Learning the pricing operators of hull-white one-factor and two-factor interest rate models using DeepONet

다국어 초록 (Multilingual Abstract)

The challenge of creating high-fidelity surrogate models for complex financial
derivatives is a significant topic in computational finance. Pricing functions for
interest rate derivatives are not simple vector mappings but are inherently math-
ematical operators, mapping an input function (the yield curve) to a scalar price.
This paper investigates the architectural superiority of Deep Operator Networks
(DeepONet) for learning these pricing operators, compared to a standard deep
learning approach. Our empirical results demonstrate that DeepONet learns the
pricing operators with a significantly higher degree of accuracy than the MLP,
achieving R-squared scores of 0.9923 (vs. MLP’s 0.9620) for the Hull-White model
and 0.9734 (vs. MLP’s 0.9367) for the G2++ model. Crucially, a detailed error
analysis reveals that DeepONet produces stable, homoscedastic errors, whereas the
MLP baseline exhibits clear signs of heteroscedasticity, indicating a structural in-
ability to robustly model the operator. These findings provide strong evidence that
an architecture specifically designed for operator learning is superior for this class
of problems, establishing a clear case for its use in building robust and accurate
surrogate models in finance.

국문 초록 (Abstract)

복잡한 금융 파생상품을 위한 고성능 서로게이트 모델(surrogate model)을 구
축하는 것은 계산 금융 분야의 중요한 주제입니다. 금리 파생상품의 가격 결정 함
수는 단순한 벡터 매핑이 아니라, 입력 함수(수익률 곡선)를 스칼라 값의 가격으로
매핑하는 본질적인 수학적 연산자(operator)입니다. 본 논문은 이러한 가격 결정
연산자를 학습하는 데 있어, 표준적인 딥러닝 접근법과 비교하여 딥 연산자 신경
망(DeepONet)이 갖는 구조적 우수성을 탐구합니다. 실증 분석 결과, DeepONet
은 MLP보다 훨씬 높은 정확도로 가격 결정 연산자를 학습했으며, Hull-White 모
델에 대해 R-제곱 점수 0.9923 (MLP: 0.9620), G2++ 모델에 대해 0.9734 (MLP:
0.9367)을 달성했습니다. 결정적으로, 상세 오차 분석을 통해 DeepONet은 안정적
인 동분산성(homoscedastic) 오차를 보인 반면, MLP 기준 모델은 명백한 이분산성
(heteroscedasticity)을 나타내어 연산자를 강건하게 모델링하는 데 구조적 한계가
있음을 보여주었습니다. 이러한 결과는 연산자 학습을 위해 특별히 설계된 아키텍처
가 이 문제 유형에 더 우수하다는 강력한 증거를 제공하며, 금융 분야에서 강건하고
정확한 서로게이트 모델을 구축하는 데 DeepONet을 사용해야 할 명확한 근거를
제시합니다.

목차 (Table of Contents)

• 1 Introduction 1
• 2 Theoretical and Literary Background 4
• 2.1 Fundamental Concepts: Term Structure and Interest Rate Derivatives 4
• 2.2 Theoretical Framework for Interest Rate Models 5
• 2.2.1 The Gaussian Affine Term Structure Model Framework 5
• 2.2.2 The Hull-White Framework: Foundations of No-Arbitrage Gaussian Models 6
• 2.3 Model Specifications: One-Factor vs. Two-Factor Models 7
• 2.3.1 The one-factor Hull-White model 7
• 2.3.2 The two-factor G2++ model 8
• 3 Methodology 10
• 3.1 Dataset Construction 10
• 3.1.1 Sourcing Realistic Input Functions from Market Data 11
• 3.1.2 Synthetic Data Generation 12
• 3.2 DeepONet Architecture and its Theoretical Foundation 14
• 3.2.1 Branch Net Architecture 17
• 3.2.2 Trunk Net Architecture 17
• 4 Numerical Experiments and Results 18
• 4.1 Model Training and Validation 18
• 4.1.1 Training Protocol 18
• 4.1.2 Model Validation Strategy 20
• 4.1.3 Training Techniques for Robustness and Generalization 20
• 4.1.4 Evaluation Metrics 21
• 4.2 Overall Performance Comparison 22
• 4.3 Detailed Prediction Accuracy 23
• 4.4 In-depth Error Analysis 5
• 4.4.1 Error Structure: Homoscedasticity vs. Heteroscedasticity 25
• 4.4.2 Error Distribution Analysis 26
• 4.5 Visualizing the Learned Pricing Operator 28
• 4.6 Computational Efficiency: Speed vs. Accuracy Trade-off Analysis 30
• 4.6.1 Performance Comparison Results 31
• 4.6.2 Efficiency Frontier Visualization 31
• 5 Discussion on Greeks 34
• 6 Conclusion 37
• Appendix A Derivations of Closed-Form Solutions 41
• A.1 The Hull-White One-Factor Model 41
• A.1.1 Zero-Coupon Bond Price 41
• A.1.2 European Bond Call Option Price 43
• A.2 The G2++ Two-Factor Model 44
• A.2.1 Zero-Coupon Bond Price 45
• A.2.2 European Bond Call Option Price 46