The Annals of Applied Probability

Perfect hedging in rough Heston models

Omar El Euch and Mathieu Rosenbaum

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Rough volatility models are known to reproduce the behavior of historical volatility data while at the same time fitting the volatility surface remarkably well, with very few parameters. However, managing the risks of derivatives under rough volatility can be intricate since the dynamics involve fractional Brownian motion. We show in this paper that surprisingly enough, explicit hedging strategies can be obtained in the case of rough Heston models. The replicating portfolios contain the underlying asset and the forward variance curve, and lead to perfect hedging (at least theoretically). From a probabilistic point of view, our study enables us to disentangle the infinite-dimensional Markovian structure associated to rough volatility models.

Article information

Ann. Appl. Probab., Volume 28, Number 6 (2018), 3813-3856.

Received: March 2017
Revised: June 2018
First available in Project Euclid: 8 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 91G20: Derivative securities 26A33: Fractional derivatives and integrals 60G22: Fractional processes, including fractional Brownian motion 60J25: Continuous-time Markov processes on general state spaces

Rough volatility rough Heston model Hawkes processes fractional Brownian motion fractional Riccati equations limit theorems forward variance curve


El Euch, Omar; Rosenbaum, Mathieu. Perfect hedging in rough Heston models. Ann. Appl. Probab. 28 (2018), no. 6, 3813--3856. doi:10.1214/18-AAP1408.

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  • [1] Abi Jaber, E. and Pulido, S. (2017). Affine Volterra processes. Working paper.
  • [2] Andersen, L. B. G. and Piterbarg, V. V. (2007). Moment explosions in stochastic volatility models. Finance Stoch. 11 29–50.
  • [3] Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J. F. (2013). Some limit theorems for Hawkes processes and application to financial statistics. Stochastic Process. Appl. 123 2475–2499.
  • [4] Bayer, C., Friz, P. and Gatheral, J. (2016). Pricing under rough volatility. Quant. Finance 16 887–904.
  • [5] Bennedsen, M., Lunde, A. and Pakkanen, M. S. (2017). Hybrid scheme for Brownian semistationary processes. Finance Stoch. 21 931–965.
  • [6] Bergomi, L. (2005). Smile dynamics II. Risk 18 67–73.
  • [7] Carr, P. and Madan, D. (1999). Option valuation using the fast Fourier transform. J. Comput. Finance 2 61–73.
  • [8] Cuchiero, C. and Teichmann, J. (2017). An affine view on rough variance models. Working paper.
  • [9] Dwass, M. (1969). The total progeny in a branching process and a related random walk. J. Appl. Probab. 6 682–686.
  • [10] El Euch, O. and Rosenbaum, M. (2016). The characteristic function of rough Heston models. Math. Finance. To appear.
  • [11] Fukasawa, M. (2011). Asymptotic analysis for stochastic volatility: Martingale expansion. Finance Stoch. 15 635–654.
  • [12] Gatheral, J., Jaisson, T. and Rosenbaum, M. (2018). Volatility is rough. Quant. Finance 18 933–949.
  • [13] Guennoun, H., Jacquier, A. and Roome, P. (2014). Asymptotic behaviour of the fractional Heston model. Available at SSRN 2531468.
  • [14] Haubold, H. J., Mathai, A. M. and Saxena, R. K. (2011). Mittag-Leffler functions and their applications. J. Appl. Math. 2011 Art. ID 298628, 51.
  • [15] Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6 327–343.
  • [16] Jaisson, T. and Rosenbaum, M. (2016). Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes. Ann. Appl. Probab. 26 2860–2882.
  • [17] Mainardi, F. (2014). On some properties of the Mittag-Leffler function $E_{\alpha}(-t^{\alpha})$, completely monotone for $t>0$ with $0<\alpha<1$. Discrete Contin. Dyn. Syst. Ser. B 19 2267–2278.
  • [18] Mathai, A. M. and Haubold, H. J. (2008). Special Functions for Applied Scientists. Springer, Berlin.
  • [19] Neuenkirch, A. and Shalaiko, T. (2016). The order barrier for strong approximation of rough volatility models. Preprint. Available at arXiv:1606.03854.
  • [20] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Yverdon. Edited and with a foreword by S. M. Nikol’skiĭ, Translated from the 1987 Russian original, Revised by the authors.