Mathematics and Financial Economics

, Volume 7, Issue 3, pp 359–403 | Cite as

Optimal posting price of limit orders: learning by trading

  • Sophie LaruelleEmail author
  • Charles-Albert Lehalle
  • Gilles Pagès


We model a trader interacting with a continuous market as an iterative algorithm that adjusts limit prices at a given rhythm and propose a procedure to minimize trading costs. We prove the \(a.s.\) convergence of the algorithm under assumptions on the cost function and give some practical criteria on model parameters to ensure that the conditions to use the algorithm are met (notably, using the co-monotony principle). We illustrate our results with numerical experiments on both simulated and market data.


Stochastic approximation Order book Limit order  Market impact Statistical learning High-frequency optimal liquidation Poisson process  Co-monotony principle 

Mathematics Subject Classification (2000)

62L20 62P05 60G55 65C05 


  1. 1.
    Abergel, F., Jedidi, A.: A mathematical approach to order book modelling. In: Abergel, F., Chakrabarti, B.K., Chakraborti, A., Mitra, M. (eds.) Econophysics of Order Driven Markets. Springer, New York (2011)CrossRefGoogle Scholar
  2. 2.
    Alfonsi, A., Fruth, A., Schied, A.: Optimal execution strategies in limit order books with general shape functions. Quant. Financ. 10(2), 143–157 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Almgren, R.F., Chriss, N.: Optimal execution of portfolio transactions. J. Risk 3(2), 5–39 (2000)Google Scholar
  4. 4.
    Avellaneda, M., Stoikov, S.: High-frequency trading in a limit order book. Quant. Financ. 8(3), 217–224 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bayraktar, E., Ludkovski, M.: Liquidation in limit order books with controlled intensity. CoRR (2011)Google Scholar
  6. 6.
    Beskos, A., Roberts, G.O.: Exact simulation of diffusions. Ann. Appl. Prob. 15(4), 2422–2444 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bouchard, B., Dang, N.-M., Lehalle, C.-A.: Optimal control of trading algorithms: a general impulse control approach. SIAM J. Financ. Math. 2, 404–438 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Duflo, M.: Algorithmes Stochastiques, vol. 23 of Mathématiques & Applications [Mathematics & Applications]. Springer, Berlin (1996)Google Scholar
  9. 9.
    Foucault, T., Kadan, O., Kandel, E.: Limit order book as a market for liquidity. Discussion Paper Series dp321, Center for Rationality and Interactive Decision Theory, Hebrew University, Jerusalem, Jan 2003Google Scholar
  10. 10.
    Guéant, O., Lehalle, C.-A., Razafinimanana, J.: High frequency simulations of an order book: a two-scales approach. In: Abergel, F., Chakrabarti, B.K., Chakraborti, A., Mitra, M. (eds.) Econophysics of Order-Driven Markets. New Economic Windows. Springer, Milan (2010)Google Scholar
  11. 11.
    Guéant, O., Fernandez-Tapia, J., Lehalle, C.-A.: Dealing with the inventory risk. Technical report (2011)Google Scholar
  12. 12.
    Guilbaud, F., Pham. H.: Optimal high-frequency trading with limit and market orders. Quant. Finac. to appear (2012)Google Scholar
  13. 13.
    Guilbaud, F., Mnif, M., Pham, H.: Numerical methods for an optimal order execution problem. J. Comput. Finan., to appear (2010)Google Scholar
  14. 14.
    Ho, T., Stoll, H.R.: Optimal dealer pricing under transactions and return uncertainty. J. Financ. Econ. 9(1), 47–73 (1981)CrossRefGoogle Scholar
  15. 15.
    Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes, vol. 288, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2003)Google Scholar
  16. 16.
    Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus, vol. 113, 2nd edn. Graduate Texts in Mathematics. Springer, New York (1991)Google Scholar
  17. 17.
    Karlin, S., Taylor, H.M.: A Second Course in Stochastic Processes. Academic, New York (1981)zbMATHGoogle Scholar
  18. 18.
    Kushner, H.J., Clark, D.S.: Stochastic Approximation Methods for Constrained and Unconstrained Systems, vol. 26 of Applied Mathematical Sciences. Springer, New York (1978)Google Scholar
  19. 19.
    Kushner, H.J., Yin, G.G.: Stochastic Approximation and Recursive Algorithms and Applications, vol. 35 of Applications of Mathematics. Stochastic Modelling and Applied Probability, 2nd edn. Springer, New York (2003)Google Scholar
  20. 20.
    Laruelle, S., Pagès, G.: Stochastic approximation with averaging innovation applied to finance. Monte Carlo Methods Appl. 18(1), 1–51 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Laruelle, S., Lehalle, C.-A., Pagès, G.: Optimal split of orders across liquidity pools: a stochastic algorithm approach. SIAM J. Finan. Math. 2(1), 1042–1076 (2011)zbMATHCrossRefGoogle Scholar
  22. 22.
    McCulloch, J.: A model of true spreads on limit order markets (2011). SSRN:
  23. 23.
    Pagès, G.: A functional co-monotony principle with an application to peacoks. Pre-pub LPMA n\(^\circ \)1536. To appear in sèmin. Proab. XLV (2010)Google Scholar
  24. 24.
    Predoiu, S., Shaikhet, G., Shreve, S.: Optimal Execution of a General One-Sided Limit-Order Book. Technical Report. Carnegie Mellon University, Pittsburgh (2010)Google Scholar
  25. 25.
    Robert, C.Y., Rosenbaum, M.: A new approach for the dynamics of ultra high frequency data: the model with uncertainty zones. J. Finan. Econ. 9(2), 344–366 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Sophie Laruelle
    • 1
    Email author
  • Charles-Albert Lehalle
    • 2
  • Gilles Pagès
    • 3
  1. 1.Laboratoire de Mathématiques Appliquées aux SystèmesChâtenay-MalabryFrance
  2. 2.Crédit Agricole Cheuvreux, CALYON GroupParis La DéfenseFrance
  3. 3.Laboratoire de Probabilités et Modèles AléatoiresParis Cedex 5France

Personalised recommendations