Regularities in stock markets

TL;DR

This paper identifies universal statistical regularities in stock market indices across six major economies, revealing exponential growth, Gaussian fluctuations, and self-similarity in daily and monthly stock behaviors.

Contribution

It provides a comprehensive analysis of stock market regularities across multiple countries, highlighting universal patterns in growth and fluctuations.

Findings

Exponential mean growth in stock values and trade volume.

Gaussian distribution of daily fluctuations with convergence.

Self-similarity in monthly average growth and decline in volatility.

Abstract

From the stock markets of six countries with high GDP, we study the stock indices, S&P 500 (NYSE, USA), SSE Composite (SSE, China), Nikkei (TSE, Japan), DAX (FSE, Germany), FTSE 100 (LSE, Britain) and NIFTY (NSE, India). The daily mean growth of the stock values is exponential. The daily price fluctuations about the mean growth are Gaussian, but with a non-zero asymptotic convergence. The growth of the monthly average of stock values is statistically self-similar to their daily growth. The monthly fluctuations of the price follow a Wiener process, with a decline of the volatility. The mean growth of the daily volume of trade is exponential. These observations are globally applicable and underline regularities across global stock markets.

Tables

Table 1. Table 1: Summary of the analysis of financial data taken from six stock markets, whose names are listed in the first column. Regularities across global markets are discernible through the numerical values of the parameters catalogued below.

Stock Index	$a$ (% per day)	$\mu$	$\sigma$	$m$ (per month)	$w$ (per month)	$\nu$ (% per day)
S&P 500 (NYSE, USA)	$0.03$	$0.032$	$1.203$	$0.006$	$-2.78\times {10}^{-6}$	$-$
SSE Composite (SSE, China)	$0.04$	$0.101$	$2.805$	$0.009$	$-3.09\times {10}^{-5}$	$0.12$
Nikkei (TSE, Japan)	$0.01$	$0.027$	$1.494$	$0.003$	$-1.02\times {10}^{-6}$	$0.53$
DAX (FSE, Germany)	$0.03$	$0.025$	$1.466$	$0.006$	$-4.42\times {10}^{-6}$	$0.02$
FTSE (LSE, Britain)	$0.01$	$0.011$	$1.165$	$0.002$	$-1.78\times {10}^{-6}$	$\mathrm{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.003}$
NIFTY (NSE, India)	$0.05$	$0.057$	$1.495$	$0.010$	$-3.41\times {10}^{-6}$	$0.04$

Equations

$$\frac{\Delta S}{S}=a\Delta t+b\Delta W,$$

$$f(\delta )=1+f_0\exp \left[-\frac{(\delta -\mu {)}^2}{2{\sigma }^2}\right],$$

Full text

Econophysics, financial markets; Brownian motion; Random walks; Time series analysis

I Introduction

The main concerns of a stock market analyst are the percentage growth rate of stock values and the susceptibility of stock prices to fluctuations. Speculation in the stock market is encouraged by the optimism that markets grow over time scales that are long enough to average out the impact of adverse fluctuations in stock values. Ever since Bachelier (1900) proposed his theory of speculation, many formal approaches have afforded continuous refinement in the analysis of financial data (see Mantegna and Stanley (2000); Sinha et al. (2011) and all references therein for a historical development of the subject). One object of such researches is to discern universal features in stock data Pan and Sinha (2008), whose statistical properties do not vary overmuch, irrespective of the market. Financial data of world markets do exhibit certain features that appear to be broadly common, and as such, markets attain some predictability. Understanding these familiar patterns inspires greater confidence in the speculative activity.

Our work here has a similar purpose. We study stock indices from the stock markets of six countries that are in the top ten of the global ranking of GDP. They are USA, China, Japan, Germany, Britain and India. Our chosen stock indices are S&P 500 (NYSE, USA) snp , SSE Composite (SSE, China) sse , Nikkei (TSE, Japan) nik , DAX (FSE, Germany) dax , FTSE 100 (LSE, Britain) fts and NIFTY (NSE, India) nif . We examine stock indices rather than individual stocks, because stock indices cover multiple companies across a wide range of sectors. Thus, stock indices reflect the overall condition of markets more comprehensively than would an individual stock Gopikrishnan et al. (1999); Plerou et al. (1999); Pan and Sinha (2008). The health of a market can be gauged fairly through the daily price movement of its stock indices, the fluctuations they undergo, and the daily volume of stocks traded. For these variables, we arrive at well-restricted ranges of numerical values across the markets we have studied. We see that the mean daily growth of the value of stock indices is exponential (Sec. II). The daily fluctuations of a stock index about its exponentially growing mean value has a Gaussian distribution, but with a non-zero asymptotic convergence for large fluctuations (Sec. III). The growth of the average monthly value of a stock index is statistically self-similar to its daily growth (Sec. IV). The monthly variance of the price follows a Wiener process Mantegna and Stanley (2000); Hull (2006), with the volatility reducing progressively in time (Sec. IV). The mean growth of the number of daily transaction of stocks is exponential (Sec. V). We present our results graphically using the data of NIFTY (NSE, India) nif , from January, 1997 to April, 2019. Similar results from five other stock markets snp ; sse ; nik ; dax ; fts are summarized in Table 1. The conclusions derived from our analysis of financial data of stock markets of six high-GDP countries are globally valid.

II The daily growth of mean stock prices

The forward relative change of a stock price, $S$, in a finite time interval, $\Delta t$, is given by Mantegna and Stanley (2000); Hull (2006)

[TABLE]

in which $\Delta W$ expresses a Wiener process Mantegna and Stanley (2000); Hull (2006) about a background exponential growth of $S$, implied by the first term on the right hand side of Eq. (1). Under an idealized volatility-free condition we set $b=0$ in Eq. (1), and then integrate it in continuous time to get a steady compounded growth of $S$. The integral solution of $S$ is exponential in time, $S=S_{0}\exp(at)$. We, however, express this volatility-free equation slightly differently as $\Delta(\ln S)=a\,\Delta t$, and fit it with the data of the stock index, NIFTY (NSE, India). The fit is shown in Fig.1, which is a linear-log plot of the movement of the daily average price of NIFTY (NSE, India) over more than two decades, from January, 1997 to April, 2019. The mean growth of $\ln S$, fitted by the least-squares method, is linear in Fig.1, demonstrating thereby that the daily mean growth of $S$ occurs exponentially.

The daily movement of five other stock indices, i.e. S&P 500 (NYSE, USA), SSE Composite (SSE, China), Nikkei (TSE, Japan), DAX (FSE, Germany) and FTSE 100 (LSE, Britain), displays the same graphical trend as in Fig.1, over time scales of two decades. In all cases, the mean relative growth rate of $S$ is $a$, whose values are provided in Table 1. Globally, $a$ is confined within a range of $0.01$% to $0.05$% per day, which seems to be surprisingly narrow and precise.

III Gaussian fluctuations in stock prices

By setting $b=0$ in Eq. (1), we have so far ignored the fluctuations about the mean exponential growth of a stock price, $S$. In reality, however, as we can see very clearly in Fig.1, stock prices fluctuate noticeably about the mean exponential growth (represented by the straight line in Fig.1). To quantify the fluctuations, we define a new variable, $\delta$, which is the daily percentage change that a stock index undergoes, with respect to the value of the previous day. Positive fluctuations are given by $\delta>0$, and negative fluctuations by $\delta<0$. The time series of such fluctuations, for the stock index, NIFTY (NSE, India), is plotted in Fig.2, which shows that positive and negative fluctuations are balanced about $\delta=0$. Most of the fluctuations are close to $\delta=0$, with a very few cast far away. The unnormalized frequency distribution of these fluctuations has a Gaussian profile in Fig.3, but with an asymptotic convergence towards unity. To model all these aspects, we put forward a Gaussian function, with unity added to it, as

[TABLE]

in which the mean of the frequency distribution, $\mu=\langle\delta\rangle$, and the standard deviation, $\sigma=\sqrt{\langle\delta^{2}\rangle-\mu^{2}}$. The values of both $\mu$ and $\sigma$, calibrated with the data of our chosen six stock indices, are listed in Table 1. With the frequency distribution clustering around $\delta=0$ in Fig.3, we expect $\mu$ to have a small value. Supporting this expectation, Table 1 shows us that $\mu$ is zero up to the first place of decimal for all the stock indices, except SSE Composite (SSE, China). Likewise, the first significant figure of $\sigma$ is unity, except for SSE Composite (SSE, China).

The theory of unbiased and random fluctuation of stock prices, described by a continuous Gaussian function, was introduced in financial markets by Bachelier (1900). The randomness of prices is a response to a ceaseless stream of unpredictable and uncorrelated information that enters the market continuously. While this process plays out on the microscopic scale of individual prices, its macroscopic imprint is manifested through a similar randomness in the market indices. Fluctuations in financial markets are, therefore, a collective macro-outcome of a diversity of microscopic causes that are independent of one another. The randomness agrees theoretically with the Gaussian function, but only if the time scale of the fluctuations is long enough Gopikrishnan et al. (1999); Plerou et al. (1999); Mantegna and Stanley (2000), such as the scale of a day in our study. On shorter time scales, however, market indices exhibit scaling properties Mantegna and Stanley (1995), leading to an inverse cubic law of stock value fluctuations Gopikrishnan et al. (1998); Pan and Sinha (2008). Such fluctuations converge more slowly towards an asymptotic limit than the Gaussian function does. Slow convergence is the hallmark of power laws, but we can mimic a slow convergence even with a Gaussian function, by adding a constant of unity to the right hand side of Eq. (2). This reconciles a continuous Gaussian function with a discrete frequency distribution, whose lower limit is practically restricted to unity.111Discrete frequency distributions in complex systems are similarly modelled by power laws that converge asymptotically to a non-zero limit Nair et al. (2012, 2014). Consequently even for large fluctuations, the unnormalized frequency distribution will asymptotically converge to unity. When $\delta\longrightarrow\infty$, the form of Eq. (2), despite being Gaussian in essence, theoretically ensures the convergence, $f(\delta)\longrightarrow 1$.

Since the Gaussian distribution in Fig.3 is based on the daily data of NIFTY (NSE, India), from January, 1997 to April, 2019, it is worth noting that nearly two decades ago, an exponential distribution of price fluctuations in NSE (India) was reported Matia et al. (2004). The reason for the exponential distribution, which is intermediate between a power-law distribution and a Gaussian distribution, was put down to conservative tendencies in the evolving Indian financial market, and the control of the state over it Matia et al. (2004). However, in the two decades since then, the Indian financial market has matured towards greater liberalization. Therefore, we are of the view that India, being a country that ranks high in terms of GDP, more faithfully reflects at present the trends of developed financial markets. Our point of view is supported by the conclusion of a study that “far from being different, the distribution of price fluctuations in Indian markets is quantitatively almost identical to that of developed markets.” Sinha and Pan (2006) The Gaussian distribution in Fig.3 is not because of the stage of development of the Indian financial market. Rather it is due to the time scale of the price fluctuations in our study, for it is recognized well by now that on longer time scales, price fluctuations assume the Gaussian form Gopikrishnan et al. (1999); Plerou et al. (1999); Mantegna and Stanley (2000); Pan and Sinha (2007).

IV The Wiener process and volatility

Looking back at Eq. (1), we consider first the basic nature of a Wiener process. If a variable, $W$, undergoes a Wiener process, then its change, $\Delta W$, in a discrete interval of time, $\Delta t$, is $\Delta W=\epsilon\sqrt{\Delta t}$ Hull (2006). Here $\epsilon$ has a standardized normal distribution with a zero mean and a standard deviation of unity Hull (2006). For different short intervals, $\Delta t$, $\Delta W$ will have independent values. Over a time interval, $T$, the mean, $\langle\Delta W\rangle=0$, and the standard deviation, $\sqrt{\langle(\Delta W)^{2}\rangle}\sim T^{1/2}$ Hull (2006).

Stock price variations follow a generalized Wiener process Hull (2006), viewed also as a geometric Brownian motion Mantegna and Stanley (2000); Sinha et al. (2011); Hull (2006). The generalized Wiener process has a non-zero mean that drifts linearly in time Hull (2006), as the first term on the right hand side of Eq. (1) suggests. About this linear mean drift, there is a stochastic component, given by the second term on the right hand side of Eq. (1), now to be read as $\Delta W=\epsilon\sqrt{\Delta t}$. On a daily time scale, $t$, the mean growth is depicted by the straight line in Fig.1, while the stochastic fluctuations (related to the volatility of the stock value Hull (2006)) are seen in the jagged features about the straight line. Likewise, all through January, 1997 to April, 2019, on a monthly time scale, which we write as $\tau$ to distinguish it from $t$, the straight line in Fig.4 shows the mean growth of the monthly average of the natural logarithm of the price of the stock index, NIFTY (NSE, India). About the mean linear growth of $\langle\ln S\rangle$, whose slope is $m$ in Fig.4, the jaggedness shows the usual stochastic behaviour — the volatility. Notwithstanding the noticeable difference in the time scales, i.e. a day for Fig.1 and a month for Fig.4, there is a statistical self-similarity between the two plots. This is a generic characteristic of all the stock indices that we have studied. Consistent with this observation, we find from Table 1 that across all stock indices, there is a high correlation coefficient of $0.987$ between $a$ and $m$.

From the NIFTY (NSE, India) index, besides estimating $\langle\ln S\rangle$ on a monthly time scale, we also estimate the standard deviation of $\ln S$ over the same month. The standard deviation, $\Sigma$, is related to the volatility of the stock price Hull (2006). This volatility is clearly revealed by the fluctuations about the linearly growing mean of $\langle\ln S\rangle$ in Fig.4. We also see in Fig.4 that the fluctuations reduce progressively as both $\tau$ and $S$ increase. Hence, the volatility of the stock price subsides with the passage of time. We can understand this phenomenon for a Wiener process by recasting Eq. (1) as $(\Delta S/S)=a\Delta t+b\epsilon\sqrt{\Delta t}$, on whose right hand side, the first term stands for the mean relative growth of $S$ and the second term stands for the fluctuations about the mean growth. In an interval of time, $\Delta t$, the mean relative growth varies linearly with $\Delta t$, while the fluctuations vary as $(\Delta t)^{1/2}$. Thus, on large time scales, the relative growth of $S$ is dominated by the linear mean growth, against which the fluctuations become much weakened. Scaling the time in months, this is exactly what we see in Fig.4. Further to this point of view, Fig.5 demonstrates how the variance of the monthly prices, $\Sigma^{2}$, generally decreases with time, $\tau$. A quantitative measure of this decline is found in the slope, $w$, of the straight line, $\Sigma^{2}=w\tau$, fitted with the data in Fig.5 by the least-squares method. With $w<0$ for the NIFTY (NSE, India) index, as well as for all the other stock indices in Table 1, stability against volatility is seen to be a general tendency of mature stock markets.

In passing, we note that Fig.5 has a prominent spike in the monthly variance. This spike corresponds to the economic recession that disrupted global markets around the year, 2008. This disruption is universally captured by a very tall spike in the monthly variance of all the stock indices we have studied.

V The daily volume of trade

A reliable yardstick of the health of a stock market is the number of daily transactions in the market Queirós (2005, 2016). The linear-log plot in Fig.6 shows an exponential mean growth of the trade volume at $0.04$% per day (which is about $15$% per annum) for the NIFTY (NSE, India) index. Volatility about the mean growth is also clearly seen in Fig.6. Other global markets display the same behaviour, going by the relative growth rate of their daily transactions in Table 1. These observations qualitatively correspond to established views about how trading volume grows with time in financial markets Pan and Sinha (2008); Queirós (2005, 2016).

The growth trends in Figs.1, 4 & 6 look similar, although the first two relate to price fluctuations (one on a daily time scale and the other on a monthly time scale), while the last relates to the fluctuations of the daily transactions. This, however, is not surprising, because, for better or for worse, price fluctuations do affect speculation in a financial market, and this in turn has a bearing on its volume of trade (Queirós, 2016). The inter-dependency between the two motivates the suggestion that the trading volume can be a suitable measure of volatility in financial markets (see Queirós (2016) and relevant references therein).

VI Conclusions

The indices of the stock markets of six countries with high GDP show that markets experience long-term growth of stock values and the amount of trade. Mature markets tend to be stable against volatility. These qualitative views are quantitatively favoured by the numerical values we present in Table 1. The closeness of the values of each parameter in Table 1 satisfies our quest for regularities in stock markets.