasticity. J. Econometr., 31: 307 - 327.
10. Calvet, L and Fisher, A. 2001. Forecasting multifractal volatility. J. Econometr., 105: 27 - 58.
11. Calvet, L and Fisher, A. 2002. Multifractality in asset returns: Theory and evidence. Rev. Econ. Statist., 84: 381 - 406.
12. Calvet, L and Fisher, A. 2004. How to forecast long-run volatility: Regime-switching and the estimation of multifractal processes. J. Financ. Econometr., 2: 49 - 83.
13. L.E. Calvet, Fisher, A.J. and Thompson, S.B., Volatility comovement: A multifrequency approach. J. Econometrics, 2006, 131, 179-215
14. Chainais, P, Riedi, R and Abry, P. 2003. " Scale invariant infinitely divisible cascades ". In In International Symposium on Physics in Signal and Image Processing, Grenoble
15. Christoffersen, P. 1998. Evaluating interval forecasts. Int. Econ. Rev., 39: 841 - 862.
16. Chung, KL. 1974. A Course in Probability Theory, 2nd, London: Academic Press.
17. Guivarc'h, Y. 1987. Remarques sur les solutions d'une équation fonctionnelle non linéaire de b. mandelbrot. C.R. Acad. Sci. Paris, 305: 139 - 141.
18. Hall, AR. 2005. Generalized method of moments,
19. Hansen, LP. 1982. Large sample properties of generalized method of moments estimators. Econometrica, 50: 1029 - 1054.
20. Kahane, JP. 1987. Positive martingales and random measures. Chin. Ann. Math., 8B: 1 - 12.
21. Kahane, JP and Peyrière, J. 1976. Sur certaines martingales de benoit mandelbrot. Adv. Math., 22: 131 - 145.
22. Kozhemyak, A. 2006. " Modélisation de séries financières à l'aide de processus invariants d'échelle. Application à la prédiction du risque ". In PhD thesis, Ecole Polytechnique
23. Kupiec, P. 1995. Techniques for verifying the accuracy of risk measurement models. J. Deriv., 2: 73 - 84.
24. Lee, SW and Hansen, BE. 1994. Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometr. Theory, 10: 29 - 52.
25. Liu, Q. 2002. An extension of a functional equation of Poincaré and Mandelbrot. Asian J. Math., 6: 145 - 168.
26. Lux, T. 2001. Turbulence in financial markets: The surprising explanatory power of simple cascade models. Quantit. Finance, 1: 632 - 640.
27. Lux, T. The multi-fractal model of asset returns: Its estimation via GMM estimation and its use for volatility forecasting. University of Kiel, 2003
28. T. Lux, The Markov-switching multifractal model of asset returns: GMM estimation and linear forecasting of volatility. J. Bus. Econ. Stat., 2008, 26, 194-210
29. Mandelbrot, BB. Fisher, A. and Calvet, L., A multifractal model of asset returns. Cowles Foundation Discussion Paper 1164, 1997
30. Mandelbrot, BB and Taylor, HM. 1967. On the distribution of stock price differences. Oper. Res., 15: 1057 - 1062.
31. Molchan, GM. 1996. Scaling exponents and multifractal dimensions for independent random cascades. Commun. Math. Phys., 179: 681 - 702.
32. Molchan, GM. 1997. Turbulent cascades: Limitations and a statistical test of the lognormal hypothesis. Phys. Fluids, 9: 2387 - 2395.
33. Muzy, JF and Bacry, E. 2002. Multifractal stationary random measures and multifractal random walks with log-innitely divisible scaling laws. Phys. Rev. E, 66: 056121 - 056150.
34. Muzy, JF, Bacry, E, Baile, R and Poggi, P. 2008. Uncovering latent singularities from multifractal scaling laws in mixed asymptotic regime. Application to turbulence. Europhys. Lett., 82 (6)
35. Muzy, JF, Bacry, E and Kozhemyak, A. 2006. Extreme values and fat tails of multifractal fluctuations. Phys. Rev. E, 73: 066114
36. Muzy, JF, Delour, J and Bacry, E. 2000. Modelling fluctuations of financial time series: From cascade process to stochastic volatility model. Eur. J. Phys. B, 17: 537 - 548.
37. Ossiander, M and Waymire, EC. 2000. Statistical estimation for multiplicative cascades. Ann. Statist., 28: 1533 - 1560.
38. Resnick, S, Samorodnitsky, G, Gilbert, A and Willinger, W. 2003. Wavelet analysis of conservative cascades. Bernoulli, 9: 97 - 135.
39. Robinson, PM. 2001. The memory of stochastic volatility models. J. Econometr., 101: 195 - 218.
40. Shorack, GR and Wellner, JA. 1986. Empirical Processes With Applications to Statistics, New York: Wiley.
41. Wick, GC. 1950. The evaluation of the collision matrix. Phys. Rev., 80: 268 - 272.