Universal behaviour of interoccurrence times between losses in financial markets: An analytical description

¹ Institut für Theoretische Physik, Justus-Liebig-Universität Giessen - 35392 Giessen, Germany
² Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex Systems 22290-180 Rio de Janeiro-RJ, Brazil
³ Santa Fe Institute - Santa Fe, NM 87501, USA

Received: 17 May 2011
Accepted: 29 July 2011

Abstract

We consider 16 representative financial records (stocks, indices, commodities, and exchange rates) and study the distribution P_Q(r) of the interoccurrence times r between daily losses below negative thresholds −Q, for fixed mean interoccurrence time R_Q. We find that in all cases, P_Q(r) follows the form P_Q(r)∝1/[(1+(q− 1)βr]^1/(q−1), where β and q are universal constants that depend only on R_Q, but not on a specific asset. While β depends only slightly on R_Q, the q-value increases logarithmically with R_Q, q=1+q₀ ln(R_Q/2), such that for R_Q→2, P_Q(r) approaches a simple exponential, P_Q(r)≅2^−r. The fact that P_Q does not scale with R_Q is due to the multifractality of the financial markets. The analytic form of P_Q allows also to estimate both the risk function and the Value-at-Risk, and thus to improve the estimation of the financial risk.