Estimation of the elasticity for CKLS model from high-frequency observations

TL;DR

This paper develops a novel high-frequency estimation method for the elasticity parameter in the CKLS diffusion model by transforming it into a CIR-type process and addressing challenges due to Feller's condition failure.

Contribution

It introduces a new estimator for CKLS elasticity based on a transformation to a CIR-type model and drift estimation via Harris recurrence, with proven asymptotic properties.

Findings

Estimator is $p$-consistent and asymptotically normal.

Explicit asymptotic variance derived for the estimator.

Stable convergence in law is invariant under measure changes.

Abstract

We investigate parametric estimation of the elasticity parameter in the CKLS diffusion based on high-frequency data. First, we transform the CKLS diffusion to a CIR-type one via a smooth state-space mapping and the general Girsanov change of measure. This transformation enables the applications of existing inference tools for CIR processes while ensuring possibilities of transferring the resulting limit theorems back to the original probability space. However, because Feller's condition fails, many existing high-frequency likelihood-based procedures cannot be applied directly, since their discretization schemes approximate likelihood terms involving the reciprocal of the process by Riemann sums that are no longer well-defined once the paths are allowed to hit zero. Instead, we estimate the drift coefficient of the transformed CIR-type model via a procedure based on its positive Harris recurrence, which is valid in the high-frequency regime. Exploiting the drift-elasticity relationship implied by the CKLS--CIR transformation, with the help of an initial estimation, we obtain an estimator of the CKLS elasticity from the CIR drift estimator in the transformed model. This yields a closed-form estimator of the elasticity parameter with an explicit asymptotic variance. We establish its $p$-consistency, stable convergence in law, and asymptotic normality. Finally, we show that stable convergence in law is invariant under equivalent changes of measure, thereby guaranteeing that the Gaussian limit remains invariant under the original measure.