Heat modulated affine stochastic volatility models for forward curve dynamics

Paper: arXiv 2409.13070

Authors: Sven Karbach

Abstract

We present a function-valued stochastic volatility model designed to capture the continuous-time evolution of forward curves in fixed-income or commodity markets. The dynamics of the (logarithmic) forward curves are defined by a Heath-Jarrow-Morton-Musiela stochastic partial differential equation modulated by an instantaneous volatility process that describes the second-order moment structure of forwards with different time-to-maturity. We propose to model the operator-valued instantaneous covariance by an affine process on the cone of positive trace-class operators with drift given by the Lyapunov operator of the Laplacian. The so defined infinite-rank stochastic volatility model is analytically tractable due to its affine structure and allows to model maturity specific risk and volatility clustering in forward markets. Furthermore, we introduce a numerically feasible spectral Galerkin approximation of the associated operator-valued generalized Riccati equations and study the robustness of the model with respect to finite-rank approximations by providing explicit error bounds on the approximation error.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly sophisticated, infinite-dimensional mathematical framework combining HJM models, operator-valued stochastic processes, and spectral Galerkin approximations with explicit error bounds. However, it is fundamentally a theoretical paper with limited empirical backtesting provided in the excerpt.

Research Flowchart

  flowchart TD
    A[Goal: Model forward curve dynamics with maturity-specific affine stochastic volatility] --> B[Model Formulation: HJMM SPDE modulated by operator-valued affine process]
    B --> C[Operator Structure: Drift driven by Laplacian Lyapunov operator on trace-class cone]
    C --> D[Numerical Method: Apply spectral Galerkin approximation to operator-valued Riccati equations]
    D --> E[Error Analysis: Derive explicit bounds for finite-rank approximations]
    E --> F[Outcome: Analytical tractability, capturing volatility clustering & maturity risk with proven robustness]