Paper: arXiv 2611.11652

Authors: Carlos Escudero, Francisco J. Luguera-Parúas, Miguel Sama

Abstract

Multi-factor approximations of rough volatility kernels affect the characteristic exponents used in Fourier-based derivative valuation. We derive an adjoint representation for the first-order sensitivity of the rough Heston characteristic exponent with respect to its memory kernel, with model parameters and the input variance curve held fixed. The Volterra–Riccati functional $\mathcal L_T(K,z)$ admits a local real-analytic extension, jointly in the kernel perturbation and the transform parameter. The Fréchet derivative of this extension with respect to the kernel in $L^1$ is represented by integration against a continuous sensitivity kernel $B_K(u,z)$, obtained from a backward linear Volterra equation and independent of the perturbation direction. This representation resolves the leading-order contributions of kernel perturbations across memory lags. On the structural kernel class, which contains the fractional kernel and its multi-factor approximations and has empty interior in $L^1$, the extension yields Hadamard directional differentiability and a first-order expansion with a quadratic $L^1$-remainder for all sufficiently close structural kernels, uniformly on compact subsets of the strip of transform parameters. The same sensitivity kernel yields multi-factor weight and decay-rate derivatives. Numerical experiments at multi-factor reference kernels assess the finite-resolution implementation and illustrate how the lag distribution of positive perturbations with equal $L^1$ mass affects the first-order response.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly complex mathematical derivation for kernel sensitivities in rough Heston models, a novel contribution to the field. While it includes numerical experiments, the empirical section is primarily illustrative rather than a rigorous backtesting exercise. The overall quality is high due to the strong theoretical foundation and novel methodology.

Research Flowchart

  flowchart TD
    A[Research Goal: Derive sensitivity of rough Heston characteristic exponent to memory kernel] --> B{Methodology: Adjoint Representation & Fréchet Derivative};
    B --> C[Mathematical Formulation: Volterra--Riccati functional L_T(K,z)];
    C --> D{Computational Process: Backward Linear Volterra Equation for Sensitivity Kernel B_K(u,z)};
    D --> E[Inputs: Model Parameters, Input Variance Curve, Multi-factor Reference Kernels];
    E --> F{Outcomes: Continuous Sensitivity Kernel B_K(u,z)};
    F --> G[Key Findings: Hadamard directional differentiability, 1st-order expansion with quadratic L1-remainder, Multi-factor weight/decay-rate derivatives, Impact of lag distribution on response];