Scaling and memory in volatility return intervals in financial markets

Abstract

For both stock and currency markets, we study the return intervals between the daily volatilities of the price changes that are above a certain threshold q. We find that the distribution function P q() scales with the mean return interval as Pq() ‫؍‬ ؊1 f(͞ ). The scaling function f(x) is similar in form for all seven stocks and for all seven currency databases analyzed, and f(x) is consistent with a power-law form, f(x) ϳ x ؊␥ with ␥ Ϸ 2. We also quantify how the conditional distribution Pq(Խ0) depends on the previous return interval 0 and find that small (or large) return intervals are more likely to be followed by small (or large) return intervals. This ''clustering'' of the volatility return intervals is a previously unrecognized phenomenon that we relate to the long-term correlations known to be present in the volatility.econophysics ͉ fluctuations ͉ extreme values ͉ long-term correlations ͉ long-term memory T he statistical properties of stock and currency market fluctuations are of importance for modeling and understanding complex market dynamics. They are also relevant for practical applications such as risk estimation and portfolio optimization (1). In particular, understanding the volatility fluctuations of financial records is of particular importance, because they are the key input of option pricing models, including the classic Black and Scholes model and the Cox, Ross, and Rubinstein binomial models that are based on estimates of the asset's volatility during the residual time of the option (2-4). Although the changes from day i Ϫ 1 to day i, ⌬p i ϵ p i Ϫ p iϪ1 , of both stock prices and currency rates are uncorrelated, their absolute values (one measure of volatility) are long-term power-law correlated (5-17). Moreover, the probability density function (pdf) of ⌬p i scales as a power law (18) ⌽(⌬p) ϳ (⌬p) Ϫ( ϩ 1) with Ϸ 3 (5, 19-21). Also, within t days after a crash, n q (t), the number of times ͉⌬p i ͉ exceeds a threshold q, follows a power-law relation n q (t) ϳ t Ϫp with p Ϸ 1 (22), a behavior similar to the Omori earthquake law.Here, we are interested in the statistical properties of large volatilities. A quantity that characterizes the occurrence of large volatilities is the return interval between two consecutive volatilities above some large threshold q (Fig. 1). We study return intervals because they are related to the rate of occurrence of volatilities that exceed a threshold q (22). Because extreme volatilities are rare, we consider also the return intervals between volatilities above intermediate thresholds. By doing this, we hope to gain insight also into the return intervals between very large volatilities that are too rare to obtain with reasonable statistics.We analyze the statistical properties of the daily return intervals of seven representative stocks and currencies obtained, respectively, from http:͞͞finance.yahoo.com and www. federalreserve.gov͞releases͞H10͞hist. We choose to study daily data records because there are intraday trends in the volatility. We report two results:(...

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