Optimal Trading under Instantaneous and Persistent Price Impact, Predictable Returns and Multiscale Stochastic Volatility

ArXiv ID: 2507.17162 “View on arXiv”

Authors: Patrick Chan, Ronnie Sircar, Iosif Zimbidis

Abstract

We consider a dynamic portfolio optimization problem that incorporates predictable returns, instantaneous transaction costs, price impact, and stochastic volatility, extending the classical results of Garleanu and Pedersen (2013), which assume constant volatility. Constructing the optimal portfolio strategy in this general setting is challenging due to the nonlinear nature of the resulting Hamilton-Jacobi-Bellman (HJB) equations. To address this, we propose a multi-scale volatility expansion that captures stochastic volatility dynamics across different time scales. Specifically, the analysis involves a singular perturbation for the fast mean-reverting volatility factor and a regular perturbation for the slow-moving factor. We also introduce an approximation for small price impact and demonstrate its numerical accuracy. We formally derive asymptotic approximations up to second order and use Monte Carlo simulations to show how incorporating these corrections improves the Profit and Loss (PnL) of the resulting portfolio strategy.

Keywords: Hamilton-Jacobi-Bellman (HJB), Stochastic Volatility, Multi-Scale Expansion, Price Impact Modeling, Optimal Portfolio Strategy, Equity

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper involves highly advanced mathematical techniques including Hamilton-Jacobi-Bellman equations, singular/regular perturbation theory for multiscale stochastic volatility, and second-order asymptotic expansions. While it uses Monte Carlo simulations to demonstrate theoretical improvements, it lacks code, backtests, statistical metrics, or implementation details required for empirical readiness.

  flowchart TD
    A["Research Goal: Optimal Portfolio with Stochastic Volatility & Price Impact"] --> B["Methodology: Multi-Scale Volatility Expansion<br>Fast (Singular) & Slow (Regular) Perturbations"]
    B --> C["Data: Simulated Price Dynamics<br>Inputting Predictable Returns, Volatility Regimes, Impact Costs"]
    C --> D["Computational Process: Solving Nonlinear HJB Equation<br>via Asymptotic Approximations"]
    D --> E["Outcome: Asymptotic Portfolio Strategy<br>Incorporating Impact & Volatility Corrections"]
    E --> F["Validation: Monte Carlo Simulations"]
    F --> G["Key Findings: <br>1. Improved PnL from Multi-Scale Corrections<br>2. Formal 2nd-Order Asymptotics"]