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Utility maximization in multivariate Volterra models

Florian Aichinger, Sascha Desmettre

TL;DR

The paper studies Merton-style portfolio optimization with power utility in multivariate rough-volatility markets, introducing both a class of multivariate affine Volterra models and a matrix-valued Volterra-Wishart volatility framework to capture cross-asset dynamics. It develops two solution paradigms: a degenerate-correlation martingale distortion method and a general-correlation convolution-resolvent verification, yielding explicit optimal strategies governed by Riccati-Volterra equations. Existence and uniqueness results for the Volterra equations underpin the theoretical foundations, while numerical experiments illustrate hedging components and roughness effects. The work extends prior single-asset and Wishart results to the multivariate rough setting, enabling explicit strategies and tractable Riccati-Volterra characterizations with practical implications for cross-asset risk management under rough volatility.

Abstract

This paper is concerned with portfolio selection for an investor with power utility in multi-asset financial markets in a rough stochastic environment. We investigate Merton's portfolio problem for different multivariate Volterra models, covering the rough Heston model. First we consider a class of multivariate affine Volterra models introduced in [E. Abi Jaber et al., SIAM J. Financial Math., 12, 369-409, (2021)]. Based on the classical Wishart model described in [N. Bäuerle and Li, Z., J. Appl. Probab., 50, 1025-1043 (2013)], we then introduce a new matrix-valued stochastic volatility model, where the volatility is driven by a Volterra-Wishart process. Due to the non-Markovianity of the underlying processes, the classical stochastic control approach cannot be applied in these settings. To overcome this issue, we provide a verification argument using calculus of convolutions and resolvents. The resulting optimal strategy can then be expressed explicitly in terms of the solution of a multivariate Riccati-Volterra equation. We thus extend the results obtained by Han and Wong to the multivariate case, avoiding restrictions on the correlation structure linked to the martingale distortion transformation used in [B. Han and Wong, H. Y., Finance Res. Lett., 39 (2021)]. We also provide existence and uniqueness theorems for the occurring Volterra processes and illustrate our results with a numerical study.

Utility maximization in multivariate Volterra models

TL;DR

The paper studies Merton-style portfolio optimization with power utility in multivariate rough-volatility markets, introducing both a class of multivariate affine Volterra models and a matrix-valued Volterra-Wishart volatility framework to capture cross-asset dynamics. It develops two solution paradigms: a degenerate-correlation martingale distortion method and a general-correlation convolution-resolvent verification, yielding explicit optimal strategies governed by Riccati-Volterra equations. Existence and uniqueness results for the Volterra equations underpin the theoretical foundations, while numerical experiments illustrate hedging components and roughness effects. The work extends prior single-asset and Wishart results to the multivariate rough setting, enabling explicit strategies and tractable Riccati-Volterra characterizations with practical implications for cross-asset risk management under rough volatility.

Abstract

This paper is concerned with portfolio selection for an investor with power utility in multi-asset financial markets in a rough stochastic environment. We investigate Merton's portfolio problem for different multivariate Volterra models, covering the rough Heston model. First we consider a class of multivariate affine Volterra models introduced in [E. Abi Jaber et al., SIAM J. Financial Math., 12, 369-409, (2021)]. Based on the classical Wishart model described in [N. Bäuerle and Li, Z., J. Appl. Probab., 50, 1025-1043 (2013)], we then introduce a new matrix-valued stochastic volatility model, where the volatility is driven by a Volterra-Wishart process. Due to the non-Markovianity of the underlying processes, the classical stochastic control approach cannot be applied in these settings. To overcome this issue, we provide a verification argument using calculus of convolutions and resolvents. The resulting optimal strategy can then be expressed explicitly in terms of the solution of a multivariate Riccati-Volterra equation. We thus extend the results obtained by Han and Wong to the multivariate case, avoiding restrictions on the correlation structure linked to the martingale distortion transformation used in [B. Han and Wong, H. Y., Finance Res. Lett., 39 (2021)]. We also provide existence and uniqueness theorems for the occurring Volterra processes and illustrate our results with a numerical study.

Paper Structure

This paper contains 18 sections, 14 theorems, 199 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Convolutions and resolvents
  3. A class of multivariate affine Volterra models
  4. The optimization problem
  5. The martingale distortion transformation
  6. The degenerate correlation case
  7. The general correlation case
  8. Comparison of the different approaches
  9. The Volterra-Wishart volatility model
  10. Market model and optimization problem
  11. The main result
  12. Numerical examples
  13. Existence and uniqueness results for the Volterra equations
  14. Solution of the Volterra Wishart process
  15. Solution of the Riccati Volterra equation
  16. ...and 3 more sections

Key Result

Lemma 2.5

AJ19 Let $K\in L_{\operatorname{loc}}^2(\mathbb{R}_+,\mathbb{R}^{m\times d})$ and let $L$ be an $\mathbb{R}^{n\times m}$-valued measure on $\mathbb{R}_+$ of locally bounded variation. Let $M$ be a $d$-dimensional continuous martingale with $\langle M\rangle_t=\int_0^t a_s ds$, $t\geq 0$, for some lo for every $t\geq 0$. In particular, taking $F\in L_{\operatorname{loc}}^1(\mathbb{R}_+)$ we may app

Figures (5)

  • Figure 1: Hedging demands for roughness level $\alpha=1$ for parameter $\gamma=0.2$ (A) and $\gamma=0.8$ (B).
  • Figure 2: Left hand side: Hedging demands $\frac{2}{1-\gamma}\psi(T-t)Q^{\top}\rho$ for risk aversion parameter $\gamma=0.8$ and roughness levels $\alpha=0.95$ (A), $\alpha=0.75$ (C) and $\alpha=0.55$ (E). Right hand side: Hedging demands $\frac{2}{1-\gamma}\psi(T-t)Q^{\top}\rho$ for risk aversion parameter $\gamma=0.2$ and roughness levels $\alpha=0.95$ (B), $\alpha=0.75$ (D) and $\alpha=0.55$ (F).
  • Figure 3: Effect of two different levels of the volatility of volatility on the hedging demand. The hedging demand depends linearly on the vol of vol $Q$.
  • Figure 4: Effect of the time horizon $T$ on the hedging demand for different levels of roughness for an investor with risk aversion $\gamma=0.2$.
  • Figure 5: Effect of the asset correlation on the hedging demand.

Theorems & Definitions (22)

  • Definition 2.1: Convolution of two functions
  • Definition 2.2: Convolution of a measurable function and a measure
  • Definition 2.3: Convolution of a measurable function and a local martingale
  • Remark 2.4
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 12 more