Mind the Cap! – Constrained Portfolio Optimisation in Heston’s Stochastic Volatility Model

ArXiv ID: 2306.11158 “View on arXiv”

Authors: Unknown

Abstract

We consider a portfolio optimisation problem for a utility-maximising investor who faces convex constraints on his portfolio allocation in Heston’s stochastic volatility model. We apply the duality methods developed in previous work to obtain a closed-form expression for the optimal portfolio allocation. In doing so, we observe that allocation constraints impact the optimal constrained portfolio allocation in a fundamentally different way in Heston’s stochastic volatility model than in the Black Scholes model. In particular, the optimal constrained portfolio may be different from the naive capped portfolio, which caps off the optimal unconstrained portfolio at the boundaries of the constraints. Despite this difference, we illustrate by way of a numerical analysis that in most realistic scenarios the capped portfolio leads to slim annual wealth equivalent losses compared to the optimal constrained portfolio. During a financial crisis, however, a capped solution might lead to compelling annual wealth equivalent losses.

Keywords: portfolio optimization, Heston stochastic volatility model, utility maximization, convex constraints, duality methods

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is highly mathematical, deriving closed-form expressions using Heston’s stochastic volatility model, Riccati ODEs, and verification theorems, with dense advanced theory. However, it lacks empirical backtesting, relying only on numerical analysis for illustration without real-world data or implementation details.

  flowchart TD
    A["Research Goal: Optimal Portfolio Allocation<br>with Constraints in Heston Stochastic Volatility"] --> B["Methodology: Duality Methods<br>to Derive Closed-Form Solution"]
    B --> C["Mathematical Process:<br>Convex Optimization &<br>Analytical Derivation"]
    C --> D["Inputs: Stochastic Volatility Parameters<br>and Convex Allocation Constraints"]
    C --> E["Inputs: Investor Utility Function<br>and Market Dynamics"]
    D --> F["Outcome: Explicit Formula for<br>Optimal Constrained Portfolio"]
    E --> F
    F --> G["Key Findings:<br>1. Constraints impact allocation differently than in Black-Scholes<br>2. Capped Portfolio is a close approximation except in crises<br>3. Significant wealth equivalent losses during financial crises"]