1. Univariate and Multivariate Characterization of Equity Volatility Salvatore Miccichè with Fabrizio Lillo, Rosario N. Mantegna http://lagash.dft.unipa.it Observatory of Complex Systems INFM - Istituto Nazionale per la Fisica della Materia - Unità di Palermo

2. Univariate and Multivariate Characterization of Equity Volatility 1) Univariate Statistical Characterization of Volatility ensemble 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 i) what are the links between volatilities in a financial market ii) what is their dynamics Outline 2) Multivariate Statistical Characterization of Volatility univariate 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”.

3. Univariate and Multivariate Characterization of Equity Volatility The Black-Scholes model describes the time behaviour of price returns: dS/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: expected price return   t is the expected price return variance.  2  t is the variance. unpredictability Therefore,  is a measure of the unpredictability of the time series S(t). volatility  is called volatility. In the Black-Scholes model  is assumed to be a constant. stochastic process itself Indeed,  can be considered as a stochastic process itself !!! Introduction

4. Univariate and Multivariate Characterization of Equity Volatility NOT locally stationary Asymptotically stationary power-law? Slowly decaying (power-law?) Autocorrelation FunctionPersistencies Rapidly decaying Autocorrelation FunctionArbitrage

5. Univariate and Multivariate Characterization of Equity Volatility The set of investigated stocks Standard&Poor’s 100 We consider the 100 most capitalized stocks traded at NYSE. 95 of them enter the Standard&Poor’s 100 (SP100) stock index. intraday INTC  11900MKG  121 synchronizehomogenize We consider high-frequency (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/homogenize the data: 100 1011 For each stock i=1,..., 100 For 1011 trading days 121950 12 intervals of 1950 seconds each TAQTAQ1995-1998 Trades And Quotes (TAQ) database maintained by NYSE (1995-1998)

6. Univariate Univariate and Multivariate Characterization of Equity Volatility

7. Empirical Facts Univariate and Multivariate Characterization of Equity Volatility lognormal power-law

8. Empirical Facts Univariate and Multivariate Characterization of Equity Volatility  (  )   -    0.3

9. Empirical Facts Univariate and Multivariate Characterization of Equity Volatility  t    1.7  t    1.7 () -() -  <1

10. Empirical Facts Univariate and Multivariate Characterization of Equity Volatility  (q)  0.85 q 0.93  (q)  0.17+0.74 q

11. Models of Stochastic Volatility Univariate and Multivariate Characterization of Equity Volatility Drift coefficient h(  ) Diffusion coefficient g(  ) What are the appropriate i) Drift coefficient h(  ) ii) Diffusion coefficient g(  ) able to reproduce the previous empirical stylized facts ? We are looking for models of stochastic volatility: dS/S =  dt +  d z d  = h(  ) dt + g(  ) d z .

12. Popular Models of Stochastic Volatility Univariate and Multivariate Characterization of Equity Volatility Hull-White dv = a (b - v) dt +  v dz  Heston Lognormal Stein-Stein dv = a (b - v) dt +  v 1/2 dz  d  = a ( b - Log  ) dt +   1/2 dz  d  = a (b -  ) dt +   dz  power-law pdf

13. Popular Models of Stochastic Volatility Univariate and Multivariate Characterization of Equity Volatility lognormal power-law

14. Univariate and Multivariate Characterization of Equity Volatility The proposed model: a Two-Region Model g(  )=1 Additive noise , L control the power-law V1 controls the log-normality

15. Univariate and Multivariate Characterization of Equity Volatility The proposed model: a Two-Region Model Lognormal Power-law Write the Fokker-Planck equation and look for the stationary solution

16. Univariate and Multivariate Characterization of Equity Volatility The proposed model: a Two-Region Model

17. Univariate and Multivariate Characterization of Equity Volatility dinamical properties of volatility are not well reproduced Nevertheless, the dinamical properties of volatility are not well reproduced by this simple model. The proposed model: a Two-Region Model  emp  4.8  emp  1.7 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, i.e.  emp  0.3  emp  0.3 in the autocorrelation function. power- law decaying autocorrelation One can prove that this simple Two-Region model admits a power- law decaying autocorrelation function with exponent:  4.8 would imply  0.9 i.e.  = 2-  1.1

18. Univariate and Multivariate Characterization of Equity Volatility Conclusions References References S. Miccichè, G. Bonanno, F. Lillo, R. N. Mantegna, Physica A, 314, 756-761, (2002)S. Miccichè, G. Bonanno, F. Lillo, R. N. Mantegna, Physica A, 314, 756-761, (2002) F. Lillo, S. Miccichè, R. N. Mantegna, cond-mat/0203442F. 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. TakayasuS. 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 This simple model reproduces the empirical pdf quite well. However, the empirical Autocorrelation function is not well reproduced.  g(  )    () We are working of making the  and  exponents independent from each other. This could be done by considering a diffusion coefficient g(  )    (multiplicative noise).

19. Multivariate Univariate and Multivariate Characterization of Equity Volatility

20. Clustering Procedure collective stochastic dynamicslinks We are looking for a possible collective stochastic dynamics and/or links between price returns / volatilities of different stocks. PRICE RETURNS / VOLATILITY CLUSTERS similarity measure Cross-Correlation Clustering Procedure based on a similarity measure: where V i are the price returns / volatilities time series.  subdominant ultrametric distance  subdominant ultrametric distance.  Hierarchical Tree (HT) and Minimum Spanning Tree (MST)  Hierarchical Tree (HT) and Minimum Spanning Tree (MST).

21. Univariate and Multivariate Characterization of Equity Volatility Hierarchical Trees: Price Returns Technology Red Financial Green Energy Blu Consumer Cyclical Brown Consumer/N.C. Yellow Health Care Gray Basic Material Violet Services Cyan Utilities Magenta Conglomerates Orange Capital Goods Indigo Transportation Maroon Each vertical lines indicates a stock. Technology Red Financial Green Energy Blu Consumer Cyclical Brown Consumer/N.C. Yellow Health Care Gray Basic Material Violet Services Cyan Utilities Magenta Conglomerates Orange Capital Goods Indigo Transportation Maroon d=0.6   =0.82

22. Univariate and Multivariate Characterization of Equity Volatility Hierarchical Trees: Volatility Volatilities are less cross- correlated than Price Returns: here clusters have an higher distance than before !! Price Returns are more clustered than Volatilities: here there are more black lines than before!! d=0.8   =0.68

23. Univariate and Multivariate Characterization of Equity Volatility Minimum Spanning Tree: Price Return Energy Technology Finance Conglomerates Consumer n.c. Services BasicMaterial Transportation BasicMaterial HealthCare 1 day = 6 h 30’ STRUCTURE GE=17

24. Univariate and Multivariate Characterization of Equity Volatility Minimum Spanning Tree: Price Return 11700’’ = 3 h 15’ 4680’’ = 1 h 18’ Epps effect The degree of cross-correlation diminishes along with the time horizon used to compute the crosscorrelation coefficients. GE=20 GE=31

25. Univariate and Multivariate Characterization of Equity Volatility Minimum Spanning Tree: Price Return 2340’’ = 39’ 1170’’ = 19’ 30’’ However, the structuring of the stocks by economic sectors still persists even for very low time horizons, … !! … i.e., the “intrasector Epps effect’’ is () faster than the usual (global) one, … !! … i.e., the “intrasector Epps effect’’ is (disappears) faster than the usual (global) one, … !! DE- STRUCTURE GE=49 GE=60 star-like

26. Univariate and Multivariate Characterization of Equity Volatility Minimum Spanning Tree: Volatility Energy Finance “Conglomerates” “Services” Consumer n.c. “Services” Technology “Technology” “Transportation” HealthCare “BasicMaterial” “BasicMaterial” DE-STRUCTURE Is this related to the fact that volatility is a long- memory process? Is 1 day not yet enough to “see” economic sectors? Is there some Epps effect for Volatility, too?? 1 day = 6 h 30’ GE=18 MER=35

27. Univariate and Multivariate Characterization of Equity Volatility Minimum Spanning Tree: Volatility (Spearman) DE-STRUCTURE The effect is also evident when using a non- parametric clustering procedure. BasicMaterial Technology Energy Services Finance 1 day = 6 h 30’

28. Conclusions References References R. Rammal, G. Toulose, M.A. Virasoro, Rev. Mod. Phys., 58, 765, (1986)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, Eur. Phys. J. B, 11, 193 (1999) R. N. Mantegna, H. E. Stanley, An introduction to Econophysics, CUP, Cambridge (2000)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)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)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 preparationG. Bonanno, G. Caldarelli, F. Lillo, S. Miccichè, N. Vanderwalle, R. N. Mantegna, in preparation Volatility - clusteringvolatility - less pronounced - long range memoryautocorrelation decaysslower 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. Univariate and Multivariate Characterization of Equity Volatility Price Return -Intradaytimeclusters -economic sectors - structuring by economic sectorspersists Epps. 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.