A monotone piecewise constant control integration approach for the two-factor uncertain volatility model

ArXiv ID: 2402.06840 “View on arXiv”

Authors: Unknown

Abstract

Option contracts on two underlying assets within uncertain volatility models have their worst-case and best-case prices determined by a two-dimensional (2D) Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) with cross-derivative terms. This paper introduces a novel ``decompose and integrate, then optimize’’ approach to tackle this HJB PDE. Within each timestep, our method applies piecewise constant control, yielding a set of independent linear 2D PDEs, each corresponding to a discretized control value. Leveraging closed-form Green’s functions, these PDEs are efficiently solved via 2D convolution integrals using a monotone numerical integration method. The value function and optimal control are then obtained by synthesizing the solutions of the individual PDEs. For enhanced efficiency, we implement the integration via Fast Fourier Transforms, exploiting the Toeplitz matrix structure. The proposed method is $\ell_{"\infty"}$-stable, consistent in the viscosity sense, and converges to the viscosity solution of the HJB equation. Numerical results show excellent agreement with benchmark solutions obtained by finite differences, tree methods, and Monte Carlo simulation, highlighting its robustness and effectiveness.

Keywords: Hamilton-Jacobi-Bellman (HJB), Uncertain Volatility Models, Fast Fourier Transform (FFT), Option Pricing, Partial Differential Equations, Options/Derivatives

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper presents advanced mathematics involving high-dimensional HJB PDEs, viscosity solutions, and analytical Green’s functions, warranting a high math complexity score. While the empirical rigor is bolstered by extensive numerical comparisons with benchmark methods and robust convergence proofs, the absence of raw datasets or explicit backtesting procedures keeps it below a perfect score.

  flowchart TD
    A["Research Goal: Solve 2D HJB PDE<br>for uncertain volatility models"] --> B["Key Methodology:<br>'Decompose & Integrate, then Optimize'"]
    B --> C["Data/Inputs:<br>Option Specs &<br>Uncertain Volatility Bounds"]
    C --> D["Computational Process:<br>1. Piecewise Constant Control<br>2. 2D Convolution (Green's Functions)<br>via Fast Fourier Transform (FFT)"]
    D --> E["Key Findings/Outcomes:<br>- ℓ∞-stable & Viscosity Consistent<br>- Fast FFT-based implementation<br>- Accurate vs Benchmarks<br>(FD, Trees, Monte Carlo)"]