Irregularity, volatility, risk, and financial market time series

Abstract

The need to assess subtle, potentially exploitable changes in serial structure is paramount in the analysis of financial data. Herein, we demonstrate the utility of approximate entropy (ApEn), a modelindependent measure of sequential irregularity, toward this goal, by several distinct applications. We consider both empirical data and models, including composite indices (Standard and Poor's 500 and Hang Seng), individual stock prices, the random-walk hypothesis, and the Black-Scholes and fractional Brownian motion models. Notably, ApEn appears to be a potentially useful marker of system stability, with rapid increases possibly foreshadowing significant changes in a financial variable.approximate entropy ͉ stock market ͉ instability ͉ random walk S eries of sequential data are pivotal to much of financial analysis. Enhanced capabilities of quantifying differences among such series would be extremely valuable. Although analysts typically track shifts in mean levels and in (several notions of) variability, in many instances, the persistence of certain patterns or shifts in an ''ensemble amount of randomness'' may provide critical information as to asset or market status. Despite this recognition, formulas to directly quantify an ''extent of randomness'' have not been utilized in market analyses, primarily because, even within mathematics itself, such a quantification technology was lacking until recently. Thus, except for settings in which egregious (changes in) sequential features or patterns presented themselves, subtler changes in serial structure would largely remain undetected.Recently, a mathematical approach and formula, approximate entropy (ApEn), was introduced to quantify serial irregularity, motivated by both application needs (1) and by fundamental questions within mathematics (2, 3). ApEn grades a continuum that ranges from totally ordered to maximally irregular ''completely random.'' The purpose of this article is to demonstrate several applications of ApEn to the evaluation of financial data.One property of ApEn that is of paramount importance in the present context is that its calculation is model-independent, i.e., prejudice-free. It is determined by joint-frequency distributions. For many assets and market indices, the development of a model that is sufficiently detailed to produce accurate forecasts of future price movements, especially sudden considerable jumps or drops, is typically very difficult (discussed below). The advantage of a model-independent measure is that it can distinguish classes of systems for a wide variety of data, applications, and models. In applying ApEn, we emphasize that, in many implementations, we are decidedly not testing for a particular (econometric) model form; we are attempting to distinguish data sets on the basis of regularity. Even if we cannot construct a relatively accurate model of the data, we can still quantify the irregularity of data, and changes thereto, straightforwardly. Of course, subsequent modeling remains of interest, although the point is that this...

Applying Approximate Entropy (ApEn) to Speculative Bubble in the Stock Market

“…Though initially developed for measuring the irregularities of a complex nonlinear system, it has been gradually introduced into finance literature as a measure of market efficiency for both stock and foreign exchange market (Pincus et al, 2004, andOh et al 2006). However, the paper for the first time attempts to use this statistics to test for speculative rational bubble in the stock markets.…”

“…The approximate entropy was introduced by Pincus et al (1991Pincus et al ( , 1997Pincus et al ( and 2004 to quantify the creation of information in a time series. Though initially developed for measuring the irregularities of a complex nonlinear system, it has been gradually introduced into finance literature as a measure of market efficiency for both stock and foreign exchange market (Pincus et al, 2004, andOh et al 2006).…”

“…In this paper, the ApEn is estimated with embedding dimension, m = 2 and the similarity measure, r = 20% of the standard deviation of the time series, similar to the previous work (Pincus et al 2004). To test the robustness of our findings several values of m and r are attempted.…”

“…First the paper introduces Approximate Entropy (ApEn), a widely used measure in statistical physics to test the special empirical properties of rational speculative bubble during three known episodes of crash across the various markets like U.S, Japan, and India respectively. The Approximate Entropy (ApEn) proposed by Pincus et al (1991Pincus et al ( , 2004, has been used in this article to quantify the likelihood of "order" (repetitiveness) in the financial time series. The three episodes of crash considered in the paper involve the US crash in October 1997, the Japan crash in December 1989, and January 2007-08 crash in India.…”

Financial Time-series Analysis: a Brief Overview

“…Comparisons are intended to be done at fixed m and r, the general ApEn(m,r) being in fact a family of parameters. In economics, the ApEn method has been shown to be a reliable estimate of the efficiency of market [25,26,27] and has been applied to various economically relevant events. For instance, the ApEn computed for the S&P 500 index has shown a drastic increase in the two-week period preceding the stock market crash of 1987.…”