Economics Research International

Volume 2014, Article ID 807580, 6 pages

http://dx.doi.org/10.1155/2014/807580

## A Complex Dynamical Analysis of the Indian Stock Market

School of Economics, University of Hyderabad, Hyderabad 500046, India

Received 17 May 2014; Revised 23 November 2014; Accepted 26 November 2014; Published 14 December 2014

Academic Editor: Laura Gardini

Copyright © 2014 Anoop Sasikumar and Bandi Kamaiah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper seeks to analyze the dynamical structure of the Indian stock market by considering two major Indian stock market indices, namely, BSE Sensex and CNX Nifty. The recurrence quantification analysis (RQA) is applied on the daily closing data of the two series during the period from January 2, 2002, to October 10, 2013. A Rolling Window of 100 and step size 21 are applied in order to see how both the series behave over time. The analysis based on three RQA measures, namely, % determinism (DET), laminarity (LAM), and trapping time (TT), provides conclusive evidence that the Indian equity market is chaotic in nature. Evidences for phase transition in the Indian equity market around the time of financial crisis are also found.

#### 1. Introduction

An equity market could be considered as complex dynamical system, with different agents as well as institutions having different time horizons in mind, carrying out transactions that result in complex patterns that are reflected in the data. A complex dynamical/chaotic system may be defined as a type of nonlinear dynamical system which could satisfactorily explain a wide range of phenomena in many natural systems, including biological and physical systems. Such systems appear apparently random in nature but indeed are part of a deterministic process. Their apparent random nature is given by their characteristic sensitivity to initial conditions that drives the system to unpredictable behavior over time. However, in a chaotic system, this nonlinear behavior is always limited by a higher deterministic structure. For this reason, there is always an underlying order in the apparent random dynamics.

Complex dynamical systems are considered to be mathematically deterministic because if the initial measurements were certain, it would be possible to derive the end point of their trajectories. Contrary to classical mechanics, chaos theory deals with nonlinear feedback forces with multiple cause and effect relationships that can produce unexpected results. In such a situation, a chaotic system cannot be understood by the simple disintegration of the whole into smaller parts.

An important characteristic of such a system is sensitive dependence on initial conditions (henceforth SDIC) implying that even a small difference in the initial conditions gives rise to widely different paths after some time interval. In a “normal” deterministic system, all nearby paths starting very close to one another remain very close in the future. Hence, a sufficiently small measurement error in the initial conditions will not affect our deterministic forecasts. On the contrary, in deterministic systems with SDIC, prediction of the future values of the variable(s) would be possible only if the initial conditions could be measured with infinite precision. This is certainly not the case.

Among the constituents of financial markets of a country, equity market plays a significant role. Due to the increased integration among financial markets in the world, they are affected by extreme events like the 2008 financial crisis. Considering the complex dynamical structure of financial markets, the usual deterministic models, even nonlinear models such as GARCH, might not be able to capture the true dynamics of the underlying process. Here, the present study seeks to analyze the dynamics of Indian capital market by applying a novel methodology. An attempt is made to analyze two major Indian equity market indices, namely, BSE Sensex and CNX Nifty, so as to study the performance of Indian capital market. Here the method used is recurrence quantification analysis proposed by Zbilut et al. [1].

#### 2. Literature Review

Application of recurrence quantification analysis (RQA) in the field of financial markets is of recent origin. Hence there are only few studies available in the literature applying this methodology to financial markets, especially equity markets. Here, we present a quick review of the available studies.

Guhathakurta et al. [2] studied three equity markets, namely, Nifty, Hong Kong AOI, and DJIA, using the RQA. They mainly employed recurrence plots (RP) to capture endogenous market crashes and found evidences for phase transitions before the occurrence of a market crash. Using the same methodology, Bastos and Caiado [3] analyzed the presence of complex dynamical structure in international stock markets. Their results suggest that the dynamics of stock prices in emerging markets is characterized by higher values of RQA measures when compared to their developed counterparts. They analyzed the behavior of stock markets during extreme financial events, such as the burst of the technology bubble, the Asian currency crisis, and the recent subprime mortgage crisis, using RQA in sliding windows. It is shown that during these events stock markets exhibit a distinctive behavior that is characterized by temporary decreases in the fraction of recurrence points contained in diagonal and vertical structures. Similarly, Bigdeli et al. [4] analyzed the dynamical properties of Iranian stock prices using recurrence quantification analysis along with other methods. They found evidences of seasonality and nonstationarity in the data analyzed. The results confirmed presence of chaotic behavior in the Iranian equity market.

O. Piskun and S. Piskun [5] studied various stock market crashes, such as DJI 1929; DJI, NYSE, and S&P500 1987; NASDAQ 2000; HSI 1994, 1997 and Spanish 1992, Portuguese 1992, British 1992, German 1992, Italian 1992, Mexican 1994, Brazilian 1999, Indonesian 1997, Thai 1997, Malaysian 1997, Philippine 1997, Russian 1998, Turkish 2001, Argentine 2002, and the 2008 financial crisis, and discussed the use of the measure “laminarity” to identify market bubbles. In a recent study, Moloney and Raghavendra [6] examined the DJIA using RQA and found evidences for phase transitions as the markets move from Bull to Bear state. There were also evidences for a nondeterministic regime when the market reaches its peaks. The authors used noise trader theory to support the findings.

From the available literature, it is evident that the RQA could be useful in capturing the dynamical properties of an equity market. A comprehensive study on the dynamical nature of the Indian equity market is yet to be carried out to the best of the author’s knowledge. The present study positions itself in this direction. The remainder of this paper is organized as follows. Section 3 presents the data and methodology used. Section 4 provides the empirical analysis and concluding remarks are given in Section 5.

#### 3. Data and Methodology

Two major indices from the Indian capital markets, that is, CNX Nifty and BSE Sensex, are selected. Daily closing data from January 2, 2002, to October 10, 2013, are collected and used for the purpose of analysis. To avoid scale difference, we apply a logarithmic transformation to both series. To analyze the dynamical structure, we apply the RQA developed by Zbilut et al. [1]. RQA is essentially quantification of the recurrence plots developed by Eckmann et al. [7]. It is employed to analyze the dynamical structure of a time series based on the property of recurrence. RQA provides various measures that could explain the dynamical nature present in the data. Here, a Rolling Window RQA is proposed so as to capture the time varying dynamics of the Indian capital market indices. Three RQA estimates, namely, percentage determinism (DET), laminarity (LAM), and trapping time (TT), are calculated using a Rolling Window.

##### 3.1. Recurrence Quantification Analysis

The RQA method which is used in the study consists of two parts: the recurrence plot (RP) developed by Eckmann et al. [7], a graphical tool that evaluates the temporal and phase space distance, and recurrence quantification analysis (RQA), a statistical quantification of RP. The basic idea of the RP is that, in a chaotic system, the nearby trajectories visit the same points in the phase space repeatedly. Here, the closeness is measured by a critical radius. A recurrence plot is used to show this behavior graphically. The advantage of the RP is that it could provide an adequate visual representation of a process that happens in the -dimensional phase space in a two-dimensional plot.

Let be a time series whose trajectories are orbiting in the phase space. If the orbit is one period, the trajectory will return to the neighbourhood of after an interval equals 1; if the orbit is two periods, it will return after an interval equals 2, and so on. Therefore, if evolves near a periodic orbit for a sufficiently long time, it will return to the neighbourhood of after some interval . The criterion of closeness requires that the difference be very small.

Computing differences , where , , and is the length of sample, the close return test detects the observations for which is smaller than a threshold value . is plotted against to observe the patterns. If the data is i.i.d., the distribution of points will be random. If the time series is deterministic, it is possible to observe horizontal line segments.

Recurrence plots are symmetrical over the main diagonal. It is based on the reconstruction of time series and an estimation of the points that are close. This closeness is measured by a critical radius so that a point is plotted as a colored pixel only if the corresponding distance is within this radius. The line segments (diagonals) parallel to main diagonal are point that move successively closer to each other in time and would not occur in random. Chaotic behavior produces very short diagonals, whereas deterministic behavior produces longer diagonals. Thus, if the analyzed time series is chaotic, then the recurrence plot shows short segments parallel to the main diagonal; on the other hand, if the series is white noise, then the recurrence plot does not show any kind of structure.

RP may be explained with the help of the examples given in Figure 1.