Paper: arXiv 2502.21252

Authors: Aram Karakhanyan, Takis Konstantopoulos, Matthew Lorig, Evgenii Samutichev

Abstract

We introduce a class of short-rate models that exhibit a ``higher for longer’’ phenomenon. Specifically, the short-rate is modeled as a general time-homogeneous one-factor Markov diffusion on a finite interval. The lower endpoint is assumed to be regular, exit or natural according to boundary classification while the upper endpoint is assumed to be regular with absorbing behavior. In this setting, we give an explicit expression for price of a zero-coupon bond (as well as more general interest rate derivatives) in terms of the transition density of the short-rate under a new probability measure, and the solution of a non-linear ordinary differential equation (ODE). We then narrow our focus to a class of models for which the transition density and ODE can be solved explicitly. For models within this class, we provide conditions under which the lower endpoint is regular, exit and natural. Finally, we study two specific models – one in which the lower endpoint is exit and another in which the lower endpoint is natural. In these two models, we give an explicit solution of transition density of the short-rate as a (generalized) eigenfunction expansion. We provide plots of the transition density, (generalized) eigenfunctions, bond prices and the associated yield curve.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework for short-rate models with absorbing and regular/exit/natural boundaries using spectral methods and Girsanov transformations. However, it is purely theoretical and mathematical, lacking real historical data backtests or empirical model calibration.

Research Flowchart

  flowchart TD
    A["Research Goal: Model short-rate derivatives<br>in a 'higher-for-longer' environment"] --> B["Framework: Time-homogeneous 1-factor Markov diffusion<br>with absorbing upper boundary"]
    B --> C["General Pricing Formula: Express ZCB prices via<br>new measure transition density & non-linear ODE"]
    C --> D["Model Specifics: Evaluate lower boundary types<br>(Exit vs. Natural) for explicit solutions"]
    D --> E["Computation: Solve transition density using<br>(generalized) eigenfunction expansions"]
    E --> F["Outcomes: Explicit ZCB prices, yield curves,<br>and transition density plots"]

Cited References (6)

  1. NIST Digital Library of Mathematical Functions (2003) — link
  2. Operator Theory for Electromagnetics: An Introduction (2001)
  3. Handbook of Brownian Motion - Facts and Formulae (1996) — link
  4. THE PARABOLIC DIFFERENTIAL EQUATIONS AND THE ASSOCIATED SEMI-GROUPS OF TRANSFORMATIONS (1952) — link
  5. Chapter 6 Spectral Methods in Derivatives Pricing (2007) — link
  6. Operator Theory for Electromagnetics (2002) — link

0 of 6 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.