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QUEUING, SOCIAL INTERACTIONS, AND THE MICROSTRUCTURE OF FINANCIAL MARKETS

Published online by Cambridge University Press:  01 April 2008

ULRICH HORST*
Affiliation:
University of British Columbia
CHRISTIAN ROTHE
Affiliation:
Humboldt University Berlin
*
Address correspondence to: Ulrich Horst, Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada; e-mail: horst@math.ubc.ca.

Abstract

We consider an agent-based model of financial markets with asynchronous order arrival in continuous time. Buying and selling orders arrive in accordance with a Poisson dynamics where the order rates depend both on past prices and on the mood of the market. The agents form their demand for an asset on the basis of their forecasts of future prices and their forecasting rules may change over time as a result of the influence of other traders. Among the possible rules are “chartist” or extrapolatory rules. We prove that when chartists are in the market, and with choice of scaling, the dynamics of asset prices can be approximated by an ordinary delay differential equation. The fluctuations around the first-order approximation follow an Ornstein–Uhlenbeck dynamics with delay in a random environment of investor sentiment.

Type
ARTICLES
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Bayraktar, E., Horst, U., and Sircar, R. (in press a) A limit theorem for financial markets with inert investors. Mathematics of Operations Research.Google Scholar
Bayraktar, E., Horst, U., and Sircar, R. (in press b) Queuing theoretic approaches to financial price fluctuations. In Linetsky, V. and Birge, J. (eds.), Handbook of Financial Engineering.Google Scholar
Blume, L. and Easley, D. (in press) Optimality and natural selection in markets. Journal of Economic Theory.Google Scholar
Böhm, V. and Wenzelburger, J. (2005) On the performance of efficient portfolios. Journal of Economic Dynamics and Control 29, 721740.Google Scholar
Brock, W. and Hommes, C. (1997) A rational route to randomness. Econometrica 65 (5), 10591096.CrossRefGoogle Scholar
Chiarella, C. and Iori, G. (2003) A simulation of the microstructure of double auctions. Quantitative Finance 2, 346353.Google Scholar
Cont, R. and Bouchaud, J.P. (2000) Herd behavior and aggregate fluctuations in financial markets. Macroeconomic Dynamics 4, 170196.Google Scholar
Day, R. and Huang, W. (1990) Bulls, bears and the market sheep. Journal of Economic Behavior and Organization 14, 299329.Google Scholar
Driver, R.D. (1977) Ordinary and Delay Differential Equations. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Evstigneev, I., Hens, T., and Schenk-Hoppé, K.R. (2006) Evolutionary stable markets. Economic Theory 27, 449468.Google Scholar
Farmer, J.D., Patelli, P., and Zovko, I.I. (2005) The predictive power of zero intelligence in financial markets. Proceedings of the National Academy of Sciences of the United States of America 102, 22542259.CrossRefGoogle ScholarPubMed
Föllmer, H. and Schweizer, M. (1993) A microeconomic approach to diffusion models for stock prices. Mathematical Finance 3, 123.CrossRefGoogle Scholar
Föllmer, H., Horst, U., and Kirman, A. (2005) Equilibria in financial markets with heterogenous agents: A probabilistic perspective. Journal of Mathematical Economics 41, 123155.CrossRefGoogle Scholar
Frankel, J.A. and Froot, K. (1986) The dollar as an irrational speculative bubble: A tale of fundamentalists and chartists. Marcus Wallenberg Papers on International Finance 1, 2755.Google Scholar
Garman, M. (1976) Market microstructure. Journal of Financial Economics 3, 257275.CrossRefGoogle Scholar
Hobson, D.G. and Rogers, L.C.G. (1998) Complete models with stochastic volatility. Mathematical Finance 8, 2748.Google Scholar
Hommes, C. (2006) Heterogeneous agent models in economics and finance. In Tesfatsion, L. and Judd, K. (eds.), Handbook of Computational Economics, Volume 2, Agent-based Computational Economics. Amsterdam: Elsevier.Google Scholar
Horst, U. (2005) Financial price fluctuations in a stock market model with many interacting agents. Economic Theory 25, 917932.CrossRefGoogle Scholar
Horst, U. and Wenzelburger, J. (in press) On non-ergodic asset prices. Economic Theory.Google Scholar
Karatzas, I. and Shreve, S.E. (1991) Brownian Motion and Stochastic Calculus. Berlin: Springer-Verlag.Google Scholar
Kazmerchuk, Y. and Wu, J. (2004) Stochastic state-dependent delay differential equations with applications in finance. Functional Differential Equations 11, 7786.Google Scholar
Kruk, L. (2003) Functional limit theorems for a simple auction. Mathematics of Operations Research 28, 716751.CrossRefGoogle Scholar
Kurtz, T. (1978) Strong approximation theorems for density dependent Markov chains. Stochastic Processes and Their Applications 6, 223240.Google Scholar
LeBaron, B. (2006) Agent-based computational finance. In Tesfatsion, L. and Judd, K. (eds.) Handbook of Computational Economics, Volume 2, Agent-Based Computational Economics. Amsterdam: Elsevier.Google Scholar
Luckock, H. (2003) A steady-state model of continuous double auction. Quantitative Finance 3, 385404.Google Scholar
Lux, T. (1995) Herd behavior, bubbles and crashes. The Economic Journal 105, 881896.CrossRefGoogle Scholar
Lux, T. (1997) Time variation of second moments from a noise trader/infection model. Journal of Economic Dynamics and Control 22, 138.Google Scholar
Lux, T. and Marchesi, M. (2000) Volatility clustering in financial markets: A microsimulation of interacting agents. International Journal of Theoretical and Applied Finance 3, 675702.Google Scholar
Mandelbaum, A and Pats, G. (1998) State-dependent stochastic networks I: Approximations and applications with continuous diffusion limits. Annals of Applied Probability 8, 569646.Google Scholar
Mandelbaum, A. W. Massey, and Reiman, M. (1998) Strong approximations for Markovian service networks. Queueing Systems 30, 149201.Google Scholar
Mendelson, H. (1982) Market behavior in a clearing house. Econometrica 50, 15051524.Google Scholar
Potters, M. and Bouchaud, J.P. (2003) More statistical properties of order books and price impact. Physica A 324, 133144.Google Scholar
Smith, E., Farmer, J.D., Gillemot, L., and Krishnamurthy, S., H., (2003) Statistical theory of continuous double auction. Quantitative Finance 3, 481514.Google Scholar
Stoica, G. (2005) A stochastic delay financial model. Proceeding of the American Mathematical Society 133(6), 1837–1841.Google Scholar