Complex Systems

There are number of systems and processes in our environment that are yet too complicated to be studied directly from the basic laws-of-nature point of view. Such processes can be found in nature itself e.g. biology, in finance markets, and in various technical and even sociatal systems, which behave in statistical manner. Therefore, it is natural to assume that these systems can be studied with statistical modelling, statistical physics and information theory methods. The aim of this kind of research is to describe measurable properties of the system in question and their dynamic behaviour.

Currently in the vast field of Complex Systems we are studying Financial Markets, Small World Networks, and Traffic Flow. In the Financial Market studies the focus is in modelling asset return, in analysing stylized facts of financial markets and modelling them through Game theory approach, in analysing correlations between financial assets through Random Matrix theory, among other things. Due to the fact that statistical physics approaches play a prominent role in these studies we call them Econophysics. In Small World Network studies we focus on various random networks and also regular graphs through simulation and analytical theory approach. These kind of network studies play an important role in Internet, WWW, stock market trading, biochemistry signaling, protein systems, epidemics, transportation and communication systems, and social interactions. In Traffic Flow studies we focus on random walk simulations to analyse collective processes in the system, e.g. traffic jam behaviour.

Statistical Analysis of Asset Returns

Researchers: Juuso Töyli, Laszlo Kullman*, Laszlo Gillemot*, Kimmo Kaski, Janos Kertesz*
* Dept. of Theoretical Physics, Budapest University of Technology and Economics

Asset returns have traditionally been modelled with methods based on the normal distribution. However, the empirical returns are characterised by stylized facts that imply non-normality. The stylised facts include heavy tails thus the empirical distribution is leptokurtic, the variances are auto-correlated although returns are not (except for very small intraday time scales), large and small returns are clustered, and there are jumps and crashes although these are typically asymmetric so that the magnitude of crashes is larger than that of jumps. Despite of these stylized facts, the shape of the distribution approaches normal distribution when the time interval is increased and the monthly returns are generally regarded as normally distributed. These characteristics suggest that the return generating stochastic process is non-linear, time dependent, and complex.

During the past century several model has been suggested but there seems to be no unanimous view. These models can be divided in time-independent and time-dependent categories. Well-known time-independent models include normal distribution, Lévy distribution, truncated Lévy distribution, generalised Lévy distribution, Student t, mixed diffusion jump, mixture normal distribution, and mixture distributions. Time dependent models contain autoregressive heteroscedastic models, stochastic volatility models assuming the volatility as stochastic process, and models based on chaos theory resulting in complex dynamics. These current models are not nevertheless able to capture the dynamics of empirical returns and the results are contradictory. The possible time-dependency of the shape of the distribution has also mainly been ignored.

The research has so far concentrated on the understanding of the return generating process. We have developed toolboxes to fit and simulate data according to the well-known time-independent models. It seems that they are able to capture the long-term distribution but not the structure of the process. We have also studied the effect of different dependencies, linear and non-linear, on the shape of the distribution and generating process. Along with these, the possible biases resulting from different terms of measurements have been researched. Finally, we have studies the changes in the return generating process when the time interval grows. Currently, it seems that the models we have used cannot completely capture the dynamics of the market. Therefore, we have also started to build artificial market in order to understand the role of micro agents in return generation process and to understand the underlying dynamics.

Stylized facts of financial market

Researchers: Marko Sysi-Aho, Juuso Töyli, Kimmo Kaski, Janos Kertesz*
* Dept. of Theoretical Physics, Budapest University of Technology and Economics

The statistical study of the stock market indices and asset returns were launched by a French mathematician Bachelier on 1900. After that there has been a growing interest on the mathematical modelling of the financial market. During the past forty years it has been found that asset returns have statistical features different from those proposed at earlier times. Especially, asset returns are not independent and not gaussian distributed. The absolute value of returns show long ranged and slowly decaying autocorrelations and the return distribution has a sharper peak and fatter tails than that of the gaussian. These well known features have lately been named as stylized facts.

Figure 15

Figure 15: Figure on the left shows the sum of i.i.d random variables that converges towards a gaussian distribution. Figure on the right describes the change of asset return distribution as the time scale is increased.

There are also other stylized facts and the term, in general, refers to features that are, on an average, common to every asset return series. The current research is feverishly trying to generate models that are capable of describing the origin of these features, as well as trying to root up high frecuency data sets containing the most fine grained price information.

An interesting and powerful model capable of producing some of the well known stylized facts is a Minority Game. The game is basically an agent based market model in which a number of players are repeatedly trying to decide between two alternatives, say whether to choose side A or B. At each time step, those who happen to be in the minority win. Evolutional features can be added to the game by replacing unsuccessful players with the clones of good players. Also the intelligence of players can be improved or impaired by defining a rule that changes the memories of the players. An index similar in nature to stock indices can be composed as a function of players choosing one of the sides.

Correlations and Taxonomy of Financial Assets

Researchers: Jukka-Pekka Onnela, Kimmo Kaski, Janos Kertesz*, Laszlo Kullmann*
* Dept. of Theoretical Physics, Budapest University of Technology and Economics

The traditional Markowitz portfolio construction problem is: Given a set of financial assets characterized by their average risk and return, what is the optimal weight of each asset, such that the overall portfolio provides the best return for a fixed level of risk. Our research concentrates on two connected fields, both closely related to the portfolio optimization problem. First, we have analyzed the temporal properties of correlations. It is reasonable to assume that there are occasions when portfolios of assets display greater correlated movements than others, such as in a rising or falling market. We have investigated some statistical properties exhibited by the correlation distribution, and indeed this is one of the findings that our study has confirmed. We are currently applying Random Matrix Theory to separate "signal" from "noise" in order to filter out the genuine information. Second, we have investigated asset taxonomy. Correlation data can be used to define a metric distance, from which we can construct a minimum spanning tree. All the stocks are now connected together, but each stock is linked only to its nearest neighbor. Consequently, a unique taxonomy of stocks emerges. We can study the dynamics of this organization as a function of time and clearly visualize the changes taking place in the stock market. We believe that our study can further the understanding of the stock market as a complex, evolving system and thus contribute towards the fundamental problem of portfolio optimization.

Figure 16a Figure 16b Figure 16c

Figure 16: Left: Contour plot of probability density for correlation coefficients. The effect of Black Monday, October 19, 1987, is clearly visible. Right: Two different taxonomies for 30 randomly picked stocks from the S&P500-index. The organization of stocks has undergone some evolution in three years.

Statistical Analysis of Small World Networks

Researchers: Jani Lahtinen, János Kertész* and Kimmo Kaski
* Dept. of Theoretical Physics, Budapest University of Technology and Economics

The small world networks are graphs, which albeit having a large amount of vertices still in average retain small distance between individual vertices relative to traversal of links. Such networks are for example the internet, WWW, stock market trading interlinking, biochemistry signaling and metabolism in protein systems, epidemics, formation of polymers, transportation systems or interlinked systems of social interactions. These real world examples can - to certain extent - be modelled either as random graphs of Erdös-Rényi, Watts-Strogatz, Barabasi etc. types, or as regular graphs like lattices and cages. Regular graphs are of great interest from the point of view of manageable analytical theory approach but because they offer means to algorithmically generate structures of smallest possible diameter (i.e. the longest distance in the graph) for a given amount of vertices. An example of such a regular graph is cage and it could perhaps be applied in developing efficient telecommunication network topologies. Our research concentrates on analysis of statistical properties of random graphs, and construction of deterministic graphs as approximations of cages and investigation of their properties.

Figure 17a Figure 17b

Figure 17: An Erdös-Rényi (left) and a Watts-Strogatz (right) graphs with 100 vertices.

Stochastic Simulation of Traffic Flow

Researchers: Kimmo Kaski and Robin Stinchcombe*
* Oxford University

Recently there has been considerable interest to investigate traffic flow using methods of statistical physics. This is interesting not only from the point of view of getting getting deeper qualitative understanding of the basic processes and principles of vehicular traffic but also because this understanding could be applied to designing a more efficient vehicular traffic system or possibly also a more efficient telecommunication system. The focus of research is through discrete one dimensional - single lane - model and by using mean field type analytical solutions and lattice based simulations to study and get insight of the steady state flow properties, formation of jams, and propagation of jam fronts etc. The variables of the model are the rates of input and output to the left end of the single lane and from the right end of the lane, respectively, as well as the single or randomly distributed rate of hopping from a lattice site to its neighbour on the right (only). As an extension to this models also a multi-lane systems, in which vehicles can overtake each other, will be studies.