More stylized facts of financial markets: leverage effect and downside correlations

https://doi.org/10.1016/S0378-4371(01)00282-5Get rights and content

Abstract

We discuss two more universal features of stock markets: the so-called leverage effect (a negative correlation between past returns and future volatility), and the increased downside correlations. For individual stocks, the leverage correlation can be rationalized in terms of a new ‘retarded’ model which interpolates between a purely additive and a purely multiplicative stochastic process. For stock indices a specific market panic phenomenon seems to be necessary to account for the observed amplitude of the effect. As for the increase of correlations in highly volatile periods, we investigate how much of this effect can be explained within a simple non-Gaussian one-factor description with time independent correlations. In particular, this one-factor model can explain the level and asymmetry of empirical exceedance correlations, which reflects the fat-tailed and negatively skewed distribution of market returns.

Introduction

The construction of adequate agent based models of financial markets is becoming a very active field of research. The hope is to understand in details collective effects in a human activity that is particularly well documented: high frequencies time series of thousands of different financial assets are now available for sophisticated statistical investigations. An important task of these investigations is to unveil several robust ‘stylized fact’ of financial markets, which consistently appear on different markets and in different periods of time, and that any candidate model should convincingly explain. Some of these stylized facts have been known for a long time, such as the absence of significant linear correlation of returns (except on short time scales) or the ‘fat tails’ in the distribution of returns. More recent results have established the long ranged nature of volatility [1], [2], [3], [4], [5], [6], [7], [8], [9], and volume correlations [10] and the related ‘multiscaling’ behaviour observed for higher order correlation functions [9], [11]. The aim of this paper is to report on two more important ‘stylized facts’ that seem to be rather universal: (a) the so-called ‘leverage’ effect (the volatility of stocks tends to increase when the price drops) [12] and (b) the apparent increase of cross correlations in highly volatility market conditions, in particular when prices significantly. We present some quantitative empirical evidence for these effects, provide simple models for their interpretation and discuss their connections. These effects are particularly important for option markets ([7], [13], [14]) and for risk management.

Section snippets

Empirical results

We will call Si(t) the price of stock i at time t, and δSi(t) the (absolute) daily price change: δSi(t)=Si(t+1)−Si(t). The relative price change will be denoted as δxi(t)=δSi(t)/Si(t). Taking δx2 as a proxy for the squared volatility, the leverage correlation function can be defined asLi(τ)=1Z〈[δxi(t+τ)]2δxi(t)〉.The coefficient Z is a normalization that we have chosen to be Z=〈δxi(t)2〉2 for reasons that will become clear below. (Note that since δx is dimensionless, Li(τ) is also dimensionless,

Cross-correlations in highly volatile periods

It is a common belief that cross-correlations between stocks actually fluctuate in time, and increase substantially in a period of high market volatility. This has been discussed in many papers—see for example [16], [17], [18], with more recent discussions, including new indicators, in [19], [20], [21], [22]. Furthermore, this increase is thought to be larger for large downward moves than for large upward moves. The dynamics of these correlations themselves, and their asymmetry, should be

Conclusion

In this paper, we have reported on two more stylized facts of financial markets, the leverage effect, and the apparent increase of cross-correlations in highly volatile periods, in particular for downside moves.

We have found that the leverage correlation is moderate and decays over a few months for individual stocks, and much stronger but decaying much faster for stock indices. In the case of individual stocks, we have found that the magnitude of this correlation can be rationalized in terms of

Acknowledgements

We thank P. Cizeau, A. Matacz and M. Meyer for their commitment in various stages of this work.

References (28)

  • Z. Ding et al.

    J. Empirical Finance

    (1993)
  • J.Y. Campbell, A.W. Lo, A.C. McKinley, The Econometrics of Financial Markets, Princeton University Press, Princeton,...
  • Y. Liu et al.

    Physica

    (1997)
  • P. Cizeau et al.

    Physica A

    (1997)
  • M. Potters et al.

    Europhys. Lett.

    (1998)
  • R. Mantegna et al.

    An introduction to Econophysics

    (1999)
  • J.-P. Bouchaud et al.

    Théorie des Risques Financiers, Aléa-Saclay, 1997; Theory of Financial Risks

    (2000)
  • J.-F. Muzy et al.

    Euro. Phys. J. B

    (2000)
  • J.-F. Muzy, J. Delour, E. Bacry, e-print...
  • G. Bonanno et al.

    Physica A

    (2000)
  • J.-P. Bouchaud et al.

    Eur. Phys. J.

    (1999)
  • G. Bekaert et al.

    Rev. Fin. Stud.

    (2000)
  • D. Backus, S. Foresi, K. Lai, L. Wu, Accounting for Biases in Black-Scholes, Working paper,...
  • J.-P. Fouque et al.

    Derivatives in Financial Markets with Stochastic Volatility

    (2000)

    • Cited by (0)

    View full text
    Copyright © 2001 Elsevier Science B.V. All rights reserved.