Observatory of Complex Systems http lagash dft unipa
Observatory of Complex Systems http: //lagash. dft. unipa. it Univariate and Multivariate Characterization of Equity Volatility Salvatore Miccichè with Fabrizio Lillo, Rosario N. Mantegna INFM - Istituto Nazionale per la Fisica della Materia - Unità di Palermo
Univariate and Multivariate Characterization of Equity Volatility Outline 1) Univariate Statistical Characterization of Volatility Investigation of the univariate (pdf and autocorrelation) properties of the 100 most capitalized stocks traded in the NYSE equity market. A simple stochastic volatility model based on a nonlinear Langevin equation with “long-memory”. 2) Multivariate Statistical Characterization of Volatility Investigation of the ensemble properties of the 100 most capitalized stocks traded in the NYSE equity market. Use a clustering procedure to understand i) what are the links between volatilities in a financial market ii) what is their dynamics
Univariate and Multivariate Characterization of Equity Volatility Introduction The Black-Scholes model describes the time behaviour of price returns: d. S/S = dt + d z where dz is a Wiener process and S is the stock price. and are two constants. On a time horizon t: • t is the expected price return • 2 t is the variance. Therefore, is a measure of the unpredictability of the time series S(t). is called volatility In the Black-Scholes model is assumed to be a constant. Indeed, can be considered as a stochastic process itself !!!
Univariate and Multivariate Characterization of Equity Volatility NOT locally stationary Asymptotically stationary Slowly decaying (power-law? ) power-law? Autocorrelation Function Persistencies Rapidly decaying Autocorrelation Function Arbitrage
Univariate and Multivariate Characterization of Equity Volatility The set of investigated stocks We consider the 100 most capitalized stocks traded at NYSE. 95 of them enter the Standard&Poor’s 100 (SP 100) stock index. Trades And Quotes (TAQ) TAQ database maintained by NYSE (1995 -1998) 1995 -1998 We consider high-frequency (intraday) intraday data. Transactions do not occur at the same time for all stocks. INTC 11900 transactions per day MKG 121 transactions per day We have to synchronize/ synchronize homogenize the data: 12 intervals of 1950 seconds each For each stock i=1, . . . , 100 For 1011 trading days
Univariate and Multivariate Characterization of Equity Volatility Univariate
Univariate and Multivariate Characterization of Equity Volatility Empirical Facts lognormal power-law
Univariate and Multivariate Characterization of Equity Volatility Empirical Facts ( ) - 0. 3
Univariate and Multivariate Characterization of Equity Volatility Empirical Facts ( ) - <1 < y(t)2> t 1. 7
Univariate and Multivariate Characterization of Equity Volatility Empirical Facts (q) 0. 85 q 0. 93 (q) 0. 17+0. 74 q
Univariate and Multivariate Characterization of Equity Volatility Models of Stochastic Volatility We are looking for models of stochastic volatility: d. S/S = dt + d z d = h( ) dt + g( ) d z. What are the appropriate i) Drift coefficient h( ) ii) Diffusion coefficient g( ) able to reproduce the previous empirical stylized facts ?
Univariate and Multivariate Characterization of Equity Volatility Popular Models of Stochastic Volatility Hull-White dv = a (b - v) dt + v dz Heston dv = a (b - v) dt + v 1/2 dz Lognormal d = a ( b - Log ) dt + 1/2 dz Stein-Stein d = a (b - ) dt + dz power-law pdf
Univariate and Multivariate Characterization of Equity Volatility Popular Models of Stochastic Volatility lognormal power-law
Univariate and Multivariate Characterization of Equity Volatility The proposed model: a Two-Region Model , L control the power-law V 1 controls the log-normality g( )=1 Additive noise
Univariate and Multivariate Characterization of Equity Volatility The proposed model: a Two-Region Model Write the Fokker-Planck equation and look for the stationary solution Lognormal Power-law
Univariate and Multivariate Characterization of Equity Volatility The proposed model: a Two-Region Model
Univariate and Multivariate Characterization of Equity Volatility The proposed model: a Two-Region Model One can prove that this simple Two-Region model admits a powerlaw decaying autocorrelation function with exponent: Nevertheless, the dinamical properties of volatility are not well reproduced by this simple model. Volatility shows an empirical pdf that has power-law tails with exponent emp 4. 8 and an empirical mean squared displacement that is asymptotically power-law with exponent emp 1. 7, 1. 7 i. e. emp 0. 3 in the autocorrelation function. 4. 8 would imply 0. 9 i. e. = 2 - 1. 1
Univariate and Multivariate Characterization of Equity Volatility Conclusions This simple model reproduces the empirical pdf quite well However, the empirical Autocorrelation function is not well reproduced. We are working of making the and exponents independent from each other. This could be done by considering a diffusion coefficient g( ) (multiplicative noise). References • S. Miccichè, G. Bonanno, F. Lillo, R. N. Mantegna, Physica A, 314, 756 -761, (2002) • F. Lillo, S. Miccichè, R. N. Mantegna, cond-mat/0203442 • S. Miccichè, G. Bonanno, F. Lillo, R. N. Mantegna, Proceedings of: "The Second Nikkey Econophysics Research Workshop and Symposium", 12 -14 November 2002, Tokio, Japan Springer Verlag, Tokio, edited by H. Takayasu
Univariate and Multivariate Characterization of Equity Volatility Multivariate
Univariate and Multivariate Characterization of Equity Volatility Clustering Procedure We are looking for a possible collective stochastic dynamics and/or links between price returns / volatilities of different stocks. PRICE RETURNS / VOLATILITY CLUSTERS Cross-Correlation Clustering Procedure based on a similarity measure: measure subdominant ultrametric distance. Hierarchical Tree (HT) and Minimum Spanning Tree (MST). where Vi are the price returns / volatilities time series.
Univariate and Multivariate Characterization of Equity Volatility Hierarchical Trees: Price Returns Each vertical lines indicates a stock. Technology Financial Energy Consumer Cyclical Consumer/N. C. Health Care Basic Material Services Utilities Conglomerates Capital Goods Transportation Red Green Blu Brown Yellow Gray Violet Cyan Magenta Orange Indigo Maroon d=0. 6 =0. 82
Univariate and Multivariate Characterization of Equity Volatility Hierarchical Trees: Volatility Price Returns are more clustered than Volatilities: here there are more black lines than before!! Volatilities are less crosscorrelated than Price Returns: here clusters have an higher distance than before !! d=0. 8 =0. 68
Univariate and Multivariate Characterization of Equity Volatility Minimum Spanning Tree: Price Return 1 day = 6 h 30’ STRUCTURE Energy GE=17 Finance Consumer n. c. Basic Material Transportation Health. Care Services Conglomerates Technology
Univariate and Multivariate Characterization of Equity Volatility Minimum Spanning Tree: Price Return 11700’’ = 3 h 15’ GE=20 Epps effect 4680’’ = 1 h 18’ GE=31 The degree of cross-correlation diminishes along with the time horizon used to compute the crosscorrelation coefficients.
Univariate and Multivariate Characterization of Equity Volatility Minimum Spanning Tree: Price Return 2340’’ = 39’ GE=49 However, the structuring of the stocks by economic sectors still persists even for very low time horizons, … !! star-like 1170’’ = 19’ 30’’ … i. e. , the “intrasector Epps effect’’ is (disappears ) faster than the usual ( (global) one, … !! DE-STRUCTURE GE=60
Univariate and Multivariate Characterization of Equity Volatility Minimum Spanning Tree: Volatility 1 day = 6 h 30’ MER=35 GE=18 Energy “Basic Finance that Material” volatility is “Basic Material” a longmemory process? Consumer n. c. Is 1 day not yet enough to “see” economic sectors? “Services” Health. Care “Transportation” Is this related to the fact “Technology” DE-STRUCTURE “Conglomerates” “Services” Technology Is there some Epps effect for Volatility, too? ?
Univariate and Multivariate Characterization of Equity Volatility Minimum Spanning Tree: Volatility (Spearman) Finance Energy Services 1 day = 6 h 30’ Basic DE-STRUCTURE Material Technology The effect is also evident when using a nonparametric clustering procedure.
Univariate and Multivariate Characterization of Equity Volatility Conclusions • Price Return - Intraday data allow for tracing the formation over time of significant clusters - These clusters correspond to the economic sectors - The structuring of the stocks by economic sectors still persists even for very low time horizons and despite the existence of the Epps effect. • Volatility - A significant clustering is shown in the hierarchical tree of volatility - However, such clustering is less pronounced than in the case of price returns. - It is not yet understood whether or not this is due to the fact that volatility is a stochastic process with long range memory and therefore its autocorrelation function decays much slower than in the case of price returns. References • R. Rammal, G. Toulose, M. A. Virasoro, Rev. Mod. Phys. , 58, 765, (1986) • R. N. Mantegna, Eur. Phys. J. B, 11, 193 (1999) • R. N. Mantegna, H. E. Stanley, An introduction to Econophysics, CUP, Cambridge (2000) • G. Bonanno, F. Lillo, R. N. Mantegna, Quantitative Finance, 1, 96 (2001) • S. Miccichè, G. Bonanno, F. Lillo, R. N. Mantegna, Physica A, 324, 66 -73, (2003) • G. Bonanno, G. Caldarelli, F. Lillo, S. Miccichè, N. Vanderwalle, R. N. Mantegna, in preparation
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