An Accurate Discretized Approach to Parameter Estimation in the CKLS Model via the CIR Framework

Paper: arXiv 2507.10041

Authors: Sourojyoti Barick

Abstract

This paper provides insight into the estimation and asymptotic behavior of parameters in interest rate models, focusing primarily on the Cox-Ingersoll-Ross (CIR) process and its extension – the more general Chan-Karolyi-Longstaff-Sanders (CKLS) framework ($α\in[“0.5,1”]$). The CIR process is widely used in modeling interest rates which possess the mean reverting feature. An Extension of CIR model, CKLS model serves as a foundational case for analyzing more complex dynamics. We employ Euler-Maruyama discretization to transform the continuous-time stochastic differential equations (SDEs) of these models into a discretized form that facilitates efficient simulation and estimation of parameters using linear regression techniques. We established the strong consistency and asymptotic normality of the estimators for the drift and volatility parameters, providing a theoretical underpinning for the parameter estimation process. Additionally, we explore the boundary behavior of these models, particularly in the context of unattainability at zero and infinity, by examining the scale and speed density functions associated with generalized SDEs involving polynomial drift and diffusion terms. Furthermore, we derive sufficient conditions for the existence of a stationary distribution within the CKLS framework and the corresponding stationary density function; and discuss its dependence on model parameters for $α\in[“0.5,1”]$.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper employs advanced stochastic calculus, SDE analysis, and asymptotic proofs, resulting in high mathematical density, but relies on theoretical derivations and simulations without providing ready-to-use code, datasets, or rigorous backtesting results.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Estimate Parameters & Analyze<br>Asymptotics in CIR/CKLS Models"] --> B["Methodology:<br>Euler-Maruyama Discretization of SDEs"]
  B --> C["Computational Process:<br>Linear Regression on<br>Discretized Equations"]
  C --> D{"Key Outcomes"}
  D --> E["Theoretical Guarantees:<br>Strong Consistency &<br>Asymptotic Normality"]
  D --> F["Boundary Analysis:<br>Unattainability at Zero/Infinity<br>via Scale/Speed Functions"]
  D --> G["Stationarity:<br>Existence & Density of<br>Stationary Distribution"]

Cited References (25)

1. Convergence of an Euler scheme for a hybrid stochastic-local volatility model with stochastic rates in foreign exchange markets (2016) — link
2. An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process (2012) — link
3. An Empirical Comparison of Alternative Models of the Short-Term Interest Rate (1992) — link
4. A Theory of the Term Structure of Interest Rates (1985)
5. Estimation for Continuous Branching Processes (1998)
6. Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model (2013) — link
7. Bayesian Estimation of CIR Model (2022) — link
8. Asymptotic properties of maximum likelihood estimator for the growth rate of a stable CIR process based on continuous time observations (2019) — link

0 of 25 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.