Econophysics

Matthias Hirtschulz

Supervisor: Dr M J Godfrey

Literature Study Essay

Abstract: The aim of this essay is to present the main findings of econophysics. (i) We describe the macroscopic statistical properties of financial markets like heavy tailed probability distributions and try to find physical systems obeying the same statistics. (ii) We introduce the Cont-Bouchaud model. This model is able to explain heavy tailed probability distributions through a herding or group behaviour mechanism.

1.Introduction

In recent years a new science called "Econophysics" has become more and more popular¹. It is one of many other new areas of study trying to apply physical methods to all kinds of disciplines. "Sociophysics", for example, is a branch of knowledge which tries to understand sociological problems from a physicists point of view [19, 20].

Econophysics, however, tries to establish contact between physics and finance in order to contribute "to a better understanding and modelling of financial markets"[15]. Most of the methods used in econophysics are statistical methods, also used in statistical physics, but there are even papers which use quantum mechanics to model financial markets [22] or apply thermodynamics to utility theory [24]. We will focus on the statistical description of financial markets here.

This very young area of study has two main approaches. The first is to understand the system we are examining, e.g. the time evolution of a market index, on a macroscopic level. That means we are using a phenomenological and experimental point of view. One observes the financial data, produced every day on different financial markets all over the world, and tries to find regular patterns or certain laws regarding the statistics of the financial data. A sound knowledge of the statistics of financial markets is important for derivate pricing, a valuation of risk of a financial portfolio and for controlling the results of a model [15].

The second approach is to attain an understanding of the system on a microscopic level. In contrast to the former case here the aim is to understand the system from the bottom to the top and find the rules that govern the interaction between the market actors or agents and from this to derive the statistics on the macroscopic level. If we find the right model it should predict the experimentally observed statistics.

The procedure described is often used in physics, e.g. in thermodynamics. In thermodynamics there is an equation derived from experimentation - the ideal gas law - , which describes the relation between macroscopic variables like pressure, volume and temperature of a gas at low density.

But this equation can also be derived from kinetic gas theory. This theory makes some assumptions about the behaviour of the particles comprising a gas.

Not only is the scientific procedure the same; there are also qualitative similarities between the systems observed in finance and thermodynamics. In both systems there are many investors or particles. The behaviour of the single agent cannot be predicted, nor can the motion of one particle in a gas, but as there are so many particles or investors respectively both systems show specific rules on a macroscopic level, because a large number averages out the individuality. Thus it is a priori possible - in spite of the big differences between human individuals - that we can find regular patterns on a macroscopic level.

This essay is organized (following the above described structure) as follows: we first present the empirical properties of some macroscopic quantities in the second chapter and then present a microscopic model of a stock market in the third chapter.

2. Empirical Statistics and Macroscopic Market Models

In this chapter we will describe the statistics of financial time series. We will consider questions about the distribution of price changes, the way these distributions change over time and look at the correlations in financial data. In this sense we will take a physicist's point of view as we observe the financial markets and compare the data to a proposed model. Therefore we can refute many models that are a priori possible [26].

The system we are examining, a market, is, for an economist, the coincidence of supply and demand. The markets we will deal with when talking about financial markets are normally stock exchanges. On these markets trade in securities (e.g. stocks, bonds or currencies) takes place every day. This produces a large amount of data, which is the basis of every experimental analysis. In this chapter we shall also have a look at the statistics of Market Indices, which describe the composite evolution of a group of selected securities. The DAX, for example, constitutes of the main 30 German companies.

2.1 Brownian motion: A macroscopic model for financial time series?

At the beginning of the 20^th century Einstein derived the diffusion equation. This equation describes the motion of suspended particles in a solvent - the so-called Brownian motion. The motion of the suspended particles is caused by particles of the solvent that are assumed to hit the suspended particles randomly in terms of direction and strength. Furthermore, it is assumed that a one-dimensional description is sufficient. These assumptions lead to the following differential equation:

where is the number of particles between x and and is the probability distribution for a particle moving the distance during the time interval .

In the case that all suspended particles are located at the origin at time t = 0 one finds that the solution of the diffusion equation is a Gaussian:

where n is the number of suspended particles. This special initial condition can be seen as analogous to the value of e.g. a stock at a time t, while f would be the probability distribution of the price changes.

We now want to compare this result to the "motion" of financial data. Is the probability distribution function (pdf) underlying the changes in financial data a Gaussian distribution? Therefore we examine the statistical properties of financial data now.

The similarity between the plot generated with a random walk and the DAX attracts attention (figure 1). One could think at first glance that there are similarities in the statistics of these datasets. In fact it would be difficult in the absence of information to say, which of the two plots is real financial data.

If, however, one wants to make a quantitative analysis and not just a rather qualitative and superficial examination, one should think about which quantities are appropriate for measuring the time (physical time, trading time or number of transactions) and price of a financial instrument (e.g. price changes or log-price changes) [18].

One often uses the log-price changes:

which has certain advantages and disadvantages we will not discuss here (see chapter 5 in [18] for more details). To examine the shape and the time evolution of the pdf, we use the daily returns of the DAX index during the period from 02.01.1975 to 05.05.2000. We divide this data into 6 equal long periods. The probability distribution of the normalized returns (as defined in ) in these six periods is shown in figure 2.

[Fig.3.7 Voit]

Fig. 1 Computer simulation of price charts as a random walk (upper panel) and comparison to the evolution of the DAX share index from January 1975 to May 1977 (lower panel). Adapted from [28].

[Fig. 5.2 in voit]

Fig. 2 Probability distributions of normalized daily returns of the DAX index in six equally long periods from 1975 to 2000. Adapted from [28].

The six distributions in figure 2 seem to be quite similar. However in the wings of the pdf there are differences. In some periods "extreme" price changes happened more often than in others. That is why one should be quite careful to analyse data over very long time scales. Figure 2 also suggests that the probability distribution does not follow a Gaussian distribution (at least in the tails). In fact there are many other investigations that show that empirical distributions are more leptokurtic. A leptokurtic distribution is characterized by a higher and narrower peak and fatter tails [18]. Therefore a Gaussian distribution cannot be the final answer and the Brownian Motion Model is refuted.

2.2 It is not a Gaussian - What else?

There have been early tests by Bachelier of the above-discussed log-normal (i.e. the log price changes follow a Gaussian distribution) statistics, but they showed a good consistency with the model of Brownian motion. This could be due to the small data sets used [28]. In 1963 Mandelbrot postulated a different distribution [13]. He observed prices of a commodity (cotton) and found data resembled a so-called stable Lévy distribution².

This distribution has fatter tails then the Gaussian distribution and is, therefore, often useful to describe multiscale phenomena where very large and small values appear [3]. In fact there are different types of stable or stable Lévy distributions characterized by their Fourier transform:

for it is a Gaussian distribution [8]. is the so-called skewness parameter (for it is a symmetric distribution).

Another property of the Lévy distributions is the stability. If you are examining statistical data you can look at them at different time scales. Hence one observes e.g. the returns of a stock on basis of minutes, hours or days. In general the pdfs corresponding to different time scales are not the same. But if the fluctuations are independent and identically distributed (iid) and have a stable distribution on a short time scale, the fluctuations on other time scales have the same distribution (more generally speaking: the sum of two iid Lévy distributed variables again follows a Lévy distribution). This stability property implies the scale invariance of the stochastic process:

Fig 5.8 voit

Fig. 3 Probability distribution of changes of the S&P 500 index. Top panel: rescaled changes of the S&P 500 index. in the figure are the price changes . Bottom panel: comparison of the data with Gaussian and stable Lévy distributions. Adapted from [28].

In other words: the probability distribution on a scale is the same as the distribution on a scale if one rescales the distribution as defined in . One also finds that all stable distributions (except the Gaussian distribution) have tails following a power-law (therefore infinite variance and fat tails):

In this context one should mention the Central Limit Theorem (CLT). It states that the probability distribution of a sum of a large number of iid variables converges towards one of the stable laws.

Apart from Mandelbrot there have been various other attempts to find Lévy distributions in financial data, e.g. in [14]. In [16] Mantegna and Stanley investigated the scaling behaviour of the American S&P500 index. If the data follows a stable Lévy distribution, different timescales should fit on the same curve just by rescaling them. As shown in figure 3 the data collapses on a single curve. This suggests "that the same mechanism operates at all time scales, and that there is a single universal distribution function characterizing it" [28].

The bottom graph in figure 3 compares the data for the timescale of one minute with a Gaussian distribution and a Lévy distribution. It shows again that a Gaussian description is not very good, while a Lévy distribution fits much better, though for large fluctuations the Lévy distribution does not fit either. The Lévy distribution predicts too many large events, which is why the idea of a truncated Lévy distribution (TLD) came about. This distribution avoids too many large events, has a finite variance (therefore converges to a Gaussian distribution for long time-scales, which is observed) and shows scaling on short time scales. The TLD describes the price-change distribution quite well but "fails to describe in a proper way the time-dependent variance observed in market data"[18](p.73) (thus the random variables are not iid and the parameters of the TLD change in time). That is why we will have a look at stationarity and correlation in financial time series in the next chapter.

2.3 Stationarity and time correlation³

A strict definition of stationarity goes as follows: A stochastic process is called stationary if its pdf is invariant under a time shift. There are, however, various other less restrictive definitions of stationarity. For example the so-called "Asymptotically stationary stochastic process" is characterized by the property that the statistics for does not depend on c if c is large. Figure 4 shows the time evolution of the volatility of price changes of the S&P 500 index.

From figure 4 it is obvious that the pdf`s volatility is strong time dependent. This fact leads to the development of autoregressive conditional heteroskedasticity (ARCH) and generalized ARCH (GARCH) processes, which we discuss in the next chapter. For independent price changes one would expect that the time evolution of the standard deviation follows

with an exponent [18]. Empirical studies often show a slightly higher value. This suggests a weak long-range correlation in volatility. To measure correlation between data one uses the autocorrelation function defined in .

An autocorrelation is a correlation of a time series with itself. That means there is only one series involved. If the value of the autocorrelation function is close to zero, we can conclude that adjacent price changes are not correlated (for distributions with zero mean). But if the

[Fig. 4.2 in introduction shows time dependent volatility]

Fig. 4 Monthly volatility of the S&P 500 index measured for the 13-year period January 1984 to December 1996. Adapted from [18].

autocorrelation function of the observed process becomes large the independence of price changes would be in doubt. The autocorrelation function is defined as:

where is the joint probability to measure at time and at time . For stochastic processes with a mean value, which is different from zero, one could use the autocovariance:

In figure 5 we see the autocorrelation function of the S&P 500 index sampled at a 1 min time scale. It shows that after about 20 min the correlations in the data vanishes.

[Fig.7.3 in introduction]

Fig. 5 Semi-log plot of the autocorrelation function for the S&P 500 index, sampled at a 1 min time scale. The straight line corresponds to exponential decay with a characteristic decay time of . Adapted from [18].

That means on the one hand we have short-range correlations in price-changes: one could say that the price changes are pair wise-independent as a good approximation on time scales of more than a few trading minutes. But on the other hand, some empirical observations show long time correlations in volatility of price changes [12] with a power law decay. The probability distribution of volatility seems to be log-normal [12] near the centre while the asymptotic behaviour follows a power law.

2.4 ARCH/GARCH processes

As we have seen the volatility of log-price changes is not constant. This has let to the development of ARCH- and GARCH-processes. They have, as the empirical data, time dependent variances. The variance at a certain instant depends on some of the past values of the random variable. The variance of an ARCH(j) process is defined as:

The random variables x are taken from a conditional pdf with zero mean. The index j controls the time memory of the process. In a GARCH(j,i) process the variance is also dependent on past values of the variance:

These processes are widely used in finance and there is a huge amount of literature about it. The ARCH-/GARCH-processes with Gaussian pdf describe the financial data quite well, which is shown in figure 6. Although the underlying pdf is a Gaussian the overall probability distribution shows fat tails due to the conditional variance.

[Fig.10.6 in Stanley introduction to?.]

Fig. 6 Comparison of the empirical pdf measured from high-frequency S&P 500 data with with the pdf of a GARCH(1,1) process characterized by and . Adapted from [18].

However ARCH/GARCH models do not explain why one should use a specific function as the underlying pdf. Furthermore these models fail to describe the scaling properties of financial data. A given GARCH(1,1) process with a certain set of parameters, which describes the data well on a one-minute time scale, fails to describe the data on a different time scale. In fact the scaling properties of a simple GARCH(1,1) process have not been found yet (no theoretical answer, only simulations). They also cannot fully describe the kurtosis⁴ of returns [1]. Furthermore there are some questions not yet answered about this type of processes [18](p.87).

2.5 Hydrodynamic Turbulence

In this chapter we want to examine the analogies and differences between the turbulent flow of a liquid and the variations in financial data. On a qualitative level financial markets and turbulence have a tempting similarity. In turbulence the so-called cascade idea was quite successful. From it Kolmogorov and Obhukov derived some of the very few exact results⁵ for the Navier-Stokes equations , which are thought to describe turbulence [28]. They stated that when energy is injected in big eddies that they break up into smaller eddies until the energy is dissipated on the smallest scale. The analogy to energy in turbulence is said to be information in financial markets [9]. Müller et al. [21] found that there are two different types of traders. The long-term traders evaluate the market on a low frequency and have a long memory. The short-term traders watch the market continuously, re-evaluate the situation and execute transactions more often but they have a shorter memory. That is why they cause different types of volatility. On a short time scale the volatility is dominated by the actions of the short-time traders, while the long-term traders cause the volatility on a long time grid. Furthermore they found, that the information flow between these two different classes of traders is asymmetric. The short-term traders react on peaks of the long-term volatility with increased trading activity, but the long-term traders often ignore the fluctuations on short time scales. This net information flow from long to short time scales is said to be the information cascade in financial markets. The claimed analogy between different quantities in turbulence and financial markets is summarized in table 1.

Whether this analogy can hold on a quantitative level we will see in this chapter. But first we have a glance at the physics of turbulence.

The control parameter, which governs the behaviour of a liquid, is the dimensionless Reynolds number R:

Where L is a typical length scale of the observed system, u the velocity of the liquid and ( positive constant, density) the kinematic viscosity. R is a measure for the complexity of the flowing fluid. For small R the motion of the fluid is laminar. When R becomes bigger the flow turns into a turbulent flow.

The Navier-Stokes equations, describing the motion of an incompressible fluid are given by:

denotes the velocity of the fluid at position and time t and P is the pressure. Each term of this equation can be interpreted: The left hand side is the total derivate of the velocity The acceleration consists of two parts, the change in speed of a particle and the change in speed due to particles moving into a small volume element. The right hand side is a force. One part is spatial change in pressure the other is friction. Nevertheless, it is impossible to derive analytical solutions for this equation. Also the numerical calculation becomes more difficult when R grows. Therefore experimental studies are sometimes used to learn about the behaviour of turbulent fluids.

For a steady flow (velocity of the main stream is constant) in an uncompressible fluid with densityit is easy to see why these three parameters () govern the behaviour of the fluid. For the motion of the fluid is restricted through the equations of motion, the boundary conditions (i.e. the geometry of the observed system and the velocity of the main stream) and the parameter . If the shape of the geometry is given⁶ (e.g. a rectangle with side ratio L:K) it is sufficient to have one of the dimensions in order to calculate all dimensions. The velocity is therefore having the form [10]. That means that the quantity is the same function dependent on for all geometrical identical systems with same Reynolds number. Flows that transform each other by simply changing the units of velocity and position are similar. "Thus flows of the same type with the same Reynolds number are similar"[10].

If we think about a Reynolds number of a given eddy (thus containing the following quantities: order of magnitude of the size of the eddy, order of magnitude of velocity in the eddy and viscosity of the fluid), we realize that a big Reynolds number implies a small viscosity. Thus the energy dissipation in big eddies is appreciable. The viscosity becomes important for length scales where . On this length scales kinetic energy is turned into heat. Consequently it is necessary to supply the system with energy (injected into the large eddies, which carry the biggest amount of kinetic energy) in order to maintain turbulence [10]. This is the reason for the energy cascade (in fully developed turbulence) described at the beginning of this chapter. This cascade generates a scaling of the moments of :

where is the velocity difference at two points in a turbulent fluid [9] in direction of . Using the analogy to financial markets we should expect the following scaling behaviour for the price changes with the time scale :

Scaling of the second moment of the price changes of S&P 500 index has been observed by R.N. Mantegna and H.E. Stanley [16]. While S. Ghashghaie and et al. [9] report scaling of the moments of price changes in a foreign exchange market. The latter authors show that the moments (up to 6^th order) scale on time scales from about five minutes up to several hours.

As an example we compare the statistics of the S&P 500 index during 1984 to 1989 and the wind velocity in the atmospheric surface layer⁷. In figure 7 you see the S&P500 chart with hourly variations.

[Fig. 11.1 in introduction?]

Fig. 7 (a) Time evolution of the S&P 500, sampled with a time resolution , over the period January 1984 to December 1989. (b) Hourly variations of the S&P 500 index in the 6-year period January 1984 to December 1989. Adapted from [18].

[Fig. 11.2 in introduction?]

Fig. 8 Time evolution of the fluid velocity in fully developed turbulence. (a) Time evolution of the wind velocity recorded in the atmosphere at extremely high Reynolds number. The time units are given in arbitrary units. (b) Velocity differences of the time series given in (a). Adapted from [18].

In figure 8 you see the wind velocity over time and the velocity differences. Both distributions seem to be quite leptokurtic, but there are obvious differences. A layman could see at first sight that these figures do not have the same origin. The time evolution of the standard deviation is also completely different (compare chapter 2.3). While the index changes are if at

all only weak correlated (), the velocity changes even show an anticorrelation with an exponent (compare figure 9).

[Fig. 11.3 in introduction?]

Fig. 9 (a) Standard deviation of the probability distribution P(Z) characterizing the increments of the S&P 500 time series plotted double logarithmically as a function of . (b) Standard deviation of the probability distribution characterizing the velocity increments (shown in fig. 8) plotted double logarithmically as a function of . Adapted from [18]

In spite of these huge differences in the statistical properties, there is some hope that the analogy can be saved. R.N. Mantegna and H.E. Stanley suggest that the statistical problems arising in this analogy can be solved by using an abstract 2.05 dimensional turbulence [17]. In my opinion, this seems to be quite speculative at the moment.

3.A Microscopic Market Model

In chapter two we characterized financial markets on a macroscopic scale and in a phenomenological manner. We did not explain the origin of the macrovariables like e.g. volatility. In this chapter we discuss one attempt to connect the microscopic structure to the macrovariables.

There are many different models⁸ to choose from. For a review of some of these models see Ref. [2, 11]. Many of these models are very complicated. One consequence is that it becomes difficult to interpret the parameters and to assess their importance. Furthermore, this fact reduces the explanatory power of the models and makes it sometimes impossible to do numerical calculations [7]. For this reason we present a model in this chapter, which allows to interpret its parameters and to predict the characteristics of pdfs found in chapter two.

The essential point of the so called Cont-Bouchaud-Model (CB-Model) [7] is group behaviour of agents (there are many other papers which extend the CB-Model [25] or do further research [4, 5] about this model). It is quite intuitive to assume that market actors interact and communicate with each other, form groups of different sizes and align their strategies. One example of such a group is represented by e.g. a fund manager, who decides to buy or sell for all of the people engaged in his investment fund. Herding among security analysts [27] is another mechanism that leads to group behaviour. Experimental observations suggest that herding behaviour and imitation are crucial and lead to large fluctuations and therefore fat-tailed pdfs [23] - a high willingness of traders to form groups or to imitate other traders can lead to avalanches of common activity.

In the CB-Model N agents labelled trade a single asset. One considers the statistics of this asset x with the price at time t. The demand of a given agent is a random variable with three possible values:

No model is provided for the process by which agent decides to sell, buy or remain inactive. The demand is modelled as a variable supposed to be random as a result of e.g. heterogeneous preferences. For simplicity agent is assumed to trade only one unit during a certain time period. The total demand (or aggregate excess demand) for x therefore is:

Furthermore, is assumed to have a symmetrical probability distribution:

Consequently the average aggregate excess demand is zero. But how can we measure the aggregate excess demand, which is "not an easily observable quantity"[7]? It is important to be able to answer this question; otherwise we could not compare the predictions of the model with reality, because most of the papers, studying the statistical properties of financial data, use price changes and not the excess demand. Furthermore, it is a common belief that with positive (negative) aggregate excess demand the prices rise (fall). Therefore one simple assumption is to connect the price changes with the aggregate excess demand via a linear relation with a constant , which is called the liquidity or market depth:

At this point the model is restricted to predict price fluctuations on short time scales (e.g. intra-day fluctuation) only, as it neglects other economic and exogenous factors (such as interest rates), which can influence the price evolution on a longer timescale. Second, a linear relation is only justified through a Taylor expansion when the aggregate excess demand is quite small.

The probability distribution of and of is in the case of iid and large due to the CLT a stable law. As the experimental studies of financial time series show that the pdf is not normally distributed, the independent agent hypothesis does not describe financial markets very well. Accordingly, group behaviour is incorporated in this model. In the CB-Model the agents form groups or clusters of different size, within which no trading takes place. The group size itself is a stochastic variable. A given group has a common strategy to buy, sell or stay out of the market. Therefore is given by:

where is the size of cluster and their common strategy. is the number of clusters. The groups are formed by establishing binary links between agents, which means that they have the same strategy. The probability that agent andare linked (undertake the same action in the market) is . For simplicity, all links are assumed to be equally probable:

Therefore the average number of links one agent has is . But should converge to a finite number for . That is the reason for the following definition of p:

where the constant c can be interpreted as the willingness of the agents to align their actions. The theory of random graphs can be used to show that for the average coalition size is of order and the average number of clusters is of order . Furthermore one can show that if is not too large the probability distribution of follows:

This model therefore leads to a value of , which is very close to the experimentally found values in chapter 3.3. That means the predictions of the CB-Model are quantitatively comparable to experimental data. This is due to the simple herding mechanism we introduced here. The kurtosis of the asset return takes the form:

where is a normalization constant with a value closed to 1 and is the average number of traders, who do not remain inactive during a given period (also called order flow). Formula allows two interesting interpretations:

1. With a smaller order flow the kurtosis becomes bigger and therefore the fluctuations increase. This explains the well-known fact that the fluctuations on less active markets are often bigger.

2. Equation provides a relation between the willingness of the traders to form groups and the fatness of the tails of the price fluctuations.

Unfortunately this model is not perfect. In fact it has some serious drawbacks:

1. The parameter c we introduced has to be smaller and close to 1 in order to avoid centralized structures and trivial statistics [2]. The model does not explain the dynamics of c.

2. The model does not explain the time evolution of clusters. Therefore it cannot explain the phenomenon of volatility clustering (large price movements are often followed by another large movement not always in the same direction) [2].

3. Due to observed cross-correlations between different stock (we did not discuss this aspect here) a model which takes only a single stock into account should not be able to explain all aspects of a financial market [15].

4. Summary and Conclusion

In chapter 2 we have seen that there are some regularities in the statistics of financial data:

• truncated Lévy distributions and self similarity on short timescales

• pdf will approach a Gaussian for long timescales

• autocorrelations only up to a few minutes

• weak long-term correlation of volatility

Surprisingly, most of the financial data follows these specific statistical properties, which are independent of the mechanism of "price" fixing (we observed very different markets). This suggests that there are some "universal" properties of the observed markets, which lead to the specific statistical behaviour. For all our recent efforts, a convincing macroscopic model which describes all different aspects correctly (including the time evolution of the pdf) has not yet been found. These difficulties could be due to the fact that traders have a time memory - in contrast to particles - and therefore change their strategies over time. The conditions under which trading takes place do also change over time and most of them are impossible to predict (think of wars for example). This is a difference between finance and physics, where laws do not change with time [6]. Consequently, predicting the future evolution of stock prices could be very difficult on longer time scales!

In chapter 3 we have shown that a simple herding mechanism leads to a model that can predict some of the observed statistics (a truncated power-law tail). But we have also explained that this herding model has some drawbacks. A microscopic model, which describes all main statistical properties, has not been found yet (although, in my opinion, it is not impossible that one day we will find a satisfying model).

After reading this essay the reader may ask what has econophysics to do with physics. In my opinion it is the general strategy we explained in detail. But unfortunately I am afraid I have to say that the analogies between particular physical systems (such as a turbulent fluid or Brownian motion) and financial markets seem to be quite superficial.

References