1. Emergent Properties of Price Processes in Artificial Markets. Sanmay Das, Tomaso Poggio and Andrew Lo. The Problem: We are studying properties of price ...
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Emergent Properties of Price Processes in Artificial Markets
Sanmay Das, Tomaso Poggio and Andrew Lo The Problem: We are studying properties of price processes that emerge from the interaction of simple agents within stylized models of financial markets, in the effort to replicate and hence explain some of the properties that are known to exist in real markets. We hope to capture some of the essential properties of real-world financial markets in our simpler models. Motivation: Financial markets are known to exhibit certain stylized properties that hold across a broad range of markets and institutions. Example of these properties are volatility clustering, the leptokurtic (“fat-tailed”) distribution of returns, power law scaling of the tail of the return distribution, and significant long range autocorrelation of absolute returns paired with no such autocorrelation in raw returns. Theoretical models in general have a hard time explaining these facts. Market simulations with artificial traders might be able to provide qualitative explanations of some of these phenomena. Previous Work: Liu et al present a detailed analysis of the time series properties of returns in a real equity market [3]. Their major findings are that return distributions are leptokurtic and fat-tailed, volatility clustering occurs (that is, big price changes are more likely to be followed by big price changes and small price changes are more likely to be followed by small price changes)1 and that the autocorrelation of absolute values of returns decays according to a power law, and is persistent over large time scales, as opposed to the autocorrelation of raw returns, which disappears rapidly. Raberto et al have developed the Genoa Artificial Stock Market, and they are able to replicate the fat tailed nature of the distribution of returns and the clustered volatility observed in real markets [5] by using an explicit model of opinion propagation and herd behavior among trading agents. Gabaix et al present a model that explains some of the phenomena observed in real markets like the distribution of returns using the observed distribution of mutual fund sizes and a derived price impact function [2]. Approach: We are using a market-making algorithm we have designed [1] to set prices in the market. This allows us to study price processes emerging from the interaction of simple trading agents with the market-maker serving as the intermediary in setting prices. This approach allows us to maintain a parsimonious model which produces interesting behavior in prices. Some traders trade based on their perception of the “true” or fundamental value of the stock, while others are liquidity traders who are assumed to trade for reasons exogenous to the model. We are able to qualitatively replicate some of the most important stylized properties of returns in financial markets, such as volatility clustering and the leptokurtic distribution of returns (see figure 1) as well as the long range autocorrelation of absolute returns and the absence of long range autocorrelation in raw returns (figure 2). Impact: Simple artificial markets populated by the kinds of trading crowds and market-makers we describe are capable of replicating some of the important time series phenomena of real financial markets. These phenomena are replicated to some extent in the artificial markets described by Lux [4] and Raberto et al [5] among others, but only with explicit models of opinion propagation and evolutionary behavior in the trading crowd. The fact that our model does not need to explicitly postulate such behavior, instead relying on the simple interaction between informed and uninformed traders, may point to an important underlying regularity of such time series phenomena. Future Work: We are attempting to explain some other phenomena observed in real markets, such as the asymmetry in positive and negative returns (the absolute value of returns conditional on them being negative is higher than the absolute value conditional on them being positive). We are also interested in extending our work on the phenomena we have modeled by focusing on the particular nature of the power laws governing the distribution of returns and the decay of autocorrelation of absolute returns and by attempting to calibrate parameters to agree with the quantitative data available about returns in real markets. 1 Liu et al are certainly not the first to discover these properties of financial time series. However, they summarize much of the work and provide detailed references, and they present novel results on the power law distribution of volatility correlation.
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Figure 1: Returns over time (left) and distribution of absolute values of returns (right) in example simulations 1
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Research Support: Research at CBCL is sponsored by grants from: Office of Naval Research (DARPA) Contract No. N00014-00-1-0907, National Science Foundation (ITR/IM) Contract No. IIS-0085836, National Science Foundation (ITR) Contract No. IIS-0112991, National Science Foundation (KDI) Contract No. DMS-9872936, and National Science Foundation Contract No. IIS-9800032. Additional support was provided by: AT&T, Central Research Institute of Electric Power Industry, Center for e-Business (MIT), DaimlerChrysler AG, Compaq/Digital Equipment Corporation, Eastman Kodak Company, Honda R&D Co., Ltd., ITRI, Komatsu Ltd., Merrill-Lynch, Mitsubishi Corporation, NEC Fund, Nippon Telegraph & Telephone, Oxygen, Siemens Corporate Research, Inc., Sumitomo Metal Industries, Toyota Motor Corporation, WatchVision Co., Ltd., and The Whitaker Foundation. References: [1] Sanmay Das. Intelligent Market-Making in Artificial Financial Markets. AI Technical Report/CBCL Memo 2003-005/226, Massachusetts Institute of Technology, 2003. [2] X. Gabaix, P. Gopikrishnan, V. Plerou, and H.E. Stanley. A Theory of Power Law Distributions in Financial Market Fluctuations. Nature, 423:267–270, 2003. [3] Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C. Peng, and H.E. Stanley. Statistical Properties of the Volatility of Price Fluctuations. Physical Review E, 60(2):1390–1400, 1999. [4] T. Lux. The Socio-Economic Dynamics of Speculative Markets: Interacting Agents, Chaos, and the Fat Tails of Return Distributions. Journal of Economic Behavior and Organization, 33:143–165, 1998. [5] M. Raberto, S. Cincotti, S.M. Focardi, and M. Marchesi. Agent-based Simulation of a Financial Market. Physica A, 299:319–327, 2001.
