Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Utility Maximization under Multivariate Fake Stationary Affine Volterra Models

Emmanuel Gnabeyeu

Abstract

This paper is concerned with Merton's portfolio optimization problem in a Volterra stochastic environment described by a multivariate fake stationary Volterra--Heston model. Due to the non-Markovianity and non-semimartingality of the underlying processes, the classical stochastic control approach cannot be directly applied in this setting. Instead, the problem is tackled using a stochastic factor solution to a Riccati backward stochastic differential equation (BSDE). Our approach is inspired by the martingale optimality principle combined with a suitable verification argument. The resulting optimal strategies for Merton's problems are derived in semi-closed form depending on the solutions to time-dependent multivariate Riccati-Volterra equations. Numerical results on a two dimensional fake stationary rough Heston model illustrate the impact of stationary rough volatilities on the optimal Merton strategies.

On Utility Maximization under Multivariate Fake Stationary Affine Volterra Models

Abstract

This paper is concerned with Merton's portfolio optimization problem in a Volterra stochastic environment described by a multivariate fake stationary Volterra--Heston model. Due to the non-Markovianity and non-semimartingality of the underlying processes, the classical stochastic control approach cannot be directly applied in this setting. Instead, the problem is tackled using a stochastic factor solution to a Riccati backward stochastic differential equation (BSDE). Our approach is inspired by the martingale optimality principle combined with a suitable verification argument. The resulting optimal strategies for Merton's problems are derived in semi-closed form depending on the solutions to time-dependent multivariate Riccati-Volterra equations. Numerical results on a two dimensional fake stationary rough Heston model illustrate the impact of stationary rough volatilities on the optimal Merton strategies.
Paper Structure (17 sections, 13 theorems, 184 equations, 4 figures)

This paper contains 17 sections, 13 theorems, 184 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: Multivariate fake stationary affine Volterra models
  3. Stabilizer and fake stationarity regimes.
  4. Formulation of the stochastic Market model
  5. Merton's portfolio problem: Utility maximisation
  6. Optimal strategy for the power utility maximization problem
  7. The degenerate correlation case
  8. The general correlation case: A verification argument
  9. Optimal strategy for the exponential utility maximization problem
  10. The degenerate correlation case
  11. The general correlation case: A verification argument
  12. Numerical experiments: The particular case of the fake stationary rough Heston volatility
  13. About the numerical scheme for the Volterra and fractional Riccati equations
  14. Numerical illustrations
  15. Proofs of the main results
  16. ...and 2 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 (4)

  • Figure 1: Graph of the stabilizer $t \to \varsigma_{\alpha_1,\lambda_1,c_1}(t)$ (left) and $30$ samples paths $t_k \mapsto V^1_{t_k}$ (right) over the time interval $[0, 1]$, for the Hurst esponent $H = 0.4$, $c_1 = 0.01$ and number of time steps $n = 600$.
  • Figure 2: 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 3: Graph of $t_k \mapsto \psi^1_{t_k}$ and $t_k \mapsto \psi^2_{t_k}$ over $[0, 1]$ with the fractional Adams algorithm, $\gamma=0.2$ and the number of time steps $n = 200$ both for Power (left) and Exponential (right) utilities functions.
  • Figure 4: Evolution of the optimal portfolio strategy for different levels of the risk aversion parameter $\gamma$ for both the Power (left) and Exponential (right) utilities functions.

Theorems & Definitions (22)

  • 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
  • Definition 3.2: Martingale optimality principle
  • Definition 3.3
  • ...and 12 more