Interacting gaps model,dynamics of order book,and stock-market fluctuations

Inspired by order-book models of financial fluctuations, we investigate the Interacting gaps model, which is the schematic one-dimensional system mimicking the order-book dynamics. We find by simulations the power-law tail in return distribution, power-law decay of volatility autocorrelation with exponent 0.5 and Hurst exponent close to 1/2. Surprisingly, when we make a mean-field approximation, i.e. replace the one-dimensional system by effectively infinite-dimensional one, we obtain analytically the return exponent 5/2, in perfect accord with one-dimensional simulations.     

Modelling fluctuations of financial time series: from cascade process to stochastic volatility model

In this paper, we provide a simple, “generic” interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that in this context 1/f power spectra, as recently observed in reference [23], naturally emerge. We then propose a simple solvable “stochastic volatility” model for return fluctuations. This model is able to reproduce most of recent empirical findings concerning financial time series: no correlation between price variations, long-range volatility correlations and multifractal statistics. Moreover, its extension to a multivariate context, in order to model portfolio behavior, is very natural. Comparisons to real data and other models proposed elsewhere are provided. Received 22 May 2000     

Comparison between volatility return intervals of the S&;P 500 index and two common models

We analyze the S&P 500 index data for the 13-year period, from January 1, 1984 to December 31, 1996, with one data point every 10 min. For this database, we study the distribution and clustering of volatility return intervals, which are defined as the time intervals between successive volatilities above a certain threshold q. We find that the long memory in the volatility leads to a clustering of above-median as well as below-median return intervals. In addition, it turns out that the short return intervals form larger clusters compared to the long return intervals. When comparing the empirical results to the ARMA-FIGARCH and fBm models for volatility, we find that the fBm model predicts scaling better than the ARMA-FIGARCH model, which is consistent with the argument that both ARMA-FIGARCH and fBm capture the long-term dependence in return intervals to a certain extent, but only fBm accounts for the scaling. We perform the Student's t-test to compare the empirical data with the shuffled records, ARMA-FIGARCH and fBm. We analyze separately the clusters of above-median return intervals and the clusters of below-median return intervals for different thresholds q. We find that the empirical data are statistically different from the shuffled data for all thresholds q. Our results also suggest that the ARMA-FIGARCH model is statistically different from the S&P 500 for intermediate q for both above-median and below-median clusters, while fBm is statistically different from S&P 500 for small and large q for above-median clusters and for small q for below-median clusters. Neither model can fully explain the entire regime of q studied.     

On the probability distribution of stock returns in the Mike-Farmer model

Recently, Mike and Farmer have constructed a very powerful and realistic behavioral model to mimick the dynamic process of stock price formation based on the empirical regularities of order placement and cancelation in a purely order-driven market, which can successfully reproduce the whole distribution of returns, not only the well-known power-law tails, together with several other important stylized facts. There are three key ingredients in the Mike-Farmer (MF) model: the long memory of order signs characterized by the Hurst index H_s, the distribution of relative order prices x in reference to the same best price described by a Student distribution (or Tsallis’ q-Gaussian), and the dynamics of order cancelation. They showed that different values of the Hurst index H_s and the freedom degree α_x of the Student distribution can always produce power-law tails in the return distribution f_r(r) with different tail exponent α_r. In this paper, we study the origin of the power-law tails of the return distribution f_r(r) in the MF model, based on extensive simulations with different combinations of the left part L(x) for x < 0 and the right part R(x) for x > 0 of f_x(x). We find that power-law tails appear only when L(x) has a power-law tail, no matter R(x) has a power-law tail or not. In addition, we find that the distributions of returns in the MF model at different timescales can be well modeled by the Student distributions, whose tail exponents are close to the well-known cubic law and increase with the timescale.     

Bounding the quality of stochastic oscillations in population models

We analyze general two-species stochastic models, of the kind generally used for the study of population dynamics. Although usually defined a priori, the deterministic version of these models can be obtained as the infinite volume limit of many stochastic models (which are necessarily defined by more parameters than the deterministic one). It is known that damped oscillations in a deterministic model usually correspond to oscillatory-like fluctuations in their deterministic counterparts. The quality of these “oscillations" depends on details of each stochastic model. We show, however, that the parameters of the deterministic system are generally enough to obtain very good bounds for the quality of “oscillations" in any of its stochastic counterparts. These bounds are shown to depend on only one dimensionless parameter.     

Microeconomics of the ideal gas like market models

We develop a framework based on microeconomic theory from which the ideal gas like market models can be addressed. A kinetic exchange model based on that framework is proposed and its distributional features have been studied by considering its moments. Next, we derive the moments of the CC model (Eur. Phys. J. B 17 (2000) 167) as well. Some precise solutions are obtained which conform with the solutions obtained earlier. Finally, an output market is introduced with global price determination in the model with some necessary modifications.     

Spread and Quote-Update Frequency of the Limit-Order Driven Sergei Maslov Model

We perform numerical simulations of the limit-order driven Sergei Maslov （SM） model and investigate the probability distribution and autocorrelation function of the bid-ask spread S and the quote-update frequency U. For the probability distribution, the model successfully reproduces the power law decay of the spread and the exponential decay of the quote-update frequency. For the autocorrelation function, both the spread and the quote-update frequency of the model decay by a power law, which is consistent with the empirical study. We obtain the power law exponent 0.54 for the spread, which is in good agreement with the real financial market.     

Modeling wealth distribution in growing markets

We introduce an auto-regressive model which captures the growing nature of realistic markets. In our model agents do not trade with other agents, they interact indirectly only through a market. Change of their wealth depends, linearly on how much they invest, and stochastically on how much they gain from the noisy market. The average wealth of the market could be fixed or growing. We show that in a market where investment capacity of agents differ, average wealth of agents generically follow the Pareto-law. In few cases, the individual distribution of wealth of every agentcould also be obtained exactly. We also show that the underlying dynamics of other well studied kinetic models of markets can be mapped to the dynamics of our auto-regressive model.     

Boundary-induced phase transitions in a space-continuous traffic model with non-unique flow-density relation

The Krauss-model is a stochastic model for traffic flow which is continuous in space. For periodic boundary conditions it is well understood and known to display a non-unique flow-density relation (fundamental diagram) for certain densities. In many applications, however, the behaviour under open boundary conditions plays a crucial role. In contrast to all models investigated so far, the high flow states of the Krauss-model are not metastable, but also stable. Nevertheless we find that the current in open systems obeys an extremal principle introduced for the case of simpler discrete models. The phase diagram of the open system will be completely determined by the fundamental diagram of the periodic system through this principle. In order to allow the investigation of the whole state space of the Krauss-model, appropriate strategies for the injection of cars into the system are needed. Two methods solving this problem are discussed and the boundary-induced phase transitions for both methods are studied. We also suggest a supplementary rule for the extremal principle to account for cases where not all the possible bulk states are generated by the chosen boundary conditions. Received 16 September 2002 / Received in final form 4 November 2002 Published online 31 December 2002     

The emergence of coordination in public good games

In physical models it is well understood that the aggregate behaviour of a system is not in one to one correspondence with the behaviour of the average individual element of that system. Yet, in many economic models the behaviour of aggregates is thought of as corresponding to that of an individual. A typical example is that of public goods experiments. A systematic feature of such experiments is that, with repetition, people contribute less to public goods. A typical explanation is that people “learn to play Nash” or something approaching it. To justify such an explanation, an individual learning model is tested on average or aggregate data. In this paper we will examine this idea by analysing average and individual behaviour in a series of public goods experiments. We analyse data from a series of games of contributions to public goods and as is usual, we test a learning model on the average data. We then look at individual data, examine the changes that this produces and see if some general model such as the EWA (Expected Weighted Attraction) with varying parameters can account for individual behaviour. We find that once we disaggregate data such models have poor explanatory power. Groups do not learn as supposed, their behaviour differs markedly from one group to another, and the behaviour of the individuals who make up the groups also varies within groups. The decline in aggregate contributions cannot be explained by resorting to a uniform model of individual behaviour. However, the Nash equilibrium of such a game is a total payment for all the individuals and there is some convergence of the group in this respect. Yet the individual contributions do not converge. How the individuals “self-organsise” to coordinate, even in this limited way remains to be explained.     

Characteristic time scales of tick quotes on foreign currency markets: an empirical study and agent-based model

Power spectrum densities for the number of tick quotes per minute (market activity) on three currency markets (USD/JPY, EUR/USD, and JPY/EUR) for periods from January 1999 to December 2000 are analyzed. We find some peaks on the power spectrum densities at a few minutes. We develop the double-threshold agent model and confirm that stochastic resonance occurs for the market activity of this model. We propose a hypothesis that the periodicities found on the power spectrum densities can be observed due to stochastic resonance.     

Role of noise in a market model with stochastic volatility

We study a generalization of the Heston model, which consists of two coupled stochastic differential equations, one for the stock price and the other one for the volatility. We consider a cubic nonlinearity in the first equation and a correlation between the two Wiener processes, which model the two white noise sources. This model can be useful to describe the market dynamics characterized by different regimes corresponding to normal and extreme days. We analyze the effect of the noise on the statistical properties of the escape time with reference to the noise enhanced stability (NES) phenomenon, that is the noise induced enhancement of the lifetime of a metastable state. We observe NES effect in our model with stochastic volatility. We investigate the role of the correlation between the two noise sources on the NES effect.     

Stylized facts from a threshold-based heterogeneous agent model

A class of heterogeneous agent models is investigated where investors switch trading position whenever their motivation to do so exceeds some critical threshold. These motivations can be psychological in nature or reflect behaviour suggested by the efficient market hypothesis (EMH). By introducing different propensities into a baseline model that displays EMH behaviour, one can attempt to isolate their effects upon the market dynamics. The simulation results indicate that the introduction of a herding propensity results in excess kurtosis and power-law decay consistent with those observed in actual return distributions, but not in significant long-term volatility correlations. Possible alternatives for introducing such long-term volatility correlations are then identified and discussed.     

Beliefs and stochastic modelling of interest rate scenario risk

We present a framework that allows for a systematic assessment of risk given a specific model and belief on the market. Within this framework the time evolution of risk is modeled in a twofold way. On the one hand, risk is modeled by the time discrete and nonlinear garch(1,1) process, which allows for a (time-)local understanding of its level, together with a short term forecast. On the other hand, via a diffusion approximation, the time evolution of the probability density of risk is modeled by a Fokker-Planck equation. Then, as a final step, using Bayes theorem, beliefs are conditioned on the stationary probability density function as obtained from the Fokker-Planck equation. We believe this to be a highly rigorous framework to integrate subjective judgments of future market behavior and underlying models. In order to demonstrate the approach, we apply it to risk assessment of empirical interest rate scenario methodologies, i.e. the application of Principal Component Analysis to the the dynamics of bonds. Received 1st August 2000     

Distribution of return intervals of extreme events

The distribution of return intervals of extreme events is studied in time series characterized by finite-term correlations with non-exponential decay. Precisely, it has been analyzed the statistics of the return intervals of extreme values of the resistance fluctuations displayed by resistors with granular structure in nonequilibrium stationary states. The resistance fluctuations are calculated by Monte Carlo simulations using a resistor network approach. It has been found that for highly disordered networks, when the auto-correlation function displays a non-exponential and non-power-law decay, the distribution of return intervals of the extreme values is a stretched exponential, with exponent independent of the threshold.     

Exactly solvable reaction diffusion models on a Cayley tree

The most general reaction-diffusion model on a Cayley tree with nearest-neighbor interactions is introduced, which can be solved exactly through the empty-interval method. The stationary solutions of such models, as well as their dynamics, are discussed. Concerning the dynamics, the spectrum of the evolution Hamiltonian is found and shown to be discrete, hence there is a finite relaxation time in the evolution of the system towards its stationary state.     

Clustering of volatility in variable diffusion processes

Increments in financial markets have anomalous statistical properties including fat-tailed distributions and volatility clustering (i.e., the autocorrelation functions of return increments decay quickly but those of the squared increments decay slowly). One of the central questions in financial market analysis is whether the nature of the underlying stochastic process can be deduced from these statistical properties. We have shown previously that a class of variable diffusion processes has fat-tailed distributions. Here we show analytically that such models also exhibit volatility clustering. To our knowledge, this is the first case where clustering of volatility is proven analytically in a model.Our results are compatible with the viewpoint that variable diffusion processes are possible models for financial markets.     

Inefficiency in Latin-American market indices

We explore the deviations from efficiency in the returns and volatility returns of Latin-American market indices. Two different approaches are considered. The dynamics of the Hurst exponent is obtained via a wavelet rolling sample approach, quantifying the degree of long memory exhibited by the stock market indices under analysis. On the other hand, the Tsallis q entropic index is measured in order to take into account the deviations from the Gaussian hypothesis. Different dynamic rankings of inefficieny are obtained, each of them contemplates a different source of inefficiency. Comparing with the results obtained for a developed country (US), we confirm a similar degree of long-range dependence for our emerging markets. Moreover, we show that the inefficiency in the Latin-American countries comes principally from the non-Gaussian form of the probability distributions.     

The grounds for time dependent market potentials from dealers’ dynamics

We apply the potential force estimation method to artificial time series of market price produced by a deterministic dealer model. We find that dealers’ feedback of linear prediction of market price based on the latest mean price changes plays the central role in the market’s potential force. When markets are dominated by dealers with positive feedback the resulting potential force is repulsive, while the effect of negative feedback enhances the attractive potential force.