- Email: [email protected]
Modeling stylized facts for financial time series
ARTICLE IN PRESS
Physica A 344 (2004) 263–266 www.elsevier.com/locate/physa
Modeling stylized facts for financial time series M.I. Krivoruchenkoa,b,, E. Alessiob, V. Frappietrob, L.J. Streckertb a
Institute for Theoretical and Experimental Physics, Theory Division B., Cheremushkinskaya 25, 117259 Moscow, Russia b Metronome-Ricerca sui Mercati Finanziari, C. so Vittorio Emanuele 84, 10121 Torino, Italy Received 11 December 2003 Available online 23 July 2004
Abstract Multivariate probability density functions of returns are constructed in order to model the empirical behavior of returns in a financial time series. They describe the well-established deviations from the Gaussian random walk, such as an approximate scaling and heavy tails of the return distributions, long-ranged volatility–volatility correlations (volatility clustering) and return–volatility correlations (leverage effect). The model is tested successfully to fit joint distributions of the 100+ years of daily price returns of the Dow Jones 30 Industrial Average. r 2004 Published by Elsevier B.V. PACS: 89.65.Gh; 89.75.Da; 02.50.Ng; 02.50.Sk Keywords: Time series; Scaling; Heavy tails; Volatility clustering; Leverage effect
The methods developed in studying complex physical systems have been successfully applied through decades to analyze financial data [1–3] and they continue to attract gradual interest [4–11]. The field of research connected to modeling financial markets and development of statistically based real-time decision systems has recently been named Econophysics. In this paper, we construct a Corresponding author. Institute for Theoretical and Experimental Physics, Theory Division B. Cheremushkinskaya 25 117259 Moscow, Russia. Tel.: +7-095-1250292; fax: +7-095-8839601. E-mail address: [email protected] (M.I. Krivoruchenko).
0378-4371/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.physa.2004.06.129
ARTICLE IN PRESS 264
M.I. Krivoruchenko et al. / Physica A 344 (2004) 263–266
phenomenological model for description of the multivariate distribution of returns in a financial time series. The random walk model proposed by Bachelier in the year 1900 [1] is equivalent to the Gaussian multivariate probability density function (PDF) of the returns xi : ! n n Y 1 1X 1 2 2 G n ðxi Þ ¼ exp s x : ð1Þ i i n=2 2 s ð2pÞ i¼1 i¼1 i The absence of the correlations, Corr½xi ; xj ¼ 0; from a time frame greater than ti ti1 ¼ 20 min [6] has been widely documented and is often cited as a support for the efficient market hypothesis [9]. The multivariate PDFs constructed in this work are extensions of the Gaussian PDFs, aimed to model the well-established deviations in the behavior of financial time series from the Gaussian random walk. The Le´vy stable truncated univariate PDFs [2,10] are known to provide, for a financial time series, (i) an approximate scaling invariance of the univariate PDFs with a slow convergence to the Gaussian behavior and (ii) the existence of heavy tails. We propose the multivariate Student PDFs, !aþn 2 n n Y Gðaþn 1X 1 a 2 2 2 Þ Sn ðxi Þ ¼ 1 þ o x ; ð2Þ i i n=2 a a i¼1 oi ðapÞ Gð Þ i¼1 2
for modeling the empirical PDFs with returns xi . The marginal PDF (2) is again PDF (2). If we integrate out all of the xi except for one, we get (2) with n ¼ 1. The tails of the distributions behave empirically like [6] dx=x4 ; and so a 3. For the Student PDF (2), we have E½xi ¼ 0 and Corr½jxi j; jxk j ¼ 2=ð2 þ pÞ ¼ 0:39: The correlation of the returns vanish as in the Gaussian random walk. The 2 2 square of the volatility equals sP i ¼ Var½xi ¼ oi a=ða 2Þ. n The cumulative returns, x ¼ i¼1 xi ; are described by ! Z n n X Y a a x dx ; ð3Þ dW ðxÞ ¼ dxd x xi Sn ðxi Þ dxi ¼ S1 O O i¼1 i¼1 P where O2 ¼ ni¼1 o2i : The variance of the x increases linearly with n; in agreement with the Gaussian random walk and in rough agreement with the empirical observations. Eq. (3) represents the scaling law for financial time series. The multivariate Student PDFs have therefore heavy tails and the exact scaling invariance from the start. These distributions can be modified further to describe two other well-established stylized facts which are (iii) long-ranged volatility–volatility correlations that are also known as volatility clustering [12] and (iv) return–volatility correlations that are also known as leverage effect [13,14]. The empirical facts show that there is a slow decay of the correlation function. An extension of the PDF (2) that has the value Corr½jxi j; jxj j which is decaying with time is rather straightforward. We use uncorrelated multivariate distributions for different groups of the returns. The analogy with the Ising model can be useful. The groups xi with the same multivariate Student PDF can be treated as domains of spins aligned in the same direction. We assign the usual probability for every such
ARTICLE IN PRESS M.I. Krivoruchenko et al. / Physica A 344 (2004) 263–266
configuration w½s1 ; . . . ; sn jb ¼ N exp b
n1 X
265
! si siþ1 ;
ð4Þ
i¼1
where si ¼ 1: The normalization constant is given by N1 ¼ 2ð2 coshðbÞÞn1 : The correlation of the absolute values of the returns equals 2=ð2 þ pÞ provided that xi and xk belong to the same domain. Otherwise the result is zero. The probability of getting the xi and xk within the same domain can be found to be wl ¼ egl where eg ¼ eb =ðeb þ eb Þo1 and l ¼ i k: The coefficient Corr½jxi j; jxk j for the modified PDF has therefore the form 2=ð2 þ pÞwl : The absence of the correlations would formally correspond to b ¼ þ1 ðg ¼ þ1Þ. This is the case when the multivariate PDF is a product of the n univariate PDFs. In order to incorporate leverage effect, we consider the o-function depending on the signs ip ¼ signðxip Þ of the lagged returns (p ¼ 1; 2; . . .). The values i ¼ 1 are assumed to be independent variables which take the two values 1 with P thepequal2 probabilities. The i dependence is modeled by oi ¼ Cð1 rðð1 nÞ=nÞ 1 p¼1 n ip Þ where C is a normalization factor. Negative recent returns ip ¼ 1 (p ¼ 1; 2; . . .) increase the volatility, so r40. The value of r is connected to the overall strength of leverage effect. Taking volatility clustering and leverage effect into account, we obtain for l ¼ i k40: Corr½jxi j; jxk j ¼ Corr½jxi j; xk ¼
2 ð1 p2Þgl wl þ p2 ðgl 1Þ ; p g0 p42
2 ð1 p2Þwl þ p2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hl ; p ðg 4 Þg 0 p2 0
ð5Þ ð6Þ
where gl ¼ E½oi ok and hl ¼ E½oi k ok can be calculated analytically. The empirical correlation functions for the 100+ years of daily price returns of the Dow Jones 30 Industrial Average are fitted using a superposition wl ¼ c1 eg1 l þ 1 1 c2 eg2 l for l40 withPc1 ¼ 0:18; c2 ¼ 0:08; g1 ¼ 1200 and g2 ¼ 233 (else c0 ¼ 0:74 and 2 g0 ¼ þ1; so that c ¼ 1): We have also used r ¼ 1 and n ¼ expð1 m¼0 m 16 Þ: The results are in a good agreement with the data. The quantitative comparison can be found in Ref. [15]. The value of c1 þ c2 ¼ 0:26 indicates that 26% of the empirical PDF consists of products of the multivariate Student PDFs. The value Corr½xi ; jxk j vanishes at i4k in agreement with the observations, since the oi depend on lagged returns only. The complete multivariate PDF of the returns is given by San ðxn ; . . . ; x1 ÞC ¼
2 X m¼0
cm
n1 X expðbm ðn 2s 1ÞÞ ð2 coshðbm ÞÞn1 s¼0
sþ1 X Y ns ;...;n0 f ¼1
Sanf nf 1 ðxnf 1 ; . . . ; xnf 1 Þ ;
ð7Þ
ARTICLE IN PRESS 266
M.I. Krivoruchenko et al. / Physica A 344 (2004) 263–266
where nsþ1 ¼ n þ 1; nXns 4 4n1 X2; and n0 ¼ 1: The marginal probability of the PDF (7) is a PDF (7) again. The authors wish to thank the Dow Jones Global Indexes for providing historical prices of the Dow Jones Averages. M.I.K. is grateful to Metronome–Ricerca sui Mercati Finanziari for kind hospitality. References [1] L. Bachelier, Theorie de la Speculation (PhD Thesis), Annales Scientifiques de l’Ecole Normale Superieure, III-17 (1900) pp. 21–86 [English Translation: Cootner (Ed.), Random Character of Stock Market Prices, Massachusetts Institute of Technology, 1964, pp. 17–78 or S. Haberman and T.A. Sibett (Eds.), History of Actuarial Science, VII, London 1995, pp. 15–78]. [2] P. Levy, The´orie de l’Addition des Variables Ale´atoires, Gauthier-Villars, Paris, 1937. [3] B.B. Mandelbrot, The Journal of Business of the University of Chicago 36 (1963) 394. [4] R. Cont, D. Sornette, J. Phys. I France 7 (1997) 431. [5] N. Vandewalle, M. Ausloos, Physica A 246 (1997) 454. [6] P. Gopikrishnan, V. Plerou, L.A. Nunes Amaral, M. Meyer, H.E. Stanley, Phys. Rev. E 60 (1999) 5305. [7] M.I. Krivoruchenko, Phys. Rev. E 70 (2004) 036102. [8] E. Alessio, A. Carbone, G. Castelli, V. Frappietro, Euro. Phys. J. B 27 (2002) 197. [9] E.F. Fama, J. Finance 25 (1970) 383. [10] R.N. Mantegna, H.E. Stanley, Nature 376 (1995) 46. [11] K. Urbanowicz, J.A. Holyst, Phys. Rev. E 67 (2003) 046218. [12] Z. Ding, C.W.J. Granger, R.F. Engle, J. Empirical Finance 1 (1993) 83. [13] F. Black, in: Proceedings of the 1976 American Statistical Association, Business and Economical Statistics Section, 1976, p. 177. [14] J.C. Cox, S.A. Ross, Journal of Financial Economics 3 (1976) 145. [15] E. Alessio, V. Frappietro, M.I. Krivoruchenko, L.J. Streckert, http://arXiv.org/abs/cond-mat/ 0310300.
Physica A 344 (2004) 263–266 www.elsevier.com/locate/physa
Modeling stylized facts for financial time series M.I. Krivoruchenkoa,b,, E. Alessiob, V. Frappietrob, L.J. Streckertb a
Institute for Theoretical and Experimental Physics, Theory Division B., Cheremushkinskaya 25, 117259 Moscow, Russia b Metronome-Ricerca sui Mercati Finanziari, C. so Vittorio Emanuele 84, 10121 Torino, Italy Received 11 December 2003 Available online 23 July 2004
Abstract Multivariate probability density functions of returns are constructed in order to model the empirical behavior of returns in a financial time series. They describe the well-established deviations from the Gaussian random walk, such as an approximate scaling and heavy tails of the return distributions, long-ranged volatility–volatility correlations (volatility clustering) and return–volatility correlations (leverage effect). The model is tested successfully to fit joint distributions of the 100+ years of daily price returns of the Dow Jones 30 Industrial Average. r 2004 Published by Elsevier B.V. PACS: 89.65.Gh; 89.75.Da; 02.50.Ng; 02.50.Sk Keywords: Time series; Scaling; Heavy tails; Volatility clustering; Leverage effect
The methods developed in studying complex physical systems have been successfully applied through decades to analyze financial data [1–3] and they continue to attract gradual interest [4–11]. The field of research connected to modeling financial markets and development of statistically based real-time decision systems has recently been named Econophysics. In this paper, we construct a Corresponding author. Institute for Theoretical and Experimental Physics, Theory Division B. Cheremushkinskaya 25 117259 Moscow, Russia. Tel.: +7-095-1250292; fax: +7-095-8839601. E-mail address: [email protected] (M.I. Krivoruchenko).
0378-4371/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.physa.2004.06.129
ARTICLE IN PRESS 264
M.I. Krivoruchenko et al. / Physica A 344 (2004) 263–266
phenomenological model for description of the multivariate distribution of returns in a financial time series. The random walk model proposed by Bachelier in the year 1900 [1] is equivalent to the Gaussian multivariate probability density function (PDF) of the returns xi : ! n n Y 1 1X 1 2 2 G n ðxi Þ ¼ exp s x : ð1Þ i i n=2 2 s ð2pÞ i¼1 i¼1 i The absence of the correlations, Corr½xi ; xj ¼ 0; from a time frame greater than ti ti1 ¼ 20 min [6] has been widely documented and is often cited as a support for the efficient market hypothesis [9]. The multivariate PDFs constructed in this work are extensions of the Gaussian PDFs, aimed to model the well-established deviations in the behavior of financial time series from the Gaussian random walk. The Le´vy stable truncated univariate PDFs [2,10] are known to provide, for a financial time series, (i) an approximate scaling invariance of the univariate PDFs with a slow convergence to the Gaussian behavior and (ii) the existence of heavy tails. We propose the multivariate Student PDFs, !aþn 2 n n Y Gðaþn 1X 1 a 2 2 2 Þ Sn ðxi Þ ¼ 1 þ o x ; ð2Þ i i n=2 a a i¼1 oi ðapÞ Gð Þ i¼1 2
for modeling the empirical PDFs with returns xi . The marginal PDF (2) is again PDF (2). If we integrate out all of the xi except for one, we get (2) with n ¼ 1. The tails of the distributions behave empirically like [6] dx=x4 ; and so a 3. For the Student PDF (2), we have E½xi ¼ 0 and Corr½jxi j; jxk j ¼ 2=ð2 þ pÞ ¼ 0:39: The correlation of the returns vanish as in the Gaussian random walk. The 2 2 square of the volatility equals sP i ¼ Var½xi ¼ oi a=ða 2Þ. n The cumulative returns, x ¼ i¼1 xi ; are described by ! Z n n X Y a a x dx ; ð3Þ dW ðxÞ ¼ dxd x xi Sn ðxi Þ dxi ¼ S1 O O i¼1 i¼1 P where O2 ¼ ni¼1 o2i : The variance of the x increases linearly with n; in agreement with the Gaussian random walk and in rough agreement with the empirical observations. Eq. (3) represents the scaling law for financial time series. The multivariate Student PDFs have therefore heavy tails and the exact scaling invariance from the start. These distributions can be modified further to describe two other well-established stylized facts which are (iii) long-ranged volatility–volatility correlations that are also known as volatility clustering [12] and (iv) return–volatility correlations that are also known as leverage effect [13,14]. The empirical facts show that there is a slow decay of the correlation function. An extension of the PDF (2) that has the value Corr½jxi j; jxj j which is decaying with time is rather straightforward. We use uncorrelated multivariate distributions for different groups of the returns. The analogy with the Ising model can be useful. The groups xi with the same multivariate Student PDF can be treated as domains of spins aligned in the same direction. We assign the usual probability for every such
ARTICLE IN PRESS M.I. Krivoruchenko et al. / Physica A 344 (2004) 263–266
configuration w½s1 ; . . . ; sn jb ¼ N exp b
n1 X
265
! si siþ1 ;
ð4Þ
i¼1
where si ¼ 1: The normalization constant is given by N1 ¼ 2ð2 coshðbÞÞn1 : The correlation of the absolute values of the returns equals 2=ð2 þ pÞ provided that xi and xk belong to the same domain. Otherwise the result is zero. The probability of getting the xi and xk within the same domain can be found to be wl ¼ egl where eg ¼ eb =ðeb þ eb Þo1 and l ¼ i k: The coefficient Corr½jxi j; jxk j for the modified PDF has therefore the form 2=ð2 þ pÞwl : The absence of the correlations would formally correspond to b ¼ þ1 ðg ¼ þ1Þ. This is the case when the multivariate PDF is a product of the n univariate PDFs. In order to incorporate leverage effect, we consider the o-function depending on the signs ip ¼ signðxip Þ of the lagged returns (p ¼ 1; 2; . . .). The values i ¼ 1 are assumed to be independent variables which take the two values 1 with P thepequal2 probabilities. The i dependence is modeled by oi ¼ Cð1 rðð1 nÞ=nÞ 1 p¼1 n ip Þ where C is a normalization factor. Negative recent returns ip ¼ 1 (p ¼ 1; 2; . . .) increase the volatility, so r40. The value of r is connected to the overall strength of leverage effect. Taking volatility clustering and leverage effect into account, we obtain for l ¼ i k40: Corr½jxi j; jxk j ¼ Corr½jxi j; xk ¼
2 ð1 p2Þgl wl þ p2 ðgl 1Þ ; p g0 p42
2 ð1 p2Þwl þ p2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hl ; p ðg 4 Þg 0 p2 0
ð5Þ ð6Þ
where gl ¼ E½oi ok and hl ¼ E½oi k ok can be calculated analytically. The empirical correlation functions for the 100+ years of daily price returns of the Dow Jones 30 Industrial Average are fitted using a superposition wl ¼ c1 eg1 l þ 1 1 c2 eg2 l for l40 withPc1 ¼ 0:18; c2 ¼ 0:08; g1 ¼ 1200 and g2 ¼ 233 (else c0 ¼ 0:74 and 2 g0 ¼ þ1; so that c ¼ 1): We have also used r ¼ 1 and n ¼ expð1 m¼0 m 16 Þ: The results are in a good agreement with the data. The quantitative comparison can be found in Ref. [15]. The value of c1 þ c2 ¼ 0:26 indicates that 26% of the empirical PDF consists of products of the multivariate Student PDFs. The value Corr½xi ; jxk j vanishes at i4k in agreement with the observations, since the oi depend on lagged returns only. The complete multivariate PDF of the returns is given by San ðxn ; . . . ; x1 ÞC ¼
2 X m¼0
cm
n1 X expðbm ðn 2s 1ÞÞ ð2 coshðbm ÞÞn1 s¼0
sþ1 X Y ns ;...;n0 f ¼1
Sanf nf 1 ðxnf 1 ; . . . ; xnf 1 Þ ;
ð7Þ
ARTICLE IN PRESS 266
M.I. Krivoruchenko et al. / Physica A 344 (2004) 263–266
where nsþ1 ¼ n þ 1; nXns 4 4n1 X2; and n0 ¼ 1: The marginal probability of the PDF (7) is a PDF (7) again. The authors wish to thank the Dow Jones Global Indexes for providing historical prices of the Dow Jones Averages. M.I.K. is grateful to Metronome–Ricerca sui Mercati Finanziari for kind hospitality. References [1] L. Bachelier, Theorie de la Speculation (PhD Thesis), Annales Scientifiques de l’Ecole Normale Superieure, III-17 (1900) pp. 21–86 [English Translation: Cootner (Ed.), Random Character of Stock Market Prices, Massachusetts Institute of Technology, 1964, pp. 17–78 or S. Haberman and T.A. Sibett (Eds.), History of Actuarial Science, VII, London 1995, pp. 15–78]. [2] P. Levy, The´orie de l’Addition des Variables Ale´atoires, Gauthier-Villars, Paris, 1937. [3] B.B. Mandelbrot, The Journal of Business of the University of Chicago 36 (1963) 394. [4] R. Cont, D. Sornette, J. Phys. I France 7 (1997) 431. [5] N. Vandewalle, M. Ausloos, Physica A 246 (1997) 454. [6] P. Gopikrishnan, V. Plerou, L.A. Nunes Amaral, M. Meyer, H.E. Stanley, Phys. Rev. E 60 (1999) 5305. [7] M.I. Krivoruchenko, Phys. Rev. E 70 (2004) 036102. [8] E. Alessio, A. Carbone, G. Castelli, V. Frappietro, Euro. Phys. J. B 27 (2002) 197. [9] E.F. Fama, J. Finance 25 (1970) 383. [10] R.N. Mantegna, H.E. Stanley, Nature 376 (1995) 46. [11] K. Urbanowicz, J.A. Holyst, Phys. Rev. E 67 (2003) 046218. [12] Z. Ding, C.W.J. Granger, R.F. Engle, J. Empirical Finance 1 (1993) 83. [13] F. Black, in: Proceedings of the 1976 American Statistical Association, Business and Economical Statistics Section, 1976, p. 177. [14] J.C. Cox, S.A. Ross, Journal of Financial Economics 3 (1976) 145. [15] E. Alessio, V. Frappietro, M.I. Krivoruchenko, L.J. Streckert, http://arXiv.org/abs/cond-mat/ 0310300.