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Markowitz portfolio selection for multivariate affine and quadratic Volterra models

Author

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  • Eduardo Abi Jaber

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Enzo Miller

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique)

  • Huyên Pham

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique)

Abstract

This paper concerns portfolio selection with multiple assets under rough covariance matrix. We investigate the continuous-time Markowitz mean-variance problem for a multivariate class of affine and quadratic Volterra models. In this incomplete non-Markovian and non-semimartingale market framework with unbounded random coefficients, the optimal portfolio strategy is expressed by means of a Riccati backward stochastic differential equation (BSDE). In the case of affine Volterra models, we derive explicit solutions to this BSDE in terms of multi-dimensional Riccati-Volterra equations. This framework includes multivariate rough Heston models and extends the results of \cite{han2019mean}. In the quadratic case, we obtain new analytic formulae for the the Riccati BSDE and we establish their link with infinite dimensional Riccati equations. This covers rough Stein-Stein and Wishart type covariance models. Numerical results on a two dimensional rough Stein-Stein model illustrate the impact of rough volatilities and stochastic correlations on the optimal Markowitz strategy. In particular for positively correlated assets, we find that the optimal strategy in our model is a `buy rough sell smooth' one.

Suggested Citation

  • Eduardo Abi Jaber & Enzo Miller & Huyên Pham, 2021. "Markowitz portfolio selection for multivariate affine and quadratic Volterra models," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-02877569, HAL.
    • Handle: RePEc:hal:cesptp:hal-02877569
    DOI: 10.1137/20M1347449
    Note: View the original document on HAL open archive server: https://hal.science/hal-02877569v4
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    References listed on IDEAS

    as
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    Cited by:

    1. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Working Papers hal-02946146, HAL.
    2. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Post-Print hal-02946146, HAL.
    3. Etienne Chevalier & Sergio Pulido & Elizabeth Zúñiga, 2021. "American options in the Volterra Heston model," Working Papers hal-03178306, HAL.
    4. Etienne Chevalier & Sergio Pulido & Elizabeth Z'u~niga, 2021. "American options in the Volterra Heston model," Papers 2103.11734, arXiv.org, revised May 2022.
    5. Etienne Chevalier & Sergio Pulido & Elizabeth Zúñiga, 2022. "American options in the Volterra Heston model," Post-Print hal-03178306, HAL.
    6. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Finance and Stochastics, Springer, vol. 26(4), pages 733-769, October.
    7. Jingtang Ma & Zhengyang Lu & Zhenyu Cui, 2022. "Delta family approach for the stochastic control problems of utility maximization," Papers 2202.12745, arXiv.org.
    8. Eduardo Abi Jaber & Eyal Neuman, 2022. "Optimal Liquidation with Signals: the General Propagator Case," Papers 2211.00447, arXiv.org.
    9. Eduardo Abi Jaber & Eyal Neuman & Moritz Vo{ss}, 2023. "Equilibrium in Functional Stochastic Games with Mean-Field Interaction," Papers 2306.05433, arXiv.org, revised Feb 2024.
    10. Eduardo Abi Jaber, 2020. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Papers 2009.10972, arXiv.org, revised May 2022.
    11. Martin Friesen & Peng Jin, 2022. "Volterra square-root process: Stationarity and regularity of the law," Papers 2203.08677, arXiv.org, revised Oct 2022.

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