[Paper Review] Optimal Portfolio under Fractional Stochastic Environment

Jean‐Pierre Fouque, Ruimeng Hu|RePEc: Research Papers in Economics|Mar 20, 2017

Stochastic processes and financial applications24 references4 citations

TL;DR

This paper develops an asymptotically optimal portfolio strategy in a non-Markovian fractional stochastic environment driven by a slowly varying fractional Ornstein-Uhlenbeck process. Using martingale distortion for power utilities, it derives a first-order approximation of the optimal value function and shows that a fixed zeroth-order trading strategy achieves asymptotic optimality for all $ H \in (0,1) $, with correction terms of order $ \delta^H $.

ABSTRACT

Rough stochastic volatility models have attracted a lot of attentions recently, in particular for the linear option pricing problem. In this paper, starting with power utilities, we propose to use a martingale distortion representation of the optimal value function for the nonlinear asset allocation problem in a (non-Markovian) fractional stochastic environment (for all Hurst index $H \in (0,1)$). We rigorously establish a first order approximation of the optimal value, where the return and volatility of the underlying asset are functions of a stationary slowly varying fractional Ornstein-Uhlenbeck process. We prove that this approximation can be also generated by a fixed zeroth order trading strategy providing an explicit strategy which is asymptotically optimal in all admissible controls. Furthermore, we extend the discussion to general utility functions, and obtain the asymptotic optimality of this fixed strategy in a specific family of admissible strategies.

Motivation & Objective

• To address the nonlinear optimal portfolio problem under a non-Markovian fractional stochastic environment with slowly varying volatility.
• To extend Merton's portfolio optimization framework to include rough fractional stochastic volatility (RFSV) models with Hurst index $ H \in (0,1) $.
• To establish rigorous asymptotic approximations of the value function and optimal strategy when the volatility factor evolves slowly (small $ \delta $).
• To demonstrate that a fixed zeroth-order trading strategy is asymptotically optimal in the case of power utilities.
• To generalize the asymptotic optimality result to a specific class of general utility functions using the same fixed strategy framework.

Proposed method

• Employs martingale distortion representation to express the optimal value function for power utilities in a fractional stochastic environment.
• Expands the martingale distortion around a frozen volatility at the initial value $ Z^{\delta,H}_0 $, yielding a first-order correction in $ \delta^H $.
• Models the slow-varying stochastic factor as a fractional Ornstein-Uhlenbeck (fOU) process with Hurst index $ H \in (0,1) $, driven by fractional Brownian motion.
• Applies the epsilon-martingale decomposition method to handle non-Markovian dynamics and derive asymptotic approximations.
• Uses a perturbation approach based on the small parameter $ \delta $, representing the time-scale separation between fast and slow factors.
• Establishes uniform bounds on key terms via assumptions on growth and integrability, ensuring the validity of the asymptotic expansion.

Experimental results

Research questions

• RQ1 Can a fixed zeroth-order trading strategy achieve asymptotic optimality in a non-Markovian fractional stochastic environment with slowly varying volatility?
• RQ2 How does the optimal value function behave under power utility when the volatility is driven by a fractional Ornstein-Uhlenbeck process with $ H \in (0,1) $?
• RQ3 What is the order of magnitude of the first-order correction to the optimal value function in the small $ \delta $ regime?
• RQ4 Under what conditions does the asymptotic approximation of the value function remain valid for general utility functions?
• RQ5 How does the Hurst index $ H $ influence the convergence rate of the asymptotic strategy?

Key findings

• The optimal value function admits a first-order approximation in $ \delta^H $, with the leading term derived via martingale distortion representation.
• The first-order correction term is of order $ \delta^H $, and this correction is uniformly bounded under the given assumptions.
• A fixed zeroth-order trading strategy is asymptotically optimal for power utilities, meaning it achieves the same leading-order performance as the true optimal strategy.
• For general utility functions, the same fixed strategy is asymptotically optimal within a specific family of admissible strategies, under appropriate boundedness conditions.
• The approximation error for the value function is of order $ \delta^{H \wedge \alpha} $, with $ \alpha $ related to the integrability of the volatility process.
• The analysis rigorously justifies the use of the epsilon-martingale decomposition in non-Markovian settings, extending its applicability beyond linear problems like option pricing.