Paper: arXiv 2611.09751

Authors: Andreas Celary, Zehra Eksi-Altay, Damir Filipovic, Paul Krühner

Abstract

Kernel ridge regression estimates the discount curve as the minimizer of a pricing error plus a penalty given by the norm of a reproducing kernel Hilbert space. By the representer theorem, the estimate lies in a finite-dimensional affine space that depends only on the kernel and on the cash flow dates of the instruments. We ask when this space is invariant under the risk-neutral drift of the discount curve, so that the estimated curves can be the states of a continuous-time arbitrage-free model. We show that this holds for every grid of cash flow dates if and only if the kernel is of the form $k(x,y)=\e^{b(x+y)}\big(c,\e^{axy}+1-c\big)$, apart from kernels whose Hilbert space has dimension at most two. For this kernel the estimated curve is an affine combination of flat discount curves, each corresponding to a constant continuously compounded yield. The drift-invariant subspaces correspond to sub-grids of the cash flow dates, and the discounted coefficients have linear risk-neutral dynamics.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly theoretical and mathematically intensive analysis of drift-invariant kernels for discount curve models. While the mathematical derivations are rigorous and novel, there is no empirical validation or backtesting, focusing purely on theoretical consistency. The novelty lies in classifying kernels that ensure arbitrage-free dynamics in continuous-time models.

Research Flowchart

  flowchart TD
    A[Research Goal: When is RKHS invariant under risk-neutral drift?] --> B{Methodology: Kernel Ridge Regression (KRR) for discount curves};
    B --> C{Inputs: Pricing error, Penalty (RKHS norm), Cash flow dates};
    C --> D[Computational Process: Representer Theorem & Drift Invariance Analysis];
    D --> E{Key Finding 1: Drift-invariant iff kernel is k(x,y) = e^(b(x+y))(c*e^(axy) + 1-c)};
    E --> F{Key Finding 2: Estimated curve is affine combination of flat discount curves; discounted coefficients have linear risk-neutral dynamics};