Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parameters estimation of a Threshold Chan-Karolyi-Longstaff-Sanders process from continuous and discrete observations

Sara Mazzonetto, Benoît Nieto

TL;DR

The paper analyzes parameter estimation for the threshold CKLS (T-CKLS) diffusion, a regime-switching CKLS-type process with piecewise-constant coefficients across multiple thresholds. It develops continuous-time MLE and QMLE for the drift, introduces discretized estimators for discrete observations—including an Itô-Tanaka based discretization—and proves consistency and asymptotic normality under ergodicity and moment conditions. A key methodological advance is the control of threshold-crossing events via hitting-time techniques, enabling high-frequency, long-time results for both drift and diffusion estimators, as well as a practical threshold-detection framework. The authors validate the theory with simulated data and apply the methods to US interest-rate data, demonstrating the presence and estimation of regime thresholds and illustrating the approach’s practical relevance for financial modeling with threshold effects.

Abstract

We consider a continuous time process that is self-exciting and ergodic, called threshold Chan-Karolyi-Longstaff-Sanders (CKLS) process. This process is a generalization of various models in econometrics, such as Vasicek model, Cox-Ingersoll-Ross, and Black-Scholes, allowing for the presence of several thresholds which determine changes in the dynamics. We study the asymptotic behavior of maximum-likelihood and quasi-maximum-likelihood estimators of the drift parameters in the case of continuous time and discrete time observations. We show that for high frequency observations and infinite horizon the estimators satisfy the same asymptotic normality property as in the case of continuous time observations. We also discuss diffusion coefficient estimation. Finally, we apply our estimators to simulated and real data to motivate considering (multiple) thresholds.

Parameters estimation of a Threshold Chan-Karolyi-Longstaff-Sanders process from continuous and discrete observations

TL;DR

The paper analyzes parameter estimation for the threshold CKLS (T-CKLS) diffusion, a regime-switching CKLS-type process with piecewise-constant coefficients across multiple thresholds. It develops continuous-time MLE and QMLE for the drift, introduces discretized estimators for discrete observations—including an Itô-Tanaka based discretization—and proves consistency and asymptotic normality under ergodicity and moment conditions. A key methodological advance is the control of threshold-crossing events via hitting-time techniques, enabling high-frequency, long-time results for both drift and diffusion estimators, as well as a practical threshold-detection framework. The authors validate the theory with simulated data and apply the methods to US interest-rate data, demonstrating the presence and estimation of regime thresholds and illustrating the approach’s practical relevance for financial modeling with threshold effects.

Abstract

We consider a continuous time process that is self-exciting and ergodic, called threshold Chan-Karolyi-Longstaff-Sanders (CKLS) process. This process is a generalization of various models in econometrics, such as Vasicek model, Cox-Ingersoll-Ross, and Black-Scholes, allowing for the presence of several thresholds which determine changes in the dynamics. We study the asymptotic behavior of maximum-likelihood and quasi-maximum-likelihood estimators of the drift parameters in the case of continuous time and discrete time observations. We show that for high frequency observations and infinite horizon the estimators satisfy the same asymptotic normality property as in the case of continuous time observations. We also discuss diffusion coefficient estimation. Finally, we apply our estimators to simulated and real data to motivate considering (multiple) thresholds.

Paper Structure

This paper contains 49 sections, 15 theorems, 123 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Outline.
  3. The framework: model and assumptions
  4. The model
  5. Likelihood and quasi-likelihood functions
  6. About our assumptions on ergodicity and moments of the stationary distribution
  7. Estimation from continuous time observations
  8. Estimators expressions
  9. Asymptotic properties: long time
  10. Estimation from discrete observations
  11. Estimators expressions
  12. Asymptotic properties: high frequency - long time
  13. Results
  14. Key elements of the proofs
  15. Numerical Experiments
  16. ...and 34 more sections

Key Result

Proposition 3.1

Let $T\in(0,\infty)$, the maximum of the likelihood $\mathcal{L}_T(\theta;\sigma,\gamma)$ is achieved at $\theta_T^{(\mathcal{L})}:=(a_T^{j,\gamma},b_T^{j,\gamma})_{j=0}^d$ with The maximum of the quasi-likelihood $q\text{-}\mathcal{L}_T(\theta)$ is achieved at $\theta_T^{(q\text{-}\mathcal{L})}:=(a_T^{j,0},b_T^{j,0})_{j=0}^d$, that is

Figures (2)

  • Figure 1: Asymptotic normality property in Theorem \ref{['Theorem_discret_cons_tcl']}, with parameters as in Table \ref{['Parameters_simu']}. We plot the theoritical distribution using \ref{['mesure_stationnaire']} and compare with the empirical distribution on $10^4$ trajectories.
  • Figure 2: The figure shows the interest rate daily data (solid line) for the time window Jan 2016 - Dec 2019, and Jan 2020 - Jan 2024. The fitted thresholds are represented by the dashed lines.

Theorems & Definitions (29)

  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Theorem 3.5
  • proof
  • Lemma 3.6: Ergodic properties
  • Remark 3.7
  • ...and 19 more