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High-Frequency Simulations of an Order Book: a Two-scale Approach

  • Charles-Albert Lehalle
  • Olivier Guéant
  • Julien Razafinimanana
Part of the New Economic Windows book series (NEW)

Abstract

Models of market microstructure at the order book scale can be split into two families:
  • First, the agent-based models [5] aiming at simulating a large number of agents, each of them having its utility function or feedback rule. The philosophy of this kind of modelling is similar to Minsky’s paradigm in artificial intelligence in the eighties: build each agent so that if you stealthily replace, one by one, each real person interacting in the market with such a virtual ersatz, you will finally obtain a full synthetic replica of a real market. The actual limits faced by this research programme are: first, the difficulty to rationalise and quantify the utility function of real persons, and then the computational capabilities of today’s computers. Last but not least, the lack of analytical results of this fully non-parametric approach is also a problem for a lot of applications. It is, for instance, usually difficult to know how to choose the parameters of such models to reach a given intra-day volatility, given sizes of jumps, average bid-ask spread, etc.

  • Second, the “zero intelligence”; models [9] aiming at reproducing stylised facts (Epps effect on correlations, signature plot of volatility, order book shapes, etc.) using random number generators for time between orders, order sizes, prices, etc. This approach is more oriented to “knowledge extraction” from existing recordings than the agent-based one. Its focus on stylized facts and our capability to emulate them using as simple as possible generators is close to the usual definition of “knowledgerd (following for instance Kolmogorov or Shannon in terms of complexity reduction). It succeeds in identifying features like short-term memory, Epps effect on correlations, signature plots for high-frequency volatility estimates, dominance of power laws [25], and the general profile of market impact [11], among others, that are now part of the usual benchmarks to validate any microscopic market model. The limits of this approach are: first, the usual stationarity assumptions that are made, and the difficulty of linking the microscopic characteristics with macroscopic ones, for instance linking characteristics of the underlying probability distributions to market volatility (even if recent advances have been made in this direction using Hawkes models [2] or usual distributions [7]). The search for such links is motivated by the fact that as they are probability-based, their diffusive limits (or equivalent) should behave similarly to usual quantitative models on a large scale (for instance Levy processes [24]).

Keywords

Real Market Order Book Noise Trader Trading Algorithm Limit Order Book 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M. Avellaneda and S. Stoikov. High-frequency trading in a limit order book. Quantitative Finance, 8(3):217–224, 2008CrossRefGoogle Scholar
  2. 2.
    E. Bacry, S. Delattre, M. Hoffman and J. F. Muzy. Deux modèles de bruit de microstructure et leur inférence statistique. Préesentation, 2009Google Scholar
  3. 3.
    B. Bouchard, N.-M. Dang and C.-A. Lehalle. Optimal control of trading algorithms: a general impulse control approach. Technical report, 2009Google Scholar
  4. 4.
    J.P. Bouchaud, M. Mezard and M. Potters. Statistical properties of stock order books: empirical results and models. 2002Google Scholar
  5. 5.
    A. Chakraborti, I. Muni Toke, M. Patriarca and F. Abergel. Econophysics: Empirical facts and agent-based models. 2009Google Scholar
  6. 6.
    K.J. Cohen, S.F. Maier, R.A. Schwartz and D.K. Whitcomb. Transaction costs, order placement strategy, and existence of the bid-ask spread. The Journal of Political Economy, 89(2):287–305, 1981CrossRefGoogle Scholar
  7. 7.
    R. Cont and A. de Larrard. Stochastic modeling of order books: high-frequency dynamics and diffusion limits. In 28th European Meeting of Statisticians, 2010Google Scholar
  8. 8.
    M.G. Daniels, D.J. Farmer, L. Gillemot, G. Iori and E. Smith. Quantitative model of price diffusion and market friction based on trading as a mechanistic random process. Physical Review Letters, 90(10):108102+, 2003CrossRefGoogle Scholar
  9. 9.
    Z. Eisler, J. Kertesz, F. Lillo and R.N. Mantegna. Diffusive behavior and the modeling of characteristic times in limit order executions. Social Science Research Network Working Paper Series, 2007Google Scholar
  10. 10.
    T. Foucault, O. Kadan and E. Kandel. Limit order book as a market for liquidity. Review of Financial Studies, 18(4), 2005Google Scholar
  11. 11.
    J. Gatheral. No-dynamic-arbitrage and market impact. Social Science Research Network Working Paper Series, 2008Google Scholar
  12. 12.
    M. González and M. Gualdani. Asymptotics for a symmetric equation in price formation. Applied Mathematics and Optimization, 59(2):233–246, 2009CrossRefGoogle Scholar
  13. 13.
    M. González and M. Gualdani. Asymptotics for a free-boundary problem in price formation. To appearGoogle Scholar
  14. 14.
    G. Gu, W. Chen and W. Zhou. Empirical shape function of limit-order books in the chinese stock market. Physica A: Statistical Mechanics and its Applications, 387(21):5182–5188, 2008CrossRefGoogle Scholar
  15. 15.
    A.G. Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1):83–90, 1971CrossRefGoogle Scholar
  16. 16.
    B. Jovanovic and A.J. Menkveld. Middlemen in Limit-Order Markets. SSRN eLibrary, 2010Google Scholar
  17. 17.
    J. Large. Measuring the resiliency of an electronic limit order book. Journal of Financial Markets, 10(1):1–25, 2007CrossRefGoogle Scholar
  18. 18.
    J.-M. Lasry and P.-L. Lions. Mean field games. Japanese Journal of Mathematics, 2(1):229–260, 2007CrossRefGoogle Scholar
  19. 19.
    P.A. Markowich, N. Matevosyan, J.F. Pietschmann and M.T. Wolfram. On a parabolic free boundary equation modeling price formation. Mathematical Models and Methods in Applied Sciences, 11(19):1929–1957, 2209Google Scholar
  20. 20.
    Official Journal of the European Union. Directive 2004/39/ec of the european parliament. Directive 2004/39/EC of the European Parliament, 2004Google Scholar
  21. 21.
    G. Pagés, S. Laruelle and C.-A. Lehalle. Optimal split of orders across liquidity pools: a stochatic algorithm approach. Technical report, 2009Google Scholar
  22. 22.
    H. Pham. Continuous-time Stochastic Control and Optimization with Financial Applications (Stochastic Modelling and Applied Probability). Springer, 1 edition, 2009Google Scholar
  23. 23.
    F.P. Schoenberg. On rescaled poisson processes and the brownian bridge. Annals of the Institute of Statistical Mathematics, 54(2):445–457, 2002CrossRefGoogle Scholar
  24. 24.
    A.N. Shiryaev. Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific Publishing Company, 1st edition, 1999Google Scholar
  25. 25.
    M. Wyart, J.-P. Bouchaud, J. Kockelkoren, M. Potters and M. Vettorazzo. Relation between bid-ask spread, impact and volatility in double auction markets. Technical report, 2006Google Scholar

Copyright information

© Springer-Verlag Italia 2011

Authors and Affiliations

  • Charles-Albert Lehalle
    • 1
  • Olivier Guéant
    • 2
  • Julien Razafinimanana
    • 3
  1. 1.Crédit Agricole CheuvreuxCourbevoieFrance
  2. 2.MFG R&DParisFrance
  3. 3.Crédit Agricole Quantitative ResearchCourbevoieFrance

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