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Markowitz portfolio selection for multivariate affine and quadratic Volterra models

Eduardo Abi Jaber (), Enzo Miller and Huyên Pham ()
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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, Statistiques et Modélisations - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique
Huyên Pham: LPSM (UMR_8001) - Laboratoire de Probabilités, Statistiques et Modélisations - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique

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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.

Keywords: Stein-Stein and Wishart models; Riccati equations; non-Markovian Heston; multi- dimensional Volterra process; rough volatility; Mean-variance portfolio theory; correlation matrices (search for similar items in EconPapers)
Date: 2021
Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-02877569v4
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Published in SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics 2021, 12 (1), pp.369-409. ⟨10.1137/20M1347449⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:cesptp:hal-02877569

DOI: 10.1137/20M1347449

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