[Paper Review] Omori law after a financial market crash

Fabrizio Lillo, Rosario N. Mantegna|arXiv (Cornell University)|Nov 14, 2001

Complex Systems and Time Series Analysis5 citations

TL;DR

This paper demonstrates that financial market crashes trigger a power-law relaxation dynamics in extreme return events—mirroring the Omori law in seismology—by showing that the decay of such events follows a power law. This behavior arises from the interplay between a fat-tailed return distribution and a power-law decay in volatility scale, challenging standard stochastic volatility models.

ABSTRACT

We study the relaxation dynamics of a financial market just after the occurrence of a crash by investigating the number of times the absolute value of an index return is exceeding a given threshold value. We show that the empirical observation of a power law evolution of the number of events exceeding the selected threshold (a behavior known as the Omori law in geophysics) is consistent with the simultaneous occurrence of (i) a return probability density function characterized by a power law asymptotic behavior and (ii) a power law relaxation decay of its typical scale. Our empirical observation cannot be explained within the framework of simple and widespread stochastic volatility models.

Motivation & Objective

• To investigate the temporal relaxation of extreme market returns after a crash.
• To determine whether the observed power-law decay of extreme events aligns with known physical laws like the Omori law in seismology.
• To examine whether standard stochastic volatility models can explain the empirical dynamics of post-crash market behavior.
• To identify the underlying statistical mechanisms driving the power-law behavior in post-crash event frequency.

Proposed method

• Empirically analyze the number of times index returns exceed a threshold after a market crash.
• Model the return probability density function as having power-law asymptotic behavior.
• Characterize the typical scale of returns as decaying according to a power law over time.
• Compare the observed relaxation dynamics with predictions from standard stochastic volatility models.
• Use empirical data to test the consistency of the Omori law with the joint presence of fat-tailed returns and decaying volatility scales.

Experimental results

Research questions

• RQ1 Does the frequency of extreme market return events decay according to a power law after a crash, as predicted by the Omori law?
• RQ2 What statistical properties of return distributions and volatility scales give rise to the observed Omori-like relaxation?
• RQ3 Can standard stochastic volatility models reproduce the empirically observed power-law decay in extreme event frequency?
• RQ4 How do fat-tailed return distributions and decaying volatility scales jointly influence post-crash market dynamics?

Key findings

• The number of extreme return events after a crash follows a power-law decay, consistent with the Omori law observed in seismology.
• The power-law behavior in event frequency is explained by the simultaneous presence of a fat-tailed return distribution and a power-law decay in the typical volatility scale.
• The empirical dynamics cannot be reproduced by simple stochastic volatility models, which fail to capture the joint scaling behavior of returns and volatility.
• The relaxation process is characterized by a long memory effect, where extreme events persist longer than predicted by standard models.