Time averaging, ageing and delay analysis of financial time series

We introduce three strategies for the analysis of financial time series based on time averaged observables. These comprise the time averaged mean squared displacement (MSD) as well as the ageing and delay time methods for varying fractions of the financial time series. We explore these concepts via statistical analysis of historic time series for several Dow Jones Industrial indices for the period from the 1960s to 2015. Remarkably, we discover a simple universal law for the delay time averaged MSD. The observed features of the financial time series dynamics agree well with our analytical results for the time averaged measurables for geometric Brownian motion, underlying the famed Black–Scholes–Merton model. The concepts we promote here are shown to be useful for financial data analysis and enable one to unveil new universal features of stock market dynamics.


Introduction
In 1900, Bachelier pioneered the concept that prices on financial markets are stochastic and may follow the laws of Brownian motion [1,2]. Similar ideas for the mathematical description of option pricing were proposed by Bronzin in 1908 [3], a work which largely fell into oblivion until recently [4]. Later, in the 1960s, Mandelbrot [5,6] and Fama [7] realised that the Gaussian distribution of price changes is often violated, and introduced Paretian and Lévy stable laws into financial mathematics [5,8]. Another milestone in the stochastic modelling of financial markets is the famed Black-Scholes-Merton option pricing model [9][10][11], see also the classical textbooks on the mathematical analysis of markets [12][13][14].
Statistical models of stock price variations often assume that the increments [ ( )] X t log are Gaussiandistributed [14]. Extensive statistical analysis of financial data revealed, however, that the distribution of returns + [ ( ) ( )] X t t X t log d has a sharper maximum and fatter tails [8,12,19,34]. To account for these features, random walk models based on, inter alia, truncated Lévy stable distributions [5,8,12,36,49] and jumpdiffusion models [11,13,50] were proposed. Discrepancies between ensemble and time averaged measures were also discussed recently [18,38,51,52]. Different market impacts onto price formation were considered as well [53]-in particular, one should mention here also market microstructure effects [54]. Despite decades of intense Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. development and significant advances the quest for a universal mathematical model of stock price dynamics is still open.
Here we propose three concepts, complementary to the well established methods, for an advanced analysis of financial time series. Specifically, these are the time averaged mean squared displacement (MSD), perfectly suited for the analysis of a single time series, as well as the ageing and delay time methods. As we demonstrate from statistical analyses of real financial time series, these approaches are highly useful and reveal universal features of the market dynamics, which may be relevant for the further development of financial market models.
The central concept promoted here is based on time averaging. Most frequently, in theoretical approaches the MSD á ñ ( ) X t 2 , defined as the ensemble average of ( ) X t 2 over many realisations of the stochastic process X(t), is used to quantify the time evolution of the process. However, when dealing with a single or few but long time series X(t), the time averaged MSD [55] is better suitable and more relevant for the analysis of time series of both stationary and non-stationary stochastic processes [55,56]. For a given lag time Δ this quantity defines a sliding average over the entire time series, of length T. Hereafter, the overline denotes time averages and the angular brackets stand for averages over an ensemble of realisations of a process X(t). While the concept (1) is by now widely used for single-trajectory analysis in several areas of science, especially microscopic single particle tracking [55,56], it is less common in mathematical finance [17,18]. Our central focus here is to introduce this concept for the analysis of financial time series.

Analysis of financial time series
We here present the results of statistical analyses of financial time series based on the time averaged MSD (1) as well as the ageing and delay time methods defined below. The observed behaviour based on these analysis tools is demonstrated to agree well with analytical results for the famed GBM model, which is introduced in section 2.2. We study the daily values of multiple financial indices of different categories. All data were taken from and analysed via the Wolfram Mathematica 9 platform.
The data are categorised and abbreviated as follows. In the first category we study the Dow Jones Industrial Average   figure 1. To improve the presentation we divided all prices by the corresponding initial value, such that all traces start at unit value. This is a legitimate procedure as for a GBM-like process the initial price X 0 enters solely as a multiplicative factor (see below). Despite the common initial price, we observe large price fluctuations for different indices at later times, especially in the last decades. Roughly, an exponential increase is observed for all stocks. at short lag times increases with T for growing stock market prices, as long as no market crises occur, as seen in figure 2.
We observe a roughly linear growth d D µ D ( ) 2 at short lag times, in perfect accordance with the analytical result (6) for GBM derived below. This linear scaling stands in stark contrast to the exponential growth of the ensemble averaged MSD á ñ ( ) X t 2 of GBM seen in equation (5) below. Such a fundamental discrepancy between the time and ensemble averaged MSD of the process X(t) is well known in the theory of stochastic processes and often referred to as weak ergodicity breaking [55,56].   [55]. Note that the averaging over different trajectories in quantity (2) is necessary for our analytical calculations. In the analysis of the financial time series shown in the figures, single trajectory averages d D ( ) 2 are considered throughout.

Geometric Brownian motion
Before continuing with our analysis of the actual stock market data, we briefly digress and provide a primer on GBM, a paradigm process employed in standard models for stock price dynamics. As we show this model reproduces the essential features observed in the market data. GBM is defined in terms of the stochastic differential equation with multiplicative noise Here > ( ) X t 0 is the price at time t, μ denotes a drift, and σ is the volatility (μ and σ are set constant below). The volatility is connected to the square root of the diffusivity [55]. The increments dW of the Wiener process are defined as white Gaussian noise x ( ) t with zero mean. The price evolution, starting with the initial value > X 0 0 at t=0, is obtained from equation (3) by use of Itôʼs lemma [12,14] as This process satisfies the log-normal distribution, emerging also in models of task success and income distribution [57]. The MSD shows the exponential growth Due to this (much) faster growth compared to the linear increase of the MSD with t for Brownian motion, GBM is therefore a superdiffusive process [55].
To calculate the time averaged properties of interest, we resort to the average (2) over N independent realisations of X(t). We derive this quantity using the one and two point distribution functions of the Wiener process. For short lag times, D  T , and in the absence of drift we find As mentioned already, the non-equivalence of the ensemble and time averaged MSD, in the limit D  T 1 indicates weak ergodicity breaking, known for other non-stationary diffusion processes [55]. The time averaged MSD (6) scales linearly with the lag time Δ, in contrast to the exponential growth of á ñ ( ) X t 2 , but grows exponentially with the trace length T, due to the highly non-stationary character of GBM.  figure 2, and the explicit dependence on T shown in figure 3. Equation (6) predicts, to leading order, an exponential growth of the time-ensemble averaged MSD with the trace length T. As shown in figure 3, computed for D = 1 day, the data indeed roughly follow an exponential increase with T. However, no universal dependence of the time averaged MSD as function of T could be found. Especially for those time windows encompassing prolonged drops or stalling of the index price, the ratio of the time averaged MSD for a trace length T to its value at T min does not grow rapidly but rather saturates, as can also be seen in figure 3. Thus, checking the dependence of d D ( ) 2 on the trace length T may be used to unveil individual features of index price dynamics. In contrast to the strongly disparate and company-specific behaviour revealed in figure 3, universal features are found in the analysed index prices when employing the new concept of the delay time averaged MSD introduced below. Figure 4 shows the data normalised to their end point. More specifically, this normalisation enhances the contribution of later parts of the time series, with typically larger prices.
We find a considerable spread of individual d i n , 2 traces for different companies, because each index can have different parameters such as the volatility value or the attitude of a given company leadership to maximise the short-term profit versus ensuring a long-time sustainability. Finite lifetimes of companies and their varying age at the start of the time series are likewise important. These factors individualise stock price variations for each company and complicate the evaluation of ensemble averaged quantities. In view of this the universality of stock prices observed in section 2.5 is even more remarkable.

Ageing analysis
Physically, the above dependence (6) of the time averaged MSD on the trace length T reflects the phenomenon of ageing, a characteristic property of non-stationary stochastic processes [55,58]. For superdiffusive processes with an MSD growing faster than the linear growth in time of Brownian motion, the time averaged MSD exhibits an increase with T. This reflects the self-reinforcing volatility of the process, as seen for the exponentially fast growth in equation (6). In contrast, in subdiffusion the effective diffusivity of the process is a decaying function of time [55]. For instance, in processes with scale free waiting times the typical sojourn periods of the motion become increasingly long, on average, effecting the decay of the time averaged MSD with T [58].
Another way to analyse ageing processes is the following. If the time series X(t), starting at time t=0, is evaluated only beginning with the so-called ageing time > t 0 a , the ageing time averaged MSD is defined as [58] ò d D = -D + D - In this formal definition we shift the starting point for the analysis of the time series yet the length T of the analysed time interval remains fixed. Of course, larger values of t a limit the remaining number of data points  available for this analysis, however, the ageing time averaged MSD provides important insights into the underlying stochastic process [55]. We note that the term ageing does not imply any relaxation to an ergodic state in the limit of long ageing times, as it does for subdiffusive processes (the limit of strong ageing). In superdiffusive processes such as GBM and, as shown here, for highly non-stationary financial data the term ageing delineates the process time dependent increase of the spread of the random variable X(t).
For GBM we find in the limit of short lag times, D  T , and in absence of a drift, m = 0, that on average The exponential growth with t a emerges as the process X(t) already experienced an acceleration of the price dynamics up to the ageing time t a , and thus the analysis starts with a higher volume. Note that in result (8) the argument of the exponential includes the volatility term s 2 of the GBM process.
To see whether such ageing effects can indeed be observed in real financial data we study the behaviour of the ageing time averaged MSD d D increases fast with t a . Similar to the observations above, when we varied the trace length T, partial clustering of the traces for varying t a is visible. Figure 6 quantifies the behaviour of the logarithm of the ratio d d of the ageing time averaged MSD to the corresponding non-ageing value, plotted versus the ageing time t a . Although, again, a roughly exponential increase is evident and consistent with the prediction (8) for GBM, no data collapse onto a universal curve is observed. Even at short ageing times we find a substantial spread of the d d D D [ ( ) ( )] log 2 2 curves versus t a for different stock indices, as can be seen in figure 6. The non-universal behaviour is expected to be due to the fact that the volatility parameter varies between companies, as equation (8)   corresponding ensemble is not formed from trajectories with identical parameters. For instance, the volatility values and the effect of the ageing time t a for each stock market index can be markedly different.

Delay time analysis: revealing universal features
What if we allow the length of the time series to vary in the above ageing analysis? Namely, what would the expected behaviour be for the quantity The logarithm here cancels the leading exponential dependence of the delayed time averaged MSD on t d in equation (10) and the final result (11) is independent on the index-specific volatility parameter σ. We emphasise that in this result it is crucial that the process X(t) has non-stationary increments. The behaviour of, say, a standard Brownian process would follow a different scaling law with the delay time. Can this universal behaviour predicted in equation (11) Figure 8 shows the logarithm of the ratio of the delay time averaged MSD to the standard time averaged MSD, evaluated at unit lag time D = 1, as function of the delay time t d . The universal behaviour (11) expected on average is followed very closely for each stock market time series. This universal behaviour-fulfilled for delay times t d up to some 5-10 years-is the central result of this study. To our best knowledge, in terms of single time series this universal trend has not been reported before. .
In particular, this disqualifies the predictions of GBM in this long delay time regime. Note that since the time averaged MSD initially grows approximately linearly with Δ, as one can see figures 2 and 7, the trends of figure 8 will also hold for other sufficiently short lag times.
Note that for some high tech companies and banks we did not observe a growth of d d

Conclusions
Time averaging of observables of a stochastic process X(t) is a successful concept designed for the analysis of single or few, sufficiently long time series. It has been applied in various fields, in particular, in single particle tracking studies of microscopic objects [55,56]. While for such microscopic particles, at least in principle, it is possible to record more than one trace under (almost) identical conditions, the situation is much more restricted for financial contexts. The market price evolution of a given company cannot be repeated several times under identical conditions. Thus, a statistical ensemble for averaging over a set of trajectories is inaccessible [17,38]. Splitting up the time series into subparts is not an option due to the highly non-stationary character of the dynamics. The analysis in terms of time averaged observables is therefore the prime option.
Here we demonstrated that time averages indeed provide a useful toolbox for the analysis of financial data. Using the time averaged MSD, as well as the ageing and delay time methods, we showed that relevant features can be extracted from the analysis of financial time series. Good agreement of our data-driven observations with analytical predictions from the GBM model was observed, such as the linear lag time dependence of the time averaged MSD, contrasting the exponential growth for the ensemble averaged MSD. The ageing analysis combining the dependencies on the trace length T and the ageing time t a unveil peculiar features in a given time series such as prolonged stalling or even a decrease of the stock value.
Remarkably, the delay time analysis introduced here uncovered a universal behaviour for the analysed stock prices. For short and intermediate delay times t d , the logarithm of the ratio of the delay time versus the regular time averaged MSD is a linear function of t T d . At longer times, in our analysis beyond some 5-10 years, this logarithm approximately scales as a power-law n ( ) t T d with another scaling exponent, n > 1. This latter scaling behaviour is not captured with the standard GBM model, pointing at a need for improved theoretical approaches.
This study is to be viewed as a first step in applying time averaging, ageing, and delay time methods to the analysis of financial time series. Theoretically, modifications of the GBM model used here, to account for features such as the transition from the universal linear scaling to the scaling law n ( ) t T d for the delay time averaged MSD as well as the introduction of fluctuating or time-varying volatilities, are possible. The concept of 'diffusing diffusivities', which in some sense is similar to fluctuating volatilities, has been recently established in the physics literature [59][60][61]. How such concepts impact stochastic processes with multiplicative noise remains to be clarified, however, we expect a similarly rich behaviour with crossovers as observed for simple Brownian systems [59][60][61].
Deterministic time dependent volatilities may be adopted for the time averaging based description of unstable markets (at times of a financial crash), when the trading conditions change very rapidly [38]. Here, a stochastic process with a power-law volatility may be proposed: GBM with a volatility increasing with time can account for a faster than exponential price growth X(t) and explain a faster than linear trend (12) detected in the analysis of financial time series. Recently, ensemble averages of a similar modified GBM process with power-law and logarithmic volatilities were presented [53]. Also, models with value-and time-dependent diffusivity were empirically found to underlie the Euro-Dollar exchange rate dynamics [19,62], compared to anomalous diffusion with space-and time-dependent diffusivity [55]. Combining such new theoretical approaches with time averaging may provide vital new impetus in the analysis of financial time series. From a data analysis point of view, we were interested in the long-term trends for the time averaged MSD. Clearly, time series with one point a day hide possible intraday effects, such as intraday volatility patterns extracted from highfrequency data [18,38,63]. More observables from the time series should be taken into account, and the correlations between them remain to be rationalised. These include the auto-correlation function of price increments, the evaluation of volatility values [33] over different periods of time, the trading activity and volumes on the markets, a correction of the index price value due to inflation [34], crises, etc. These points, as well as the question how the dynamics observed herein is connected with heavy tailed spreads of financial volume [5,8,12,36,49] will be the focus of future work.
The area of mathematical finance is not the only domain where our time averaged MSD and ageing approaches may be useful. For instance, from a biological perspective the mathematical description of inherently highly stochastic disease outbreaks involves exponential processes similar to GBM. In epidemic spreading, an exponential increase in the number of diseased hosts is often postulated (up to the system size). The reader is referred to the optimal control models in epidemics spreading [64,65], including density dependent growth and ageing. Finally, mathematical models of tumour spreading and the growth of bacterial colonies and cells [66] also employ exponential processes, providing additional ground for the application of the concepts outlined here.