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 $\alpha_x$ of the Student distribution can always produce power-law tails in the return distribution $f(r)$ with different tail exponent $\alpha_r$. In this paper, we study the origin of the power-law tails of the return distribution $f(r)$ in the MF model, based on extensive simulations with different combinations of the left part $f_L(x)$ for $x0$ of $f(x)$. We find that power-law tails appear only when $f_L(x)$ has a power-law tail, no matter $f_R(x)$ 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.Comment: 16 Elsart pages including 1 table and 5 figure

Physicists attempt to scale the ivory towers of finance

Physicists have recently begun doing research in finance, and even though this movement is less than five years old, interesting and useful contributions have already emerged. This article reviews these developments in four areas, including empirical statistical properties of prices, random-process models for price dynamics, agent-based modeling, and practical applications.Comment: 13 pages, 5 figure

Emotional persistence in online chatting communities

How do users behave in online chatrooms, where they instantaneously read and write posts? We analyzed about 2.5 million posts covering various topics in Internet relay channels, and found that user activity patterns follow known power-law and stretched exponential distributions, indicating that online chat activity is not different from other forms of communication. Analysing the emotional expressions (positive, negative, neutral) of users, we revealed a remarkable persistence both for individual users and channels. I.e. despite their anonymity, users tend to follow social norms in repeated interactions in online chats, which results in a specific emotional "tone" of the channels. We provide an agent-based model of emotional interaction, which recovers qualitatively both the activity patterns in chatrooms and the emotional persistence of users and channels. While our assumptions about agent's emotional expressions are rooted in psychology, the model allows to test different hypothesis regarding their emotional impact in online communication.Comment: 34 pages, 4 main and 12 supplementary figure

Static and dynamic factors in an information-based multi-asset artificial stock market

3noAn information-based multi-asset artificial stock market characterized by different types of stocks and populated by heterogeneous agents is presented. In the market, agents trade risky assets in exchange for cash. Beside the amount of cash and of stocks owned, each agent is characterized by sentiments and agents share their sentiments by means of interactions that are determined by sparsely connected networks. A central market maker (clearing house mechanism) determines the price processes for each stock at the intersection of the demand and the supply curves. Single stock price processes exhibit volatility clustering and fat-tailed distribution of returns whereas multivariate price process exhibits both static and dynamic stylized facts, i.e., the presence of static factors and common trends. Static factors are studied making reference to the cross-correlation of returns of different stocks. The common trends are investigated considering the variance–covariance matrix of prices. Results point out that the probability distribution of eigenvalues of the cross-correlation matrix of returns shows the presence of sectors, similar to those observed on real empirical data. As regarding the dynamic factors, the variance–covariance matrix of prices point out a limited number of assets prices series that are independent integrated processes, in close agreement with the empirical evidence of asset price time series of real stock markets. These results remarks the crucial dependence of statistical properties of multi-assets stock market on the agents’ interaction structure.partially_openopenPonta, Linda*; Pastore, Stefano; Cincotti, SilvanoPonta, Linda; Pastore, Stefano; Cincotti, Silvan

Financial power laws: Empirical evidence, models, and mechanism

Financial markets (share markets, foreign exchange markets and others) are all characterized by a number of universal power laws. The most prominent example is the ubiquitous finding of a robust, approximately cubic power law characterizing the distribution of large returns. A similarly robust feature is long-range dependence in volatility (i.e., hyperbolic decline of its autocorrelation function). The recent literature adds temporal scaling of trading volume and multi-scaling of higher moments of returns. Increasing awareness of these properties has recently spurred attempts at theoretical explanations of the emergence of these key characteristics form the market process. In principle, different types of dynamic processes could be responsible for these power-laws. Examples to be found in the economics literature include multiplicative stochastic processes as well as dynamic processes with multiple equilibria. Though both types of dynamics are characterized by intermittent behavior which occasionally generates large bursts of activity, they can be based on fundamentally different perceptions of the trading process. The present chapter reviews both the analytical background of the power laws emerging from the above data generating mechanism as well as pertinent models proposed in the economics literature. --