Gambling on the Budapest stock exchange

The statistical properties of the total yield are analyzed for an assembly of gamblers in an erratic period on the Budapest stock exchange. Random trading results in a log-normal limit distribution of a surprisingly large width, while the simplest profit realizing strategy narrows down the peak around a positive average value. The effect of transaction costs, the statistics of extremes, and patterns of successful trading are also investigated. In spite of the very simple approach, we present strong indications that large trading activity (e.g. day trading) is a rather risky way of capital investment. A comparison with the yield distribution of 32 public investment funds in the given period does not reflect the presence of a sophisticated investment strategy in the background. Received 5 May 2000     

The Nasdaq crash of April 2000: Yet another example of log-periodicity in a speculative bubble ending in a crash

The Nasdaq Composite fell another % on Friday the 14'th of April 2000 signaling the end of a remarkable speculative high-tech bubble starting in spring 1997. The closing of the Nasdaq Composite at 3321 corresponds to a total loss of over 35% since its all-time high of 5133 on the 10'th of March 2000. Similarities to the speculative bubble preceding the infamous crash of October 1929 are quite striking: the belief in what was coined a “New Economy” both in 1929 and presently made share-prices of companies with three digits price-earning ratios soar. Furthermore, we show that the largest draw downs of the Nasdaq are outliers with a confidence level better than 99% and that these two speculative bubbles, as well as others, both nicely fit into the quantitative framework proposed by the authors in a series of recent papers. Received 3 May 2000     

A stability criterion for financial markets

Using the theory of random cluster models, we give a stability criterion for financial markets with random communications between agents. Received 25 September 1999 and Received in final form 2 October 1999     

On the physical relevance of max- and log-max-selfsimilar distributions

This work emphasizes the special role played by max-semistable and log-max-semistable distributions as relevant statistical models of various observable and “internal” variables in Physics. Some of their remarkable properties (chiefly self-similarity) are displayed in some detail. One of their characteristic features is a log-periodic variation of the scale parameter which appears in the stable extreme value distributions. Received 29 November 1999 and Received in final form 24 March 2000     

How popular is your paper? An empirical study of the citation distribution

Numerical data for the distribution of citations are examined for: (i) papers published in 1981 in journals which are catalogued by the Institute for Scientific Information (783,339 papers) and (ii) 20 years of publications in Physical Review D, vols. 11-50 (24,296 papers). A Zipf plot of the number of citations to a given paper versus its citation rank appears to be consistent with a power-law dependence for leading rank papers, with exponent close to -1/2. This, in turn, suggests that the number of papers with x citations, N(x), has a large-x power law decay , with . Received: 12 May 1998 / Accepted: 12 May 1998     

Economic returns of research: the Pareto law and its implications

At what level should government or companies support research? This complex multi-faceted question encompasses such qualitative bonus as satisfying natural human curiosity, the quest for knowledge and the impact on education and culture, but one of its most scrutinized component reduces to the assessment of economic performance and wealth creation derived from research. Many studies report evidences of positive economic benefits derived from basic research [#!Martin!#,#!NAS!#]. In certain areas such as biotechnology, semi-conductor physics, optical communications [#!Ehrenreich!#], the impact of basic research is direct while, in other disciplines, the path from discovery to applications is full of surprises. As a consequence, there are persistent uncertainties in the quantification of the exact economic returns of public expenditure on basic research. This gives little help to policy makers trying to determine what should be the level of funding. Here, we suggest that these uncertainties have a fundamental origin to be found in the interplay between the intrinsic “fat tail” power law nature of the distribution of economic returns, characterized by a mathematically diverging variance, and the stochastic character of discovery rates. In the regime where the cumulative economic wealth derived from research is expected to exhibit a long-term positive trend, we show that strong fluctuations blur out significantly the short-time scales: a few major unpredictable innovations may provide a finite fraction of the total creation of wealth. In such a scenario, any attempt to assess the economic impact of research over a finite time horizon encompassing only a small number of major discoveries is bound to be highly unreliable. New tools, developed in the theory of self-similar and complex systems [#!Dubrulleetal!#] to tackle similar extreme fluctuations in Nature [#!Mandelbrot!#], can be adapted to measure the economic benefits of research, which is intimately associated to this large variability. Received 26 October 1998 and Received in final form 27 October 1998     

Are citations of scientific papers a case of nonextensivity?

The distribution N(x) of citations of scientific papers has recently been illustrated (on ISI and PRE data sets) and analyzed by Redner (Eur. Phys. J. B 4, 131 (1998)). To fit the data, a stretched exponential () has been used with only partial success. The success is not complete because the data exhibit, for large citation count x, a power law (roughly for the ISI data), which, clearly, the stretched exponential does not reproduce. This fact is then attributed to a possibly different nature of rarely cited and largely cited papers. We show here that, within a nonextensive thermostatistical formalism, the same data can be quite satisfactorily fitted with a single curve (namely, [0pt] for the available values of x. This is consistent with the connection recently established by Denisov (Phys. Lett. A 235, 447 (1997)) between this nonextensive formalism and the Zipf-Mandelbrot law. What the present analysis ultimately suggests is that, in contrast to Redner's conclusion, the phenomenon might essentially be one and the same along the entire range of the citation number x. Received 13 April 1999     

Stretched exponential distributions in nature and economy: “fat tails” with characteristic scales

To account quantitatively for many reported “natural” fat tail distributions in Nature and Economy, we propose the stretched exponential family as a complement to the often used power law distributions. It has many advantages, among which to be economical with only two adjustable parameters with clear physical interpretation. Furthermore, it derives from a simple and generic mechanism in terms of multiplicative processes. We show that stretched exponentials describe very well the distributions of radio and light emissions from galaxies, of US GOM OCS oilfield reserve sizes, of World, US and French agglomeration sizes, of country population sizes, of daily Forex US-Mark and Franc-Mark price variations, of Vostok (near the south pole) temperature variations over the last 400 000 years, of the Raup-Sepkoski's kill curve and of citations of the most cited physicists in the world. We also discuss its potential for the distribution of earthquake sizes and fault displacements. We suggest physical interpretations of the parameters and provide a short toolkit of the statistical properties of the stretched exponentials. We also provide a comparison with other distributions, such as the shifted linear fractal, the log-normal and the recently introduced parabolic fractal distributions. Received: 20 January 1998 / Received in final form: 27 January 1998 / Accepted: 6 February 1998     

“String” formulation of the dynamics of the forward interest rate curve

We propose a formulation of the term structure of interest rates in which the forward curve is seen as the deformation of a string. We derive the general condition that the partial differential equations governing the motion of such string must obey in order to account for the condition of absence of arbitrage opportunities. This condition takes a form similar to a fluctuation-dissipation theorem, albeit on the same quantity (the forward rate), linking the bias to the covariance of variation fluctuations. We provide the general structure of the models that obey this constraint in the framework of stochastic partial (possibly non-linear) differential equations. We derive the general solution for the pricing and hedging of interest rate derivatives within this framework, albeit for the linear case (we also provide in the appendix a simple and intuitive derivation of the standard European option problem). We also show how the “string” formulation simplifies into a standard N-factor model under a Galerkin approximation. Received: 30 January 1998 / Revised: 12 February 1998 / Accepted: 16 February 1998     

Domino effect for world market fluctuations

In order to emphasize cross-correlations for fluctuations in major market places, series of up and down spins are built from financial data. Patterns frequencies are measured, and statistical tests performed. Strong cross-correlations are emphasized, proving that market moves are collective behaviors. Received 15 January 2000     

”Direct” causal cascade in the stock market

We use wavelets to decompose the volatility (standard deviation) of intraday (S&P500) return data across scales. We show that when investigating two-point correlation functions of the volatility logarithms across different time scales, one reveals the existence of a causal information cascade from large scales (i.e. small frequencies) to fine scales. We quantify and visualize the information flux across scales. We provide a possible interpretation of our findings in terms of market dynamics. Received: 9 January 1998 / Received in final form and accepted: 13 January 1998     

Delay, feedback and quenching in financial markets

An asset whose price exhibits geometric Brownian motion is analysed. The basic Brownian motion model is modified to account for the effects of market delay and investor feedback. A Langevin equation model is appropriate. When the feedback coupling is sufficiently strong, the market dynamics switches from a slow random walk behaviour to a rapid unstable behaviour with a fast time scale characteristic of the market delay. The unstable runaway behaviour is subsequently quenched by investors deserting a collapsing market or saturating a booming one. This quenching effect is sufficient to ensure long term bounding of the asset price. A form of market sabotage is demonstrated in which investors can push the market from a stable to an unstable regime. Received 24 February 2000     

Statistical mechanics of money

In a closed economic system, money is conserved. Thus, by analogy with energy, the equilibrium probability distribution of money must follow the exponential Boltzmann-Gibbs law characterized by an effective temperature equal to the average amount of money per economic agent. We demonstrate how the Boltzmann-Gibbs distribution emerges in computer simulations of economic models. Then we consider a thermal machine, in which the difference of temperatures allows one to extract a monetary profit. We also discuss the role of debt, and models with broken time-reversal symmetry for which the Boltzmann-Gibbs law does not hold. The instantaneous distribution of money among the agents of a system should not be confused with the distribution of wealth. The latter also includes material wealth, which is not conserved, and thus may have a different (e.g. power-law) distribution. Received 22 June 2000     

Fundamental framework for “technical analysis” of market prices

Starting from the characterization of the past time evolution of market prices in terms of two fundamental indicators, price velocity and price acceleration, we construct a general classification of the possible patterns characterizing the deviation or defects from the random walk market state and its time-translational invariant properties. The classification relies on two dimensionless parameters, the Froude number characterizing the relative strength of the acceleration with respect to the velocity and the time horizon forecast dimensionalized to the training period. Trend-following and contrarian patterns are found to coexist and depend on the dimensionless time horizon. The classification is based on the symmetry requirements of invariance with respect to change of price units and of functional scale-invariance in the space of scenarii. This “renormalized scenario” approach is fundamentally probabilistic in nature and exemplifies the view that multiple competing scenarii have to be taken into account for the same past history. Empirical tests are performed on about nine to thirty years of daily returns of twelve data sets comprising some major indices (Dow Jones, SP500, Nasdaq, DAX, FTSE, Nikkei), some major bonds (JGB, TYX) and some major currencies against the US dollar (GBP, CHF, DEM, JPY). Our “renormalized scenario” exhibits statistically significant predictive power in essentially all market phases. In contrast, a trend following strategy and following strategy perform well only on different and specific market phases. The value of the “renormalized scenario” approach lies in the fact that it always selects the best of the two, based on a calculation of the stability of their predicted market trajectories. Received 3 October 1999     

Apparent multifractality in financial time series

We present a exactly soluble model for financial time series that mimics the long range volatility correlations known to be present in financial data. Although our model is asymptotically `monofractal' by construction, it shows apparent multiscaling as a result of a slow crossover phenomenon on finite time scales. Our results suggest that it might be hard to distinguish apparent and true multifractal behavior in financial data. Our model also leads to a new family of stable laws for sums of correlated random variables. Received 30 June 1999     

Hierarchical structure in financial markets

I find a hierarchical arrangement of stocks traded in a financial market by investigating the daily time series of the logarithm of stock price. The topological space is a subdominant ultrametric space associated with a graph connecting the stocks of the portfolio analyzed. The graph is obtained starting from the matrix of correlation coefficient computed between all pairs of stocks of the portfolio by considering the synchronous time evolution of the difference of the logarithm of daily stock price. The hierarchical tree of the subdominant ultrametric space associated with the graph provides a meaningful economic taxonomy. Received 24 March 1999 and Received in final form 28 June 1999     

A kinetic approach to some quasi-linear laws of macroeconomics

Some previous works have presented the data on wealth and income distributions in developed countries and have found that the great majority of population is described by an exponential distribution, which results in idea that the kinetic approach could be adequate to describe this empirical evidence. The aim of our paper is to extend this framework by developing a systematic kinetic approach of the socio-economic systems and to explain how linear laws, modelling correlations between macroeconomic variables, may arise in this context. Firstly we construct the Boltzmann kinetic equation for an idealised system composed by many individuals (workers, officers, business men, etc.), each of them getting a certain income and spending money for their needs. To each individual a certain time variable amount of money is associated - this meaning him/her phase space coordinate. In this way the exponential distribution of money in a closed economy is explicitly found. The extension of this result, including states near the equilibrium, give us the possibility to take into account the regular increase of the total amount of money, according to the modern economic theories. The Kubo-Green-Onsager linear response theory leads us to a set of linear equations between some macroeconomic variables. Finally, the validity of such laws is discussed in relation with the time reversal symmetry and is tested empirically using some macroeconomic time series. Received 25 February 2002 / Received in final form 11 July 2002 Published online 19 November 2002     

Inverse cubic law for the distribution of stock price variations

The probability distribution of stock price changes is studied by analyzing a database (the Trades and Quotes Database) documenting every trade for all stocks in three major US stock markets, for the two year period January 1994 - December 1995. A sample of 40 million data points is extracted, which is substantially larger than studied hitherto. We find an asymptotic power-law behavior for the cumulative distribution with an exponent , well outside the Lévy regime . Received: 23 April 1998 / Revised and Accepted: 24 April 1998