Random matrix application to correlations amongst the volatility of assets

Abstract

In this paper, we apply tools from random matrix theory (RMT) to estimates of correlations across the volatility of various assets in the S&P 500. The volatility inputs are estimated by modelling price fluctuations as a GARCH(1,1) process. The corresponding volatility correlation matrix is then constructed. It is found that the distribution of a significant number of eigenvalues of the volatility correlation matrix matches with the analytical result from RMT. Furthermore, the empirical estimates of short- and long-range correlations amongst eigenvalues, which are within RMT bounds, match with the analytical results for the Gaussian Orthogonal ensemble of RMT. To understand the information content of the largest eigenvectors, we estimate the contribution of the Global Industry Classification Standard industry groups to each eigenvector. In comparison with eigenvectors of correlation matrix for price fluctuations, only few of the largest eigenvectors of the volatility correlation matrix are dominated by a single industry group. We also study correlations between ‘volatility returns’ and log-volatility to find similar results.

Keywords:

• Random matrix theory
• Volatility correlation
• Eigenvalue distribution

JEL Classifications:

• C10
• C58

Acknowledgements

We would like to thank the managing Editor and two anonymous referees for their constructive comments and suggestions, which have led to improvement of the contents of our paper. We also thank Samuel Vazquez and John Nieminen for their insightful comments and suggestions. It is a also a pleasure to thank Jean-Philippe Bouchaud for pointing out connections to his related work in Chicheportiche and Bouchaud (2015). Ajay Singh would like to thank Robert Myers for his support and encouragement in this work. Ajay Singh also thanks Apurva Narayan and Heidar Moradi for several interesting discussions. This research was supported in part by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Note that the volatility for the market mode is not precisely the volatility of the market itself. We further comment on this in section 5.1.

2 In summary, four types of correlation matrices are discussed in the paper: the return correlation matrix defined as in (5(5) ), the volatility correlation matrix given by (11(11) ), the correlation matrix for volatility returns in (C3(C3) ) and the correlation matrix for log-volatility (C2(C2) ).

3 For few time series, we observe that either they are non-stationary or the GARCH(1,1) is not a good model to estimate the volatility. Hence, these time series are also not considered in the paper.

4 It is worth mentioning that unit root tests are preformed to verify the stationarity condition before fitting the data into the GARCH structure (Hyndman and Khandakar 2008, Hyndman et al. 2013).

5 We thank Samuel Vazquez for pointing this out to us.