Finance and Stochastics

, Volume 22, Issue 2, pp 241–280 | Cite as

The microstructural foundations of leverage effect and rough volatility

  • Omar El Euch
  • Masaaki Fukasawa
  • Mathieu Rosenbaum


We show that typical behaviors of market participants at the high frequency scale generate leverage effect and rough volatility. To do so, we build a simple microscopic model for the price of an asset based on Hawkes processes. We encode in this model some of the main features of market microstructure in the context of high frequency trading: high degree of endogeneity of market, no-arbitrage property, buying/selling asymmetry and presence of metaorders. We prove that when the first three of these stylized facts are considered within the framework of our microscopic model, it behaves in the long run as a Heston stochastic volatility model, where a leverage effect is generated. Adding the last property enables us to obtain a rough Heston model in the limit, exhibiting both leverage effect and rough volatility. Hence we show that at least part of the foundations of leverage effect and rough volatility can be found in the microstructure of the asset.


Market microstructure High frequency trading Leverage effect Rough volatility Hawkes processes Limit theorems Heston model Rough Heston model 

Mathematics Subject Classification (2010)

60F17 60G55 91G70 

JEL Classification

C58 G10 G14 



We thank Neil Shephard for inspiring discussions and Jim Gatheral and Kasper Larsen for very relevant comments. Omar El Euch and Mathieu Rosenbaum gratefully acknowledge the financial support of the ERC Grant 679836 Staqamof and the Chair Analytics and Models for Regulation.


  1. 1.
    Abergel, F., Lehalle, C.A., Rosenbaum, M.: Understanding the stakes of high-frequency trading. J. Trading 9(4), 49–73 (2014) CrossRefGoogle Scholar
  2. 2.
    Aït-Sahalia, Y., Cacho-Diaz, J., Laeven, R.J.: Modeling financial contagion using mutually exciting jump processes. J. Financ. Econ. 117, 585–606 (2015) CrossRefGoogle Scholar
  3. 3.
    Aït-Sahalia, Y., Fan, J., Li, Y.: The leverage effect puzzle: disentangling sources of bias at high frequency. J. Financ. Econ. 109, 224–249 (2013) CrossRefGoogle Scholar
  4. 4.
    Almgren, R., Chriss, N.: Optimal execution of portfolio transactions. J. Risk 3, 5–40 (2001) CrossRefGoogle Scholar
  5. 5.
    Bacry, E., Dayri, K., Muzy, J.F.: Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data. Eur. Phys. J. B 85(5), 1–12 (2012) CrossRefGoogle Scholar
  6. 6.
    Bacry, E., Delattre, S., Hoffmann, M., Muzy, J.F.: Modelling microstructure noise with mutually exciting point processes. Quant. Finance 13, 65–77 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bacry, E., Delattre, S., Hoffmann, M., Muzy, J.F.: Some limit theorems for Hawkes processes and application to financial statistics. Stoch. Process. Appl. 123, 2475–2499 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bacry, E., Jaisson, T., Muzy, J.F.: Estimation of slowly decreasing Hawkes kernels: application to high frequency order book modelling. Quant. Finance 16, 1179–1201 (2016) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bacry, E., Mastromatteo, I., Muzy, J.F.: Hawkes processes in finance. Mark. Microstruct. Liq. 01, 1550005 (2015) CrossRefGoogle Scholar
  10. 10.
    Bauwens, L., Hautsch, N.: Dynamic latent factor models for intensity processes. CORE Discussion Paper No. 2003/103 (2004). Available online at
  11. 11.
    Bayer, C., Friz, P., Gatheral, J.: Pricing under rough volatility. Quant. Finance 16, 887–904 (2016) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bekaert, G., Wu, G.: Asymmetric volatility and risk in equity markets. Rev. Financ. Stud. 13, 1–42 (2000) CrossRefGoogle Scholar
  13. 13.
    Black, F.: Studies of stock price volatility changes. In: Proceedings of the 1976 Meeting of the Business and Economic Statistics Section, pp. 171–181. American Statistical Association, Washington D.C. (1976) Google Scholar
  14. 14.
    Bollerslev, T., Chou, R.Y., Kroner, K.F.: ARCH modeling in finance: a review of the theory and empirical evidence. J. Econom. 52, 5–59 (1992) CrossRefzbMATHGoogle Scholar
  15. 15.
    Bollerslev, T., Litvinova, J., Tauchen, G.: Leverage and volatility feedback effects in high-frequency data. J. Financ. Econom. 4, 353–384 (2006) CrossRefGoogle Scholar
  16. 16.
    Bowsher, C.G.: Modelling security market events in continuous time: intensity based, multivariate point process models. J. Econom. 141, 876–912 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Brennan, M.J., Chordia, T., Subrahmanyam, A., Tong, Q.: Sell-order liquidity and the cross-section of expected stock returns. J. Financ. Econ. 105, 523–541 (2012) CrossRefGoogle Scholar
  18. 18.
    Brunnermeier, M.K., Pedersen, L.H.: Market liquidity and funding liquidity. Rev. Financ. Stud. 22, 2201–2238 (2009) CrossRefGoogle Scholar
  19. 19.
    Campbell, J.Y., Hentschel, L.: No news is good news: an asymmetric model of changing volatility in stock returns. J. Financ. Econ. 31, 281–318 (1992) CrossRefGoogle Scholar
  20. 20.
    Chavez-Demoulin, V., Davison, A.C., McNeil, A.J.: A point process approach to value-at-risk estimation. Quant. Finance 5, 227–234 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Christie, A.A.: The stochastic behavior of common stock variances: value, leverage and interest rate effects. J. Financ. Econ. 10, 407–432 (1982) CrossRefGoogle Scholar
  22. 22.
    Corradi, V.: Reconsidering the continuous time limit of the \(\mathrm {GARCH}(1, 1)\) process. J. Econom. 96, 145–153 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Dayri, K., Rosenbaum, M.: Large tick assets: implicit spread and optimal tick size. Mark. Microstruct. Liq. 01, 1550003 (2015) CrossRefGoogle Scholar
  24. 24.
    Ding, Z., Granger, C.W., Engle, R.F.: A long memory property of stock market returns and a new model. J. Empir. Finance 1, 83–106 (1993) CrossRefGoogle Scholar
  25. 25.
    Duan, J.C.: Augmented \(\mathrm{GARCH}(p, q)\) process and its diffusion limit. J. Econom. 79, 97–127 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Embrechts, P., Liniger, T., Lin, L.: Multivariate Hawkes processes: an application to financial data. J. Appl. Probab. 48, 367–378 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Engle, R.F., Ng, V.K.: Measuring and testing the impact of news on volatility. J. Finance 48, 1749–1778 (1993) CrossRefGoogle Scholar
  28. 28.
    Errais, E., Giesecke, K., Goldberg, L.R.: Affine point processes and portfolio credit risk. SIAM J. Financ. Math. 1, 642–665 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Figlewski, S., Wang, X.: Is the leverage effect a leverage effect? Preprint (2000). Available online at:
  30. 30.
    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, 1293–1314 (2015) MathSciNetCrossRefGoogle Scholar
  31. 31.
    French, K.R., Schwert, G.W., Stambaugh, R.F.: Expected stock returns and volatility. J. Financ. Econ. 19, 3–29 (1987) CrossRefGoogle Scholar
  32. 32.
    Gatheral, J., Jaisson, T., Rosenbaum, M.: Volatility is rough. Quant. Finance (2018 forthcoming). Available online at: arXiv:1410.3394
  33. 33.
    Guennoun, H., Jacquier, A., Roome, P., Shi, F.: Asymptotic behaviour of the fractional Heston model. Preprint (2014). Available online at:
  34. 34.
    Hagan, P.S., Kumar, D., Lesniewski, A.S., Woodward, D.E.: Managing smile risk. Wilmott Mag. 1, 84–108 (2002) Google Scholar
  35. 35.
    Hardiman, S.J., Bercot, N., Bouchaud, J.P.: Critical reflexivity in financial markets: a Hawkes process analysis. Eur. Phys. J. B 86(10), 1–9 (2013) MathSciNetCrossRefGoogle Scholar
  36. 36.
    Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag-Leffler functions and their applications. J. Appl. Math. 2011, 298628 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Hawkes, A.G.: Point spectra of some mutually exciting point processes. J. R. Stat. Soc. B 33, 438–443 (1971) MathSciNetzbMATHGoogle Scholar
  38. 38.
    Hawkes, A.G., Oakes, D.: A cluster process representation of a self-exciting process. J. Appl. Probab. 11, 493–503 (1974) MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Hendershott, T., Seasholes, M.S.: Market maker inventories and stock prices. Am. Econ. Rev. 97, 210–214 (2007) CrossRefGoogle Scholar
  40. 40.
    Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6, 327–343 (1993) CrossRefGoogle Scholar
  41. 41.
    Ho, T., Stoll, H.R.: Optimal dealer pricing under transactions and return uncertainty. J. Financ. Econ. 9, 47–73 (1981) CrossRefGoogle Scholar
  42. 42.
    Hull, J., White, A.: The pricing of options on assets with stochastic volatilities. J. Finance 42, 281–300 (1987) CrossRefzbMATHGoogle Scholar
  43. 43.
    Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2013) zbMATHGoogle Scholar
  44. 44.
    Jaisson, T., Rosenbaum, M.: Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Probab. 25, 600–631 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Jaisson, T., Rosenbaum, M., et al.: Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes. Ann. Appl. Probab. 26, 2860–2882 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Jakubowski, A., Mémin, J., Pagès, G.: Convergence en loi des suites d’intégrales stochastiques sur l’espace 1 de Skorokhod. Probab. Theory Relat. Fields 81, 111–137 (1989) CrossRefzbMATHGoogle Scholar
  47. 47.
    Kurtz, T.G., Protter, P.: Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19, 1035–1070 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Lehalle, C.A., Laruelle, S.: Market Microstructure in Practice. World Scientific, Singapore (2013) CrossRefGoogle Scholar
  49. 49.
    Lindner, A.M.: Continuous time approximations to GARCH and stochastic volatility models. In: Mikosch, T., et al. (eds.) Handbook of Financial Time Series, pp. 481–496. Springer, Berlin (2009) CrossRefGoogle Scholar
  50. 50.
    Mainardi, F.: On some properties of the Mittag-Leffler function. Preprint (2014). Available online at: arXiv:1305.0161
  51. 51.
    Mathai, A.M., Haubold, H.J.: Special Functions for Applied Scientists, vol. 4. Springer, Berlin (2008) CrossRefzbMATHGoogle Scholar
  52. 52.
    Mytnik, L., Salisbury, T.S.: Uniqueness for Volterra-type stochastic integral equations. Preprint (2015). Available online at: arXiv:1502.05513
  53. 53.
    Nelson, D.B.: ARCH models as diffusion approximations. J. Econom. 45, 7–38 (1990) MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Nelson, D.B.: Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–370 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999) CrossRefzbMATHGoogle Scholar
  56. 56.
    Rodríguez, M.J., Ruiz, E.: Revisiting several popular GARCH models with leverage effect: differences and similarities. J. Financ. Econom. 10, 637–668 (2012) CrossRefGoogle Scholar
  57. 57.
    Stein, E.M., Stein, J.C.: Stock price distributions with stochastic volatility: an analytic approach. Rev. Financ. Stud. 4, 727–752 (1991) CrossRefGoogle Scholar
  58. 58.
    Tayal, R., Thomas, S.: Measuring and explaining the asymmetry of liquidity. Preprint (2012). Available online at:
  59. 59.
    Veraar, M.: The stochastic Fubini theorem revisited. Stoch. Int. J. Probab. Stoch. Process. 84, 543–551 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Wang, C.D., Mykland, P.A.: The estimation of leverage effect with high-frequency data. J. Am. Stat. Assoc. 109, 197–215 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Wu, G., Xiao, Z.: A generalized partially linear model of asymmetric volatility. J. Empir. Finance 9, 287–319 (2002) CrossRefGoogle Scholar
  62. 62.
    Zakoian, J.M.: Threshold heteroskedastic models. J. Econ. Dyn. Control 18, 931–955 (1994) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Omar El Euch
    • 1
  • Masaaki Fukasawa
    • 2
  • Mathieu Rosenbaum
    • 1
  1. 1.CMAPÉcole PolytechniquePalaiseau CedexFrance
  2. 2.Graduate School of Engineering ScienceOsaka UniversityToyonaka, OsakaJapan

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