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On the mean-variance problem through the lens of multivariate fake stationary affine Volterra dynamics

Emmanuel Gnabeyeu

Abstract

We investigate the continuous-time Markowitz mean-variance portfolio selection problem within a multivariate class of fake stationary affine Volterra models. In this non-Markovian and non-semimartingale market framework with unbounded random coefficients, the classical stochastic control approach cannot be directly applied to the associated optimization task. Instead, the problem is tackled using a stochastic factor solution to a Riccati backward stochastic differential equation (BSDE). The optimal feedback control is characterized by means of this equation, whose explicit solutions is derived in terms of multi-dimensional Riccati-Volterra equations. Specifically, we obtain analytical closed-form expressions for the optimal portfolio policies as well as the mean-variance efficient frontier, both of which depend on the solution to the associated multivariate Riccati-Volterra system. To illustrate our results, numerical experiments based on a two dimensional fake stationary rough Heston model highlight the impact of rough volatilities and stochastic correlations on the optimal Markowitz strategies.

On the mean-variance problem through the lens of multivariate fake stationary affine Volterra dynamics

Abstract

We investigate the continuous-time Markowitz mean-variance portfolio selection problem within a multivariate class of fake stationary affine Volterra models. In this non-Markovian and non-semimartingale market framework with unbounded random coefficients, the classical stochastic control approach cannot be directly applied to the associated optimization task. Instead, the problem is tackled using a stochastic factor solution to a Riccati backward stochastic differential equation (BSDE). The optimal feedback control is characterized by means of this equation, whose explicit solutions is derived in terms of multi-dimensional Riccati-Volterra equations. Specifically, we obtain analytical closed-form expressions for the optimal portfolio policies as well as the mean-variance efficient frontier, both of which depend on the solution to the associated multivariate Riccati-Volterra system. To illustrate our results, numerical experiments based on a two dimensional fake stationary rough Heston model highlight the impact of rough volatilities and stochastic correlations on the optimal Markowitz strategies.

Paper Structure

This paper contains 16 sections, 9 theorems, 137 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: Multivariate fake stationary affine Volterra models
  3. Stabilizer and fake stationarity regimes.
  4. Formulation of the Market model
  5. Markowitz portfolio selection: Mean-variance optimization problem.
  6. An intuition from the degenerate multidimensional correlation structure
  7. Optimal strategy in the general multivariate correlation structure
  8. Numerical experiments: The particular case of the fake stationary rough Heston volatility
  9. About the numerical scheme for the Volterra and fractional Riccati equations
  10. Numerical illustrations
  11. Proofs of the main results
  12. Proof of Proposition \ref{['prop:existence_riccati_sto']}
  13. Proof of Theorem \ref{['thm:inner']}
  14. Supplementary material and Proofs.
  15. Existence of solutions for inhomogeneous Riccati-Volterra equations
  16. ...and 1 more sections

Key Result

Proposition 2.4

Let $(V_t)_{t \geq 0}$ be a solution to the scaled Volterra square root equation in its form VolSqrt_ starting from any random variable $V_0\in L^2(\Omega, \mathcal{F}, \mathbb{P})$. Then, a necessary and sufficient condition for the relations eq:fs1_ to be satisfied is that for $i=1,\ldots,d$ and the couple $(v_0^i, \varsigma^i(t))$, where $v_0^i = \text{Var}(V_0^i)$ must satisfy the functional e

Figures (9)

  • Figure 1: Graph of the stabilizer $t \to \varsigma_{\alpha_1,\lambda_1,c_1}(t)$ (left) and $5$ samples paths $t_k \mapsto V^1_{t_k}$ (right) over the time interval $[0, 1]$, for the Hurst esponent $H = 0.1$, $c_1 = 0.01$ and number of time steps $n = 600$.
  • Figure 2: Graph of the stabilizer $t \to \varsigma_{\alpha_2,\lambda_2,c_2}(t)$ (left) and $5$ samples paths $t_k \mapsto V^2_{t_k}$ (right) over the time interval $[0, 1]$, for the Hurst esponent $H = 0.4$, $c_2 = 0.03$ and number of time steps $n = 600$.
  • Figure 3: Graph of $t_k \mapsto \text{Var}(V^1_{t_k}, M)$ and $t_k \mapsto \mathbb{E}[\sigma^2(V^1_{t_k},M)]$ over $[0, 1]$, $c_1 = 0.01$ and $n = 600$.
  • Figure 4: Graph of $t_k \mapsto \text{Var}(V^2_{t_k}, M)$ and $t_k \mapsto \mathbb{E}[\sigma^2(V^2_{t_k},M)]$ over $[0, 1]$, $c_2 = 0.03$ and $n = 600$.
  • Figure 5: Graph of $t_k \mapsto \psi_{t_k}$ with risk premium parameter $\theta = (3.6,3.0)$ and correlation $|\rho_i| \leq \frac{1}{2}$.
  • ...and 4 more figures

Theorems & Definitions (15)

  • Remark 2.1
  • Definition 2.2: Fake Stationarity Regimes
  • Example 2.3
  • Proposition 2.4: Fake stationary Volterra square root process.
  • Definition 2.5
  • Example 2.6
  • Theorem 2.7
  • Definition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • ...and 5 more