Actuarial Notes

Calibrating Interest Rate Models

By Rohana Ambagaspitiya and Charles Ford · Society of Actuaries Research Institute · 2023

Study guide: Exam INV-201

A research report, dated October 2023 and sponsored by the SOA's Quantitative Finance and Investment Curriculum Committee, that introduces practitioners to the selection and calibration of stochastic interest rate models. It presents six continuous-time interest rate models and their calibration, with R code and data examples, and is aimed at the investment and valuation actuary who must choose a model for a purpose, use real-world or risk-neutral scenarios correctly, and calibrate and validate the result. Its scope is narrow: default-free bond yields in a single economy.

On the syllabus

  • Exam INV-201 — Sections 1.1-4.3, excluding 4.1.2; outcomes 2g-2i.

Executive Summary

1 Introduction

  • 1.1 R Setup
  • 1.2 Introduction to Scenarios
  • 1.3 Interest Rate Markets
  • 1.4 Market-Consistent Models
  • 1.5 The Market Price of Risk
  • 1.6 Parameter Uncertainty

2 Three Continuous-Time Interest Rate Models

  • 2.1 The Vasicek Model
    • 2.1.1 Simulating paths of the Vasicek model: Euler-Maruyama discretization
    • 2.1.2 Simulating paths of the Vasicek model: Transition Density Method
  • 2.2 Cox-Ingersoll-Ross (CIR) Model
    • 2.2.1 Simulating paths of the CIR Model: Euler-Maruyama discretization
    • 2.2.2 Simulating paths of the CIR model: Transition density method
  • 2.3 The Two-Factor Vasicek model with correlated factors
    • 2.3.1 Simulating paths of the two-factor Vasicek model: Transition density method

3 Calibration Techniques

  • 3.1 The Vasicek model: real-world calibration
    • 3.1.1 Maximum Likelihood Estimator Method
    • 3.1.2 Long Term Quantile Method
  • 3.2 The Vasicek model: risk-neutral calibration
  • 3.3 The CIR model: real-world calibration
    • 3.3.1 Euler method
    • 3.3.2 Maximum likelihood estimate
  • 3.4 The generalized method of moments
  • 3.5 The CIR model: risk-neutral calibration
  • 3.6 The two-factor Vasicek model calibration

4 No-Arbitrage Models

  • 4.1 Hull-white models
    • 4.1.1 One factor Hull-white model
    • 4.1.2 The two-factor Hull-white model
    • 4.1.3 Hull-white model calibration
  • 4.2 Yield curve interpolation: one-factor model
    • 4.2.1 Fitting an n degree polynomial for r_{0,t}
    • 4.2.2 Fitting a Nelson-Siegel Curve to r_{0,t}
  • 4.3 Calibration of the One-Factor Hull-White Model
  • 4.4 Calibration of the Two-Factor Hull-White Model

5 Model Validation

  • 5.1 Data and Assumptions
  • 5.2 Investigate the Data
  • 5.3 Recalibrate
  • 5.4 Validate
  • 5.5 Model Governance

6 Conclusion

7 Acknowledgments

References

Appendix A Zero coupon bond prices under one-factor Vasicek model

Appendix B Zero coupon bond prices under the two-factor Vasicek model

Sources