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Transfer Learning Across Fixed-Income Product Classes

Nicolas Camenzind, Damir Filipovic

TL;DR

This work develops a transfer-learning framework to jointly estimate multiple fixed-income discount curves across product classes by extending kernel ridge regression to a vector-valued RKHS with separable kernels. The approach introduces a graph-regularized norm that promotes smooth spread curves across product classes, and provides a decomposition of the RKHS norm to ensure a valid separable kernel. A Gaussian-process interpretation enables uncertainty quantification and connects KR estimators to posterior means. Empirically, the method improves extrapolation in data-sparse regions—demonstrated via a masking experiment using US government bonds and SOFR swaps—while preserving fit quality in well-sampled regions, with practical implications for multi-currency pricing and cross-product risk assessment.

Abstract

We propose a framework for transfer learning of discount curves across different fixed-income product classes. Motivated by challenges in estimating discount curves from sparse or noisy data, we extend kernel ridge regression (KR) to a vector-valued setting, formulating a convex optimization problem in a vector-valued reproducing kernel Hilbert space (RKHS). Each component of the solution corresponds to the discount curve implied by a specific product class. We introduce an additional regularization term motivated by economic principles, promoting smoothness of spread curves between product classes, and show that it leads to a valid separable kernel structure. A main theoretical contribution is a decomposition of the vector-valued RKHS norm induced by separable kernels. We further provide a Gaussian process interpretation of vector-valued KR, enabling quantification of estimation uncertainty. Illustrative examples show how transfer learning tightens confidence intervals compared to single-curve estimation. An extensive masking experiment demonstrates that transfer learning significantly improves extrapolation performance.

Transfer Learning Across Fixed-Income Product Classes

TL;DR

This work develops a transfer-learning framework to jointly estimate multiple fixed-income discount curves across product classes by extending kernel ridge regression to a vector-valued RKHS with separable kernels. The approach introduces a graph-regularized norm that promotes smooth spread curves across product classes, and provides a decomposition of the RKHS norm to ensure a valid separable kernel. A Gaussian-process interpretation enables uncertainty quantification and connects KR estimators to posterior means. Empirically, the method improves extrapolation in data-sparse regions—demonstrated via a masking experiment using US government bonds and SOFR swaps—while preserving fit quality in well-sampled regions, with practical implications for multi-currency pricing and cross-product risk assessment.

Abstract

We propose a framework for transfer learning of discount curves across different fixed-income product classes. Motivated by challenges in estimating discount curves from sparse or noisy data, we extend kernel ridge regression (KR) to a vector-valued setting, formulating a convex optimization problem in a vector-valued reproducing kernel Hilbert space (RKHS). Each component of the solution corresponds to the discount curve implied by a specific product class. We introduce an additional regularization term motivated by economic principles, promoting smoothness of spread curves between product classes, and show that it leads to a valid separable kernel structure. A main theoretical contribution is a decomposition of the vector-valued RKHS norm induced by separable kernels. We further provide a Gaussian process interpretation of vector-valued KR, enabling quantification of estimation uncertainty. Illustrative examples show how transfer learning tightens confidence intervals compared to single-curve estimation. An extensive masking experiment demonstrates that transfer learning significantly improves extrapolation performance.
Paper Structure (27 sections, 8 theorems, 69 equations, 13 figures, 2 tables)

This paper contains 27 sections, 8 theorems, 69 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Transfer Learning Problem Formulation
  3. Gaussian Process View
  4. Vector-valued Gaussian processes
  5. Gaussian Process View for Separable Kernels
  6. A Workable Class of Separable Kernels
  7. Standard Fixed-Income Products
  8. Coupon Bonds
  9. Interest-Rate Swaps
  10. Cross-Currency Swaps
  11. Empirical Analysis
  12. Data and Hyperparameter Selection
  13. Illustrative Yield and Forward Curves
  14. Masking Experiment
  15. Conclusion
  16. ...and 12 more sections

Key Result

Theorem 2.1

There exists a unique solution of the vector-valued KR problem eq:krr_obj, which is given by $\bar{h} = \sum_{j=1}^N K(\cdot,x_j) \beta_j$ where $\beta=(\beta_1,\dots,\beta_N)\in {\mathbb R}^{A\times N}$ takes the form for the block diagonal matrix $\bm\Lambda=\operatorname{diag}(\Lambda_1,\dots,\Lambda_A)\in\mathbb{R}^{M\times M}$ with $\Lambda_a=\operatorname{diag}(\lambda/\omega_{a,1},\dots,\l

Figures (13)

  • Figure 1: Schematic cash flows of a floating--floating XCCY swap
  • Figure 2: Maturities of US government bonds
  • Figure 3: Bid--Ask yield and swap spreads
  • Figure 4: LOOCV RMSE heatmaps
  • Figure 5: Example day 2024-06-14
  • ...and 8 more figures

Theorems & Definitions (29)

  • Theorem 2.1
  • Example 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1: vector-Valued Gaussian Process
  • Remark 3.2
  • Theorem 3.3
  • Remark 3.4
  • Definition 3.5
  • Example 4.1
  • ...and 19 more