A switching microstructure model for stock prices

  • Donatien HainautEmail author
  • Stephane Goutte


This article proposes a microstructure model for stock prices in which parameters are modulated by a Markov chain determining the market behaviour. In this approach, called the switching microstructure model (SMM), the stock price is the result of the balance between the supply and the demand for shares. The arrivals of bid and ask orders are represented by two mutually- and self-excited processes. The intensities of these processes converge to a mean reversion level that depends upon the regime of the Markov chain. The first part of this work studies the mathematical properties of the SMM. The second part focuses on the econometric estimation of parameters. For this purpose, we combine a particle filter with a Markov chain Monte Carlo algorithm. Finally, we calibrate the SMM with two and three regimes to daily returns of the S&P 500 and compare them with a non switching model.


Hawkes process Switching process Microstructure 

JEL Classifications

C58 G1 



We thank for its support the Chair “Data Analytics and Models for insurance” of BNP Paribas Cardif, hosted by ISFA (Université Claude Bernard, Lyon France). We also thank the two anonymous referees and the editor, Ulrich Horst, for their recommandations.


  1. 1.
    Ait-Sahalia, Y., Cacho-Diaz, J., Laeven, R.J.A.: Modeling financial contagion using mutually exciting jump processes. J. Financ. Econ. 117(3), 586–606 (2015)CrossRefGoogle Scholar
  2. 2.
    Al-Anaswah, N., Wilfing, B.: Identification of speculative bubbles using state-space models with Markov-switching. J. Bank. Finance 35(5), 1073–1086 (2011)CrossRefGoogle Scholar
  3. 3.
    Bacry, E., Delattre, S., Hoffmann, M., Muzy, J.F.: Modelling microstructure noise with mutually exciting point processes. Quant. Finance 13(1), 65–77 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bacry, E., Delattre, S., Hoffmann, M., Muzy, J.F.: Scaling limits for Hawkes processes and application to financial statistics. Stoch. Process. Appl. 123(7), 2475–2499 (2013)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bacry, E., Muzy, J.F.: Hawkes model for price and trades high-frequency dynamics. Quant. Finance 14(7), 1147–1166 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bacry, E., Mastromatteo, I., Muzy, J.F.: Hawkes processes in finance. Mark. Microstruct. Liq. 1(1), 1–59 (2015)CrossRefGoogle Scholar
  7. 7.
    Bacry, E., Muzy, J.F.: Second order statistics characterization of Hawkes processes and non-parametric estimation. IEEE Trans. Inf. Theory 62(4), 2184–2202 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bormetti, G., Calcagnile, L.M., Treccani, M., Corsi, F., Marmi, S., Lillo, F.: Modelling systemic price cojumps with Hawkes factor models. Quant. Finance 15(7), 1137–1156 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bouchaud, J.P.: Price impact. In: Cont, R. (ed.) Encyclopedia of Quantitative Finance. Wiley, Hoboken (2010)Google Scholar
  10. 10.
    Bouchaud, J.P., Farmer, J.D., Lillo, F.: How markets slowly diggest changes in supply and demand. In: Hens, T., Reiner, K., Schenk-Hoppé. (eds.) Handbook of Financial Markets. Elsevier, New York (2009)Google Scholar
  11. 11.
    Bowsher, C.G.: Modelling security markets in continuous time: intensity based, multivariate point process models. Economics Discussion Paper No. 2002- W22, Nuffield College, Oxford (2002)Google Scholar
  12. 12.
    Branger, N., Kraft, H., Meinerding, C.: Partial information about contagion risk, self-exciting processes and portfolio optimization. J. Econ. Dyn. Control 39, 18–36 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chavez-Demoulin, V., McGill, J.A.: High-frequency financial data modeling using Hawkes processes. J. Bank. Finance 36, 3415–3426 (2012)CrossRefGoogle Scholar
  14. 14.
    Cont, R., Kukanov, A., Stoikov, S.: The price impact of order book events. J. Financ. Econ. 12(1), 47–88 (2013)Google Scholar
  15. 15.
    Da Fonseca, J., Zaatour, R.: Hawkes process: fast calibration, application to trade clustering, and diffusive limit. J. Futures Mark. 34(6), 548–579 (2014)Google Scholar
  16. 16.
    Doucet, A., Godsill, S., Andrieu, C.: On sequential Monte Carlo sampling methods for Bayesian filtering. Stat. Comput. 10, 197–208 (2000)CrossRefGoogle Scholar
  17. 17.
    Errais, E., Giesecke, K., Goldberg, L.: Affine point processes and portfolio credit risk. SIAM J. Financ. Math. 1, 642–665 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Filimonov, V., Sornette, D.: Apparent criticality and calibration issues in the Hawkes self-excited point process model: application to high-frequency financial data. Quant. Finance 15(8), 1293–1314 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gatumel, M., Ielpo, F.: The number of regimes across asset returns: identification and economic value. Int. J. Theor. Appl. Finance 17(06), 25 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Guidolin, M., Timmermann, A.: Economic implications of bull and bear regimes in UK stock and bond returns. Econ. J. 115, 11–143 (2005)CrossRefGoogle Scholar
  21. 21.
    Guidolin, M., Timmermann, A.: International asset allocation under regime switching, skew, and kurtosis preferences. Rev. Financ. Stud. 21(2), 889–935 (2008)CrossRefGoogle Scholar
  22. 22.
    Hainaut, D.: A model for interest rates with clustering effects. Quant. Finance 16(8), 1203–1218 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hainaut, D.: A bivariate Hawkes process for interest rate modeling. Econ. Model. 57, 180–196 (2016)CrossRefGoogle Scholar
  24. 24.
    Hainaut, D.: Clustered Lévy processes and their financial applications. J. Comput. Appl. Math. 319, 117–140 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hainaut, D., MacGilchrist, R.: Strategic asset allocation with switching dependence. Ann. Finance 8(1), 75–96 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hardiman, S.J., Bouchaud, J.P.: Branching ratio approximation for the self-exciting Hawkes process. Phys. Rev. E 90(6), 628071–628076 (2014)CrossRefGoogle Scholar
  27. 27.
    Hautsch, N.: Modelling Irregularly Spaced Financial Data. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  28. 28.
    Hawkes, A.: Point sprectra of some mutually exciting point processes. J. R. Stat. Soc. Ser. B 33, 438–443 (1971)zbMATHGoogle Scholar
  29. 29.
    Hawkes, A.: Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hawkes, A., Oakes, D.: A cluster representation of a self-exciting process. J. Appl. Probab. 11, 493–503 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Horst, U., Paulsen, M.: A law of large numbers for limit order books. Math. Oper. Res. (2017).
  32. 32.
    Jaisson, T., Rosenbaum, M.: Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Probab. 25(2), 600–631 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kelly, F., Yudovina, E.: A Markov model of a limit order book: thresholds, recurrence, and trading strategies. Math. Oper. Res. (2017).
  34. 34.
    Kyle, A.S.: Continuous auction and insider trading. Econometrica 53, 1315–1335 (1985)CrossRefzbMATHGoogle Scholar
  35. 35.
    Large, J.: Measuring the resiliency of an electronic limit order book. Working Paper, All Souls College, University of Oxford (2005)Google Scholar
  36. 36.
    Lee, K., Seo, B.K.: Modeling microstructure price dynamics with symmetric Hawkes and diffusion model using ultra-high-frequency stock data. J. Econ. Dyn. Control 79, 154–183 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Protter, P.E.: Stochastic Integration and Differential Equations. Springer, Berlin (2004)zbMATHGoogle Scholar
  38. 38.
    Wang, T., Bebbington, M., Harte, D.: Markov-modulated Hawkes process with stepwise decay. Ann. Inst. Stat. Math. 64, 521–544 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Statistics, Bio-statistics and Actuarial Science (ISBA)Université Catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Laboratoire d’Economie Dionysien (LED)Université Paris 8 and Paris School of Business, PSBSaint-Denis CedexFrance

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