Chapter Introduction

Transcription

1 Chapter Introduction Research on stock market volatility is central for the regulation of financial institutions and for financial risk management. Its implications for economic, social and public welfare issues have a great significance. In recent times, a lot of activity has been witnessed in the field of econophysics (Fama (1970), Mantegna and Stanely (2000), Weron and Przybylowicz (2000)). Various concepts of mathematics and physics have been applied to study financial time series both for short and long range studies. In the recent past some new mathematical methods such as wavelets, fractal and multifractal are being used for classification and understanding these systems (Gencay et al. (2002), Arneodo et al. (2002), Razdan (2004)). It has been found that financial time series (like stock indices, currency exchange rates etc.) have either fractal or multifractal features (Ausloos and Ivanova (2001), Razdan (2002)). Another important feature that has been observed in recent times is that stock market indices can be represented by fractional Brownian motion instead of a classical Brownian motion. The similarity between the fluid turbulence and financial market is well known. The information transfer in financial markets is similar to energy flow in hydrodynamics (Vandewalle and Ausloos (1998)). The accuracy of measured financial risk models depends on the assumptions about the statistical properties of asset prices. The statistical properties of data during financial crisis are very different from those in normal markets. This will require considerable adjustment of the existent regulatory risk monitoring techniques which were developed for non extreme financial market behavior. Analyzing the time series statistically has been problem of considerable recent interest. With the surge of data outpouring from various fields such as biology, geophysics and finance, it is becoming imperative to use proper mathematical tools for classification and understanding these systems. For example, in the field of DNA sequence analysis, there exists immense mathematical literature (Bernaola et al. (2000), Arneodo et al. (2002)). More recently (as of late 1995 and beyond), Wall Street analysts have started using calculus based methods for their own analysis. 119

2 These advances have motivated certain groups of physicists and applied mathematicians to try some of their own classical methodologies to understand the dynamics of the stock market fluctuations. A tremendous amount of work using wavelet applications (Muzy et al. (2001)), L evy distributions, spectral analysis, and multifractal models (Mandelbrot (1997)) (to name a few) have been used as tools to understand the mathematical nature of the financial time series. The basic goal of all these methodologies is to understand the mathematical properties of long term memory versus short term memory (long range versus short range correlations) of the market. The mathematics and the images derived from fractal geometry exploded the world in 1970s and 1980s. It is difficult to think of an area of science that has not been influenced by fractal geometry. Along with providing new insight in mathematics and science, fractal geometry helped us see the world around us. The Hurst exponent occurs in several areas of applied mathematics including fractals and chaos theory, long memory process and spectral analysis. Hurst exponent estimation has been applied in areas ranging from biophysics to computer networking (Hurst (1951), Moody and Wu (1995), Weron and Przybylowicz (2000)). The Hurst exponent is directly related to the fractal dimension which gives a measure of roughness of a surface. Estimating the Hurst exponent for a data set provides a measure of whether the data is a pure random walk or has underlying trends. Another way to state this is that a random process with an underlying trend has some degree of autocorrelation. When the autocorrelation has a very long (or mathematically infinite) decay this kind of Gaussian process is sometimes referred to as a long memory process. Standard statistical analysis begins by assuming that the system under study is primarily random, i.e., the underlying components have many parts and the interaction of those components is so complex that a deterministic explanation is not possible. In this set up the system is usually modeled by a random walk-type process, which implies that the events measured are independently and identically distributed. In other words, the events must not influence one another and they must all be equally likely to occur. However there exists strong evidence that price processes of financial assets do not fall in this category. 120

3 In general, much research ((Fama (1970), Mandelbrot (1997), Weron and Przybyloeicz (2000), Muzy et al. (2001), Gencay et al. (2002), Cajuerio and Tabak (2004)) has been carried out to measure the fluctuations of the market indices. For example, a lot of work has been done in efforts to detect trends in S&P 500 (for various different time periods).very recently Cajueiro and Tabak (2004) have worked for calculating Hurst exponent for testing the assertion in the financial markets that emerging markets are becoming more efficient over time by calculating the Hurst exponent. Similarly, Weron and Przybyloeicz (2000) have studied the electricity price dynamics through Hurst analysis. Fractal dimension of the S&P 500 index has been studied by Bayraktar et al. (2004). They have used wavelet analysis. Siddiqi and Ahmad (2001) have studied the behavior of the Saudi Stock market through instruments of Black Scholes world. There has also been some work done on currency exchange dynamics (Vandewalle and Ausloos (1998)). In most of these cases, the correlation content has been measured for long term periods. This chapter deals with the fluctuations in the market indices. Here we present study of financial data set of BSE 100 and Nifty 50 (the Indian stock exchanges). In particular we have examined the BSE 100 over 17.5 years, taken at one day interval and Nifty 50 over the 2 years, taken at every 5 minute interval. We estimated the Hurst exponent using wavelets and Rescaled Range (R/S) method Financial Market Data Sets There are many kinds of financial markets like equity, insurance, mutual funds and commodity markets but the most important one is the stock market which deals in shares. The sample employed in our study consists of daily closing of the indices of BSE 100 and Nifty 50. For BSE100, we consider the daily closings from 06 January 1986 to 07 August This generates 4096 points for the said period. In case of Nifty50, the data length is from 29 August 2002 to 12 August We study this data over every 5 minutes return, generating in total data points. This chapter has been divided into five sections. In the first section, we review some literature and the work done in the field of econophysics to study the behavior of financial markets by the application of Hurst exponent. In the second section, we present term and notation used in subsequent analysis. In the third section, we have 121

4 estimated the Hurst exponent for two Indian stock markets BSE 100 and Nifty 50 by using two different approaches, one with the Rescaled Range method and secondly, by using the Benoit software. In the fourth section, we have compared the results obtained for Hurst exponent and fractal dimension. We give the post analysis conclusion in the last section. The results of the work are published in Kumar and Manchanda (2009) Preliminaries We recall here that any set is said to be self similar if it is the union of disjoint subsets that can be obtained from with a scaling, translation and rotation. This self similarity often implies an infinite multiplication of details, which creates irregular structures Daily return The return is the profit or loss in buying share of stock (or some other traded item), holding it for some period of time and then selling it. The most common way to calculate a return is the log return. ( ) ( ) ( ) (5.1) This equation calculates log return for a share of stock that is purchased, held for days, and then sold. Here and are the prices at time t (when we sell the stock) and (when we purchased the stock, days ago). Taking the log of price removes most of the market trend from the return calculations. Another way to calculate the return is to use the percentage return, which is as shown in Equation (5.2). ( ) ( ) (5.2) Equations (5.1) and (5.2) yield similar results. An n-day return time series is created by dividing the given time series data into a set of blocks and calculating log return (equation 5.1) for, the price at the start of the block and the price at the end of the block. We considered the 1-day return time series. 122

5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (5.3) 5.3. Estimation of Hurst Exponent In this section, we estimate the Hurst exponent of BSE 100 and Nifty 50 using Rescaled Range (R/S) method and Benoit Rescaled Range Method We denote ( ) as the closing of the index on a time and ( ) as the logarithmic return denoted by The ( ) ( ( ) ( )). statistic is the range of partial sums of deviations of time series from its mean, rescaled by its standard deviation. Consider a sample of continuously compounded asset returns * + and let denote the sample mean where is the time span considered. The statistic is then given by ( ), ( ) ( )- (5.4) where is the usual standard deviation estimator ( ). (5.5) Hurst found that the rescaled range, R/S, for many records in time is very well described by the following empirical relation: ( ) ( ). (5.6) In practice for a given window length, we subdivide the input series in a number of intervals of length w, measure ( ) and ( ) in each interval and calculate ( )( ) as the average ratio ( ) ( ). This process is repeated for a number of window lengths, and the logarthims of ( )( ) are plotted versus the logarithms of. The plot follows a straight line whose slope equals the Hurst exponent. The 123

6 fractal dimension of the trace is then calculated from the relationship between the Hurst exponent and the fractal dimension: where denotes the fractal dimension estimated from the Rescaled Range method. The values of the Hurst exponent range between 0 and 1. A value of 0.5 indicates a true random walk (a Brownian time series). In a random walk there is no correlation between any element and future element. A Hurst exponent value H,, indicates persistent behavior (e.g., a positive autocorrelation). If there is any increase from step to there will probably be an increase from time step to. The same is true for decrease, where a decrease will tend to follow a decrease. A Hurst exponent value will exist for a time series with anti-persistent behavior (or negative autocorrelation). Here an increase will tend to be followed by a decrease or decrease will be followed by an increase. This behavior is also sometimes is called mean reversion Benoit Benoit software enables to measure the fractal dimension and/or Hurst exponent of the data sets. It includes ruler, box, information, perimeter-area, and mass for analysis of self-similar patterns and R/S power-spectral analysis, variogram, roughness-length, and wavelets for analysis of self-affine traces. It is particularly useful to find orders and patterns in seemingly chaotic data where traditional statistical approaches of analysis fail. It is widely used in diverse disciplines like physics, chemistry, biology, economics and medicine. We used the wavelet method in Benoit to find Hurst exponent for the financial time series data of BSE 100 and Nifty Hurst Analysis We perform the estimation of the Hurst exponent for time windows with 4096 data values (log returns) for the BSE. This data is processed on one day return basis. NSE data for 2 years is processed on every 5 minute return basis, thus taking data values in total. First R/S is calculated over the single time window over the full time range. Then the successive calculations were performed by subdividing the data in respective halves at each stage of calculation. We developed a C program to carry out the above calculations. 124

7 Figure 5.1: This plot shows the daily closing of 4096 data points of BSE100. Figure 5.2: This Plot shows the log returns of 4096 data points of BSE

8 Figure 5.3: This plot shows the values of Nifty 50 for the first half data values (from 1 to 16384) over every five minutes return. Figure 5.4: This plot shows the daily closing of second half data values (from to 32768) of Nifty

9 Figure 5.5: This plot shows the log of daily returns for first data points of Nifty 50. Figure 5.6: This plot shows the log returns for second half (from data values16385 to 32768) data points of Nifty

10 5.4. Results Table 1. Results for BSE Region Size R/S Avg. log2(region size) log2(r/s Avg.) H.E. for BSE=

11 Table2. Results for NSE Region Size R/S Avg. log2(region Size) log2(r/s Avg.) H.E.for NSE=

12 Table 3. Rescaled Range Method BSE 100 Nifty 50 Hurst Exponent Fractal Dimension Table 4. Benoit (Wavelet method) BSE 100 Nifty 50 Hurst Exponent Fractal Dimension Conclusion We compared the results calculated using rescaled range method with those obtained from Benoit. Using Rescaled Range (R/S) method the values of the Hurst exponents obtained suggests an anti persistent behavior. But the graphs of two stock markets BSE 100 and Nifty 50 shows a persistent behavior of the data under consideration and this assertion is also supported by the values of Hurst exponents those obtained with Benoit. For the estimation of Hurst exponent, using Benoit is a better approach than Rescaled Range (R/S) method. ************* 130