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Deep Joint Learning valuation of Bermudan Swaptions

Francisco Gómez Casanova, Álvaro Leitao, Fernando de Lope Contreras, Carlos Vázquez

TL;DR

The paper tackles pricing Bermudan swaptions under a one-factor Linear Gauss Markov (LGM) model using a Differential Artificial Neural Network (DANN) trained on Monte Carlo–style noisy labels. It introduces interdependent Backward DANNs to encode the optimal early-exercise policy and a Forward DANN for price estimation, enhanced by joint learning with coterminal European swaptions via automatic adjoint differentiation. By embedding coterminal European prices as exact targets, the model benefits from additional supervision, improving accuracy and reducing variance. Numerical experiments show substantial reductions in pricing error and improved robustness across parameter variations, with additional gains when pricing at future dates; the approach is extensible to other derivatives and capable of yielding Greeks efficiently.

Abstract

This paper addresses the problem of pricing involved financial derivatives by means of advanced of deep learning techniques. More precisely, we smartly combine several sophisticated neural network-based concepts like differential machine learning, Monte Carlo simulation-like training samples and joint learning to come up with an efficient numerical solution. The application of the latter development represents a novelty in the context of computational finance. We also propose a novel design of interdependent neural networks to price early-exercise products, in this case, Bermudan swaptions. The improvements in efficiency and accuracy provided by the here proposed approach is widely illustrated throughout a range of numerical experiments. Moreover, this novel methodology can be extended to the pricing of other financial derivatives.

Deep Joint Learning valuation of Bermudan Swaptions

TL;DR

The paper tackles pricing Bermudan swaptions under a one-factor Linear Gauss Markov (LGM) model using a Differential Artificial Neural Network (DANN) trained on Monte Carlo–style noisy labels. It introduces interdependent Backward DANNs to encode the optimal early-exercise policy and a Forward DANN for price estimation, enhanced by joint learning with coterminal European swaptions via automatic adjoint differentiation. By embedding coterminal European prices as exact targets, the model benefits from additional supervision, improving accuracy and reducing variance. Numerical experiments show substantial reductions in pricing error and improved robustness across parameter variations, with additional gains when pricing at future dates; the approach is extensible to other derivatives and capable of yielding Greeks efficiently.

Abstract

This paper addresses the problem of pricing involved financial derivatives by means of advanced of deep learning techniques. More precisely, we smartly combine several sophisticated neural network-based concepts like differential machine learning, Monte Carlo simulation-like training samples and joint learning to come up with an efficient numerical solution. The application of the latter development represents a novelty in the context of computational finance. We also propose a novel design of interdependent neural networks to price early-exercise products, in this case, Bermudan swaptions. The improvements in efficiency and accuracy provided by the here proposed approach is widely illustrated throughout a range of numerical experiments. Moreover, this novel methodology can be extended to the pricing of other financial derivatives.
Paper Structure (17 sections, 11 equations, 11 figures, 2 tables, 2 algorithms)

This paper contains 17 sections, 11 equations, 11 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Linear Gauss Markov model
  4. Valuation of European swaptions
  5. Valuation of Bermudan swaptions
  6. Further details about the financial product
  7. Deep learning for pricing Bermudan swaptions under LGM model
  8. DANNs design for the Cancellable IRS
  9. Training set generation
  10. Joint learning for Cancellable IRS: adding European swaptions as outputs
  11. Numerical experiments
  12. Configuration
  13. Test base cases
  14. Impact of joint learning
  15. Impact of number of samples and Monte Carlo paths per sample
  16. ...and 2 more sections

Figures (11)

  • Figure 3: DANN structure considering multiple outputs, i.e., integrating the joint learning approach.
  • Figure 4: Pricing differences in basis points of Test Case I: Plain DANN (left) and DANN with joint learning (right).
  • Figure 5: Pricing differences in basis points of Test Case II: Plain DANN (left) and DANN with joint learning (right).
  • Figure 6: Pricing differences in basis points of Test Case III: Plain DANN (left) and DANN with joint learning (right).
  • Figure 7: Pricing differences in basis points of Test Case IV: Plain DANN (left) and DANN with joint learning (right).
  • ...and 6 more figures