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From microscopic price dynamics to multidimensional rough volatility models

Part of: Applications Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2021

Mathieu Rosenbaum  and
Mehdi Tomas
Mathieu Rosenbaum*
Affiliation:
École Polytechnique
Mehdi Tomas*
Affiliation:
École Polytechnique
*
*Postal address: CMAP, École Polytechnique, Route de Saclay, 91120 Palaiseau, France.
**Postal address: CMAP & LadHyX, École Polytechnique, Route de Saclay, 91120 Palaiseau, France. Email address: [email protected]
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Abstract

Rough volatility is a well-established statistical stylized fact of financial assets. This property has led to the design and analysis of various new rough stochastic volatility models. However, most of these developments have been carried out in the mono-asset case. In this work, we show that some specific multivariate rough volatility models arise naturally from microstructural properties of the joint dynamics of asset prices. To do so, we use Hawkes processes to build microscopic models that accurately reproduce high-frequency cross-asset interactions and investigate their long-term scaling limits. We emphasize the relevance of our approach by providing insights on the role of microscopic features such as momentum and mean-reversion in the multidimensional price formation process. In particular, we recover classical properties of high-dimensional stock correlation matrices.

Keywords

Rough volatilitymultidimensional processesmicrostructureHawkes processeslimit theoremshigh-dimensional correlation matrices

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60F17: Functional limit theorems; invariance principles 60G55: Point processes 62P05: Applications to actuarial sciences and financial mathematics
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 2 , June 2021 , pp. 425 - 462
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J.-F. (2013). Modelling microstructure noise with mutually exciting point processes. Quant. Finance 13, 65–77.CrossRefGoogle Scholar
Bacry, E., Mastromatteo, I. and Muzy, J.-F. (2015). Hawkes processes in finance. Market Microstructure Liquidity 1, 1550005.CrossRefGoogle Scholar
Bayer, C., Friz, P. and Gatheral, J. (2016). Pricing under rough volatility. Quant. Finance 16, 887–904.CrossRefGoogle Scholar
Benzaquen, M., Mastromatteo, I., Eisler, Z. and Bouchaud, J.-P. (2017). Dissecting cross-impact on stock markets: an empirical analysis. J. Statist. Mech. Theory Experiment 2017, 23406.CrossRefGoogle Scholar
Cuchiero, C. and Teichmann, J. (2019). Markovian lifts of positive semidefinite affine Volterra-type processes. Decisions Econom. Finance 42, 407–448.CrossRefGoogle Scholar
Da Fonseca, J. and Zhang, W. (2019). Volatility of volatility is (also) rough. J. Futures Markets 39, 600–611.CrossRefGoogle Scholar
Dandapani, A., Jusselin, P. and Rosenbaum, M. (2019). From quadratic Hawkes processes to super-Heston rough volatility models with Zumbach effect. Preprint. Available at https://arxiv.org/abs/1907.06151.Google Scholar
El Euch, O., Fukasawa, M. and Rosenbaum, M. (2018). The microstructural foundations of leverage effect and rough volatility. Finance Stoch. 22, 241–280.CrossRefGoogle Scholar
El Euch, O., Gatheral, J. and Rosenbaum, M. (2018). Roughening Heston. Available at https://ssrn.com/abstract=3116887.Google Scholar
El Euch, O., Gatheral, J., Radoičić, R. and Rosenbaum, M. (2020). The Zumbach effect under rough Heston. Quant. Finance 20, 235–241.CrossRefGoogle Scholar
Gatheral, J., Jaisson, T. and Rosenbaum, M. (2018). Volatility is rough. Quant. Finance 18, 933–949.CrossRefGoogle Scholar
Hardiman, S. J., Bercot, N. and Bouchaud, J.-P. (2013). Critical reflexivity in financial markets: a Hawkes process analysis. Europ. Phys. J. B 86, 442.CrossRefGoogle Scholar
Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. J. R. Statist. Soc. B [Statist. Methodology] 33, 438–443.Google Scholar
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.CrossRefGoogle Scholar
Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Prob. 11, 493–503.CrossRefGoogle Scholar
Higham, N. J. (2008). Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia, PA.CrossRefGoogle Scholar
Horvath, B., Muguruza, A. and Tomas, M. (2019). Deep learning volatility. Available at https://ssrn.com/abstract=3322085.Google Scholar
Jaber, E. A., Cuchiero, C., Larsson, M. and Pulido, S. (2019). A weak solution theory for stochastic Volterra equations of convolution type. Preprint. Available at https://arxiv.org/abs/1909.01166.Google Scholar
Jacod, J. and Shiryaev, A. (2013). Limit Theorems for Stochastic Processes. Springer, Berlin, Heidelberg.Google Scholar
Jaisson, T. and Rosenbaum, M. (2015). Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Prob. 25, 600–631.CrossRefGoogle Scholar
Jaisson, T. and Rosenbaum, M. (2016). Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes. Ann. Appl. Prob. 26, 2860–2882.CrossRefGoogle Scholar
Jusselin, P. and Rosenbaum, M. (2018). No-arbitrage implies power-law market impact and rough volatility. Preprint. Available at https://arxiv.org/abs/1805.07134.Google Scholar
Laloux, L., Cizeau, P., Bouchaud, J.-P. and Potters, M. (1999). Noise dressing of financial correlation matrices. Phys. Rev. Lett. 83, 1467.CrossRefGoogle Scholar
Livieri, G., Mouti, S., Pallavicini, A. and Rosenbaum, M. (2018). Rough volatility: evidence from option prices. IISE Trans. 50, 767–776.CrossRefGoogle Scholar
Reigneron, P.-A., Allez, R. and Bouchaud, J.-P. (2011). Principal regression analysis and the index leverage effect. Physica A 390, 3026–3035.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (2013). Continuous martingales and Brownian motion. Springer, Berlin, Heidelberg.Google Scholar
Veraar, M. (2012). The stochastic Fubini theorem revisited. Stochastics 84, 543–551.CrossRefGoogle Scholar