Paper: arXiv 2506.23876

Authors: 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.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a strong theoretical contribution by extending Dupire’s local volatility framework to Cheyette interest rate models using perturbation methods and probabilistic techniques. However, it is primarily a theoretical and analytical paper with limited empirical backtesting provided in the excerpt.

Research Flowchart

  flowchart TD
    A["<b>Research Goal</b><br>Approximate local volatility function<br>for Cheyette interest rate models"] --> B["<b>Inputs & Framework</b><br>Short rate options & Bachelier<br>implied variance derivatives"]
    
    B --> C["<b>Dupire Extension</b><br>Derive implicit Dupire-like expression<br>for short rate option local volatility"]
    
    C --> D["<b>Approximation Methods</b><br>Apply perturbation methods &<br>probabilistic techniques"]
    
    D --> E["<b>Explicit Formula</b><br>Derive analytical local volatility formula<br>for single & multi-factor models"]
    
    E --> F["<b>Key Outcome</b><br>Practical, efficient tool for<br>interest rate model calibration"]

Cited References (11)

  1. Non-Parametric Local Volatility Swaption Formula Revisited (2016) — link
  2. Interest Rate Models Enhanced with Local Volatility (2016) — link
  3. Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing (2008) — link
  4. A Theory of Volatility (2001) — link
  5. Continuous Univariate Distributions, Volume 1 (1994) — link
  6. A unified approach to local and stochastic volatility expansion (2015)
  7. Interest Rate Modeling. Volume 2: Term Structure Models (2010)
  8. Brownian Motion and Stochastic Calculus (1987) — link
and 3 more
  1. Stochastic differential equations and diffusion processes (1981) — link
  2. where u ( t ) is the function defined in (49). Introducing the vector Y ( t ) = (cid:7) y 1, t y 2, t (cid:8) rewrite equations (A1) and (A2) in the matrix form
  3. where P = (cid:5) 1 + 2 ρβ/α 1 + ρα/β (cid:6)

0 of 11 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.