Transition from lognormal to χ2-superstatistics for financial time series

doi:10.1016/j.physa.2016.02.057

Physica A: Statistical Mechanics and its Applications

Volume 453, 1 July 2016, Pages 173-183

https://doi.org/10.1016/j.physa.2016.02.057 Get rights and content

Highlights

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For the first time a transition from one superstatistics to another is described, as a function of the time scale considered.
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Relevant example system is financial time series of share price returns on various time scales, good quantitative agreement with data.
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New model interpolating between lognormal and chi-square superstatistics is introduced.

Abstract

Share price returns on different time scales can be well modelled by a superstatistical dynamics. Here we provide an investigation which type of superstatistics is most suitable to properly describe share price dynamics on various time scales. It is shown that while $χ^{2}$ -superstatistics works well on a time scale of days, on a much smaller time scale of minutes the price changes are better described by lognormal superstatistics. The system dynamics thus exhibits a transition from lognormal to $χ^{2}$ superstatistics as a function of time scale. We discuss a more general model interpolating between both statistics which fits the observed data very well. We also present results on correlation functions of the extracted superstatistical volatility parameter, which exhibits exponential decay for returns on large time scales, whereas for returns on small time scales there are long-range correlations and power-law decay.

Introduction

Many well established concepts in mathematical finance (such as the Black–Scholes model) are based on the assumption that an index or a stock price follows a geometric Brownian motion, and as consequence the log returns of these processes are Gaussian distributed. But nowadays it is well known that the log returns of realistic stock prices are typically non-Gaussian with fat tails [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. Such behaviour can be well captured by superstatistical models [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. The basic idea of this method borrowed from nonequilibrium statistical mechanics is to regard the time series as a superposition of local Gaussian processes weighted with a process of a slowly changing variance parameter, often called $β$ . This approach has been applied to many areas of complex systems research, including turbulence, high energy scattering processes, heterogeneous nonequilibrium systems, and econophysics (see e.g. Ref. [11] for a short review). In finance early applications of the superstatistics concept were worked out by Duarte Queiros et al. [6], [7] and Ausloos et al. [5]. Van der Straeten and Beck [3] analysed daily closing prices of the Dow Jones Industrial Average index (DJI) and the SP 500 index. They verified that both log-normal superstatistics and $χ^{2}$ superstatistics result in good approximations. Biro and Rosenfeld [4] also studied the data sets of the Dow Jones index and verified that the distribution of log returns is well fitted by a Tsallis distribution. Katz and Li Tian [1] showed that the probability distributions of daily leverage returns of 520 North American industrial companies during the 2006–2012 financial crisis comply with the $q$ -Gaussian distribution which can be generated by $χ^{2}$ superstatistics. They also verified in Ref. [2] that the Tsallis entropic parameter $q$ obtained by direct fitting to $q$ -Gaussians coincides with the $q$ obtained from the shape parameters of the $χ^{2}$ distribution fitted to the histogram of the volatility of the returns. Gerig, Vicente and Fuentes [8] consider a similar model that indicates that the volatility of intraday returns is well described by the $χ^{2}$ distribution, see also Ref. [9] for related work in this direction.

In this paper, we will carefully analyse for various data sets of historical share prices which type of superstatistics is best suited to model the dynamics. While Tsallis statistics ( $= q$ -statistics) is known to be equivalent to $χ^{2}$ superstatistics [13], [22], there are other types of superstatistics, such as lognormal superstatistics and inverse $χ^{2}$ superstatistics [10], which are known to be different from $q$ -statistics (though all these different statistics generate similar distributions if the variance of the fluctuations in $β$ is small [13]). We show that in our analysis $χ^{2}$ -superstatistics appears best suitable to describe the daily price changes, whereas on much smaller time scales of minutes lognormal superstatistics seems preferable. We analyse the relevant time scale of the changes in the superstatistical parameter $β$ and present results for the decay of correlations in $β$ . For small return time scales, correlation functions exhibit power law decay and there are long memory effects. In the final section, we develop a synthetic stochastic model that fits the data well. This is kind of a hybrid model interpolating between lognormal and $χ^{2}$ -superstatistics.

This paper is organized as follows. In Section 2 we look at share price returns on large (daily) time scales. In Section 3 we do a similar analysis on small (minute) time scales. In Section 4 we investigate correlations of the superstatistical volatility parameter on both time scales. In Section 5 the hybrid model is introduced. Our final concluding remarks are given in Section 6.

Section snippets

Superstatistics of log-returns of share prices on a large time scale

Non-equilibrium system dynamics can often be regarded as a superposition of a local equilibrium dynamics and a slowly fluctuating process of some variance variable $β$ [13]. These types of ‘superstatistical’ nonequilibrium models are also useful for financial time series [6], [7]. In this article, the empirical data we use as an example is the historical stock prices of Alcoa Inc(AA), which is an American company that engages in the production and management of primary aluminium, fabricated

Short time scales

Let us extend our analysis to returns on much smaller time scales. A change of statistics as a function of the time scale considered is a common phenomenon for many complex systems, see e.g. Refs. [24], [25] for work in this direction. Hence it is interesting to also consider return data on much smaller time scales (say, minutes), and see what is similar and what is different as compared to the analysis of the previous section. Let $s_{i}$ be the stock price for every recorded minute, in our

Correlation functions

For the development of a suitable dynamical model, it is very important to look not only at probability densities but also on correlation functions and memory effects [16], [17], [18], [19], [20], [21]. In our case there are two types of correlation functions: the one of the original data $u_{i}$ , $C_{u} (t) = \frac{1}{N - t} \sum_{i = 1}^{N - t} u_{i} u_{i + t} - {〈 u_{i} 〉}^{2}$ and those of the volatility parameter $β_{k}$ , $C_{β} (t) = \frac{1}{n - t} \sum_{k = 1}^{n - t} β_{k} β_{k + t} - {〈 β_{k} 〉}^{2} .$ Fig. 9, Fig. 10, Fig. 11, Fig. 12 show $C_{u} (t) / C_{u} (0)$ and $C_{β} (t) / C_{β} (0)$ , both for the daily returns as well

Synthetic model

Based on the results of the previous sections, it is desirable to construct a simple superstatistical dynamical model that incorporates the possibility of both lognormal and $χ^{2}$ superstatistics on different scales, and allows for different decay patterns of correlation functions.

Here we propose the following model. We start from a linear superstatistical Langevin equation $\dot{u} = - γ u + σ L (t)$ where $L (t)$ is Gaussian white noise and the ‘inverse temperature’ $β$ , in accordance with Einstein’s theory of

Conclusion

Many investigations of complex systems in the past have focused on the application of a particular statistics, for example $q$ -statistics [22], and then studying the effect of varying system parameters, which may change the entropic index $q$ . Here we have shown that for financial time series it is sometimes useful to consider broader classes of statistics and even proceed from one class of superstatistics to another when the scale or other system parameters under consideration is changed. The

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View full text

Transition from lognormal to χ2-superstatistics for financial time series

Highlights

Abstract

Introduction

Section snippets

Superstatistics of log-returns of share prices on a large time scale

Short time scales

Correlation functions

Synthetic model

Conclusion

Physica A

Physica A

Physica A

Physica A

J. Empir. Finance

Physica A

Res. Int. Bus. Finance

Physica A

Superstatistical fluctuations in time series: Applications to share-price dynamics and turbulence

Phys. Rev. E

Dynamical model and nonextensive statistical mechanics of a market index on large time windows

Phys. Rev. E

On the emergence of a generalized Gamma distribution. Application to traded volume in financial markets

Europhys. Lett.

On the connection between financial processes with stochastic volatility and nonextensive statistical mechanics

Eur. Phys. J. B

Model for non-Gaussian intraday stock returns

Phys. Rev. E

Transition from lognormal to $χ^{2}$ -superstatistics for financial time series