Direct Evidence for Inversion Formula in Multifractal Financial Volatility Measure
JIANG Zhi-Qiang1,2,3, ZHOU Wei-Xing 1,2,3,4,5
1School of Business, East China University of Science and Technology, Shanghai 2002372School of Science, East China University of Science and Technology, Shanghai 2002373Research Center for Econophysics, East China University of Science and Technology, Shanghai 2002374Engineering Research Center of Process Systems Engineering (Ministry of Education), East China University of Science and Technology, Shanghai 2002375Research Center on Fictitious Economics and Data Science, Chinese Academy of Sciences, Beijing 100190
Direct Evidence for Inversion Formula in Multifractal Financial Volatility Measure
JIANG Zhi-Qiang1,2,3, ZHOU Wei-Xing 1,2,3,4,5
1School of Business, East China University of Science and Technology, Shanghai 2002372School of Science, East China University of Science and Technology, Shanghai 2002373Research Center for Econophysics, East China University of Science and Technology, Shanghai 2002374Engineering Research Center of Process Systems Engineering (Ministry of Education), East China University of Science and Technology, Shanghai 2002375Research Center on Fictitious Economics and Data Science, Chinese Academy of Sciences, Beijing 100190
摘要The inversion formula for conservative multifractal measures was unveiled mathematically a decade ago, which is however not well tested in real complex systems. We propose to verify the inversion formula using high-frequency turbulent financial data. We construct conservative volatility measure based on minutely S&P 500 index from 1982 to 1999 and its inverse measure of exit time. Both the direct and inverse measures exhibit nice multifractal nature, whose scaling ranges are not irrelevant. Empirical investigation shows that the inversion formula holds in financial markets.
Abstract：The inversion formula for conservative multifractal measures was unveiled mathematically a decade ago, which is however not well tested in real complex systems. We propose to verify the inversion formula using high-frequency turbulent financial data. We construct conservative volatility measure based on minutely S&P 500 index from 1982 to 1999 and its inverse measure of exit time. Both the direct and inverse measures exhibit nice multifractal nature, whose scaling ranges are not irrelevant. Empirical investigation shows that the inversion formula holds in financial markets.
JIANG Zhi-Qiang;;ZHOU Wei-Xing;;. Direct Evidence for Inversion Formula in Multifractal Financial Volatility Measure[J]. 中国物理快报, 2009, 26(2): 28901-028901.
JIANG Zhi-Qiang, , ZHOU Wei-Xing, ,. Direct Evidence for Inversion Formula in Multifractal Financial Volatility Measure. Chin. Phys. Lett., 2009, 26(2): 28901-028901.
[1] Jensen M H 1999 Phys. Rev. Lett. 83 76 [2] Biferale L, Cencini M, Vergni D and Vulpiani A 1999 Phys. Rev. E 60 R6295 [3] Simonsen I, Jensen M H and Johansen A 2002 Eur.Phys. J. B 27 583 [4] McCauley J L 1990 Phys. Rep. 189 225 [5] Frisch U 1996 Turbulence: The Legacy of A. N.Kolmogorov (Cambridge: Cambridge University Press) [6] Anselmet F, Gagne Y, Hopfinger E J and Antonia R A 1984 J. Fluid Mech. 140 63 [7] Vandewalle N and Ausloos M 1998 Eur. Phys. J. B 4 257 [8] Ivanova K and Ausloos M 1999 Eur. Phys. J. B 8 665 [9] Calvet L and Fisher A 2002 Rev. Econ. Stat. 84 381 [10] Schmitt F G 2005 Phys. Lett. A 342 448 [11] Hastings M B 2002 Phys. Rev. Lett. 88 055506 [12] Biferale L, Cencini M, Lanotte A S, Vergni D and VulpianiA 2001 Phys. Rev. Lett. 87 124501 [13] Biferale L, Cencini M, Lanotte A S and Vergni D 2003 Phys. Fluids 15 1012 [14] Roux S and Jensen M H 2004 Phys. Rev. E 69016309 [15] Beaulac S and Mydlarski L 2004 Phys. Fluids 16 2126 [16] Pearson B R and van de Water W 2005 Phys. Rev. E 71 036303 [17] Zhou W X, Sornette D and Yuan W K 2006 Physica D 214 55 [18] Xu J L, Zhou W X, Liu H F, Gong X, Wang F C and Yu Z H2006 Phys. Rev. E 73 056308 [19] Mandelbrot B B and Riedi R H 1997 Adv. Appl. Math. 18 50 [20] Riedi R H and Mandelbrot B B 1997 Adv. Appl. Math. 19 332 [21] Mantegna R N and Stanley H E 2000 An Introduction toEconophysics: Correlations and Complexity in Finance (Cambridge:Cambridge University Press) [22] Jiang Z Q and Zhou W X 2007 Physica A 381343 [23] Jiang Z Q and Zhou W X 2008 Physica A 3874881 [24] Cizeau P, Liu Y H, Meyer M, Peng C K and Stanley H E 1997 Physica A 245 441 [25] Liu Y H, Gopikrishnan P, Cizeau P, Meyer M, Peng C K andStanley H E 1999 Phys. Rev. E 60 1390 [26] L'vov V S, Podivilov E, Pomyalov A, Procaccia I andVandembroucq D 1998 Phys. Rev. E 58 1811 [27] Jensen M H, Johansen A, Petroni F and Simonsen I 2004 Physica A 340 678 [28] Jensen M H, Johansen A and Simonsen I 2003 Int. J.Modern Phys. B 17 4003
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