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Explicit local volatility formula for Cheyette-type interest rate models

Alexander Gairat, Vyacheslav Gorovoy, Vadim Shcherbakov

Abstract

This paper addresses the approximation of the local volatility function in the Cheyette interest rate model. Its main contribution is an explicit analytical formula for approximating local volatility, derived by extending the classical Dupire framework to interest rate models. In particular, an implicit Dupire-like expression for local volatility is first derived for options written on the short rate. This expression is then approximated using a combination of perturbation methods and probabilistic techniques, resulting in a formula expressed in terms of time and strike derivatives of the Bachelier implied variance. The final formula naturally extends to multi-factor Cheyette models and provides a practical tool for model calibration.

Explicit local volatility formula for Cheyette-type interest rate models

Abstract

This paper addresses the approximation of the local volatility function in the Cheyette interest rate model. Its main contribution is an explicit analytical formula for approximating local volatility, derived by extending the classical Dupire framework to interest rate models. In particular, an implicit Dupire-like expression for local volatility is first derived for options written on the short rate. This expression is then approximated using a combination of perturbation methods and probabilistic techniques, resulting in a formula expressed in terms of time and strike derivatives of the Bachelier implied variance. The final formula naturally extends to multi-factor Cheyette models and provides a practical tool for model calibration.

Paper Structure

This paper contains 21 sections, 5 theorems, 140 equations, 4 figures.

Table of Contents

  1. Introduction
  2. One-factor Cheytte model
  3. Results
  4. The main result
  5. Implicit formula for local volatility
  6. First-Order approximation
  7. Third-Order adjustment
  8. Proofs
  9. Proof of Theorem \ref{['ImplicitFormula']}
  10. Proof of Theorem \ref{['thm:first_order']}
  11. Proof of Theorem \ref{['thm:linear']}
  12. Multi-factor model
  13. Example: the two-factor model
  14. Calibration of the Cheyette model to swaptions
  15. Swaption-based calibration framework
  16. ...and 6 more sections

Key Result

Theorem 1

The local volatility $\sigma^2(T,k)$ satisfies the equation where $C(T, k)$ is the non-discounted option price given by C_t(T,k) and

Figures (4)

  • Figure 1: Ten-year implied volatility curve and adjusted approximations.
  • Figure 2: Ten-year implied volatility curve and 1F and 2F cases.
  • Figure 3: Implied forward volatilities $\mathrm{IV}_f$ derived from swaption volatilities $\mathrm{IV}_S$.
  • Figure 4: Model-implied swaption volatilities compared to market swaption volatilities.

Theorems & Definitions (12)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Theorem 2: First-Order Approximation
  • Remark 4: Assumptions
  • Theorem 3: Third-Order Correction
  • Lemma 1
  • proof : Proof of Lemma \ref{['IG']}
  • Theorem 4
  • ...and 2 more