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Deep Calibration of Interest Rates Model

Mohamed Ben Alaya, Ahmed Kebaier, Djibril Sarr

TL;DR

It is shown that covariances are more suited to the problem than correlations due to the effects of the unfeasible backpropagation phenomenon, and the calibration based on deep learning outperforms the classic calibration method used as a benchmark.

Abstract

For any financial institution, it is essential to understand the behavior of interest rates. Despite the growing use of Deep Learning, for many reasons (expertise, ease of use, etc.), classic rate models such as CIR and the Gaussian family are still widely used. In this paper, we propose to calibrate the five parameters of the G2++ model using Neural Networks. Our first model is a Fully Connected Neural Network and is trained on covariances and correlations of Zero-Coupon and Forward rates. We show that covariances are more suited to the problem than correlations due to the effects of the unfeasible backpropagation phenomenon, which we analyze in this paper. The second model is a Convolutional Neural Network trained on Zero-Coupon rates with no further transformation. Our numerical tests show that our calibration based on deep learning outperforms the classic calibration method used as a benchmark. Additionally, our Deep Calibration approach is designed to be systematic. To illustrate this feature, we applied it to calibrate the popular CIR intensity model.

Deep Calibration of Interest Rates Model

TL;DR

It is shown that covariances are more suited to the problem than correlations due to the effects of the unfeasible backpropagation phenomenon, and the calibration based on deep learning outperforms the classic calibration method used as a benchmark.

Abstract

For any financial institution, it is essential to understand the behavior of interest rates. Despite the growing use of Deep Learning, for many reasons (expertise, ease of use, etc.), classic rate models such as CIR and the Gaussian family are still widely used. In this paper, we propose to calibrate the five parameters of the G2++ model using Neural Networks. Our first model is a Fully Connected Neural Network and is trained on covariances and correlations of Zero-Coupon and Forward rates. We show that covariances are more suited to the problem than correlations due to the effects of the unfeasible backpropagation phenomenon, which we analyze in this paper. The second model is a Convolutional Neural Network trained on Zero-Coupon rates with no further transformation. Our numerical tests show that our calibration based on deep learning outperforms the classic calibration method used as a benchmark. Additionally, our Deep Calibration approach is designed to be systematic. To illustrate this feature, we applied it to calibrate the popular CIR intensity model.

Paper Structure

This paper contains 38 sections, 1 theorem, 22 equations, 28 figures, 5 tables.

Table of Contents

  1. Introduction
  2. A systematic approach with Deep Calibration (DC) for interest rate models
  3. Indirect Deep Calibration of the G2++ model
  4. Covariances and Correlations as OQOI
  5. Data set construction
  6. Step 1: Choice of reference parameters
  7. Step 2: Extension of reference parameters and random draws
  8. Step 3: Computing covariances and correlations for ZCs and FWDs
  9. Step 4: Scaling transformations
  10. Step 5: Splitting the data set
  11. The neural network architecture
  12. Direct Deep Calibration of the G2++ model
  13. ZCs as OQOI
  14. Data set construction
  15. Computing ZC rates:
  16. ...and 23 more sections

Key Result

Theorem 3.1

In a feedforward neural network architecture, if the derivatives of the inputs with respect to the targets vanish, then the derivatives of the loss function with respect to the weights also vanish.

Figures (28)

  • Figure 1: Covariances of ZC rates for random sets of parameters.
  • Figure 2: Correlations of FWD rates for random sets of parameters
  • Figure 3: FCN Architecture of the Indirect Deep Calibration
  • Figure 4: Example of expectations of simulated ZC rate curves for different dates. $K_x=0.07$, $K_y=0.09$, $\sigma_x=0.09$, $\sigma_y=0.09$ and $\rho=-0.99$.
  • Figure 5: CNN Architecture of the Direct Deep Calibration
  • ...and 23 more figures

Theorems & Definitions (2)

  • Theorem 3.1: Unfeasible backpropagation
  • proof : Proof of Theorem \ref{['theo:unfeasbackprop']}.