Markov-switching threshold stochastic volatility models with regime changes

Ahmed Ghezal; Mohamed balegh; Imane Zemmouri; Ahmed Ghezal; Mohamed balegh; Imane Zemmouri

doi:10.3934/math.2024192

AIMS Mathematics

2024, Volume 9, Issue 2: 3895-3910. doi: 10.3934/math.2024192

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Markov-switching threshold stochastic volatility models with regime changes

1.
Department of Mathematics, Abdelhafid Boussouf University Center of Mila, Algeria
2.
Department of Mathematics, College of Science and Arts, Muhayil, King Khalid University, Abha 61413, Saudi Arabia
3.
Department of Mathematics, University of Annaba, Elhadjar $ 23 $, Annaba, Algeria

Received: 16 November 2023 Revised: 18 December 2023 Accepted: 05 January 2024 Published: 11 January 2024
MSC : 60G10, 62F12

This paper introduces a comprehensive class of models known as Markov-Switching Threshold Stochastic Volatility (MS-TSV) models, specifically designed to address asymmetry and the leverage effect observed in the volatility of financial time series. Extending the classical threshold stochastic volatility model, our approach expresses the parameters governing log-volatility as a function of a homogeneous Markov chain with a finite state space. The primary goal of our proposed model is to capture the dynamic behavior of volatility driven by a Markov chain, enabling the accommodation of both gradual shifts due to economic forces and sudden changes caused by abnormal events. Following the model's definition, we derive several probabilistic properties of the MS-TSV models, including strict (or second-order) stationarity, causality, ergodicity, and the computation of higher-order moments. Additionally, we provide the expression for the covariance function of the squared (or powered) process. Furthermore, we establish the limit theory for the Quasi-Maximum Likelihood Estimator (QMLE) and demonstrate the strong consistency of this estimator. Finally, a simulation study is presented to assess the performance of the proposed estimation method.
- Markov chain,
- threshold stochastic volatility,
- stationarity,
- QMLE
Citation: Ahmed Ghezal, Mohamed balegh, Imane Zemmouri. Markov-switching threshold stochastic volatility models with regime changes[J]. AIMS Mathematics, 2024, 9(2): 3895-3910. doi: 10.3934/math.2024192

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Abstract

This paper introduces a comprehensive class of models known as Markov-Switching Threshold Stochastic Volatility (MS-TSV) models, specifically designed to address asymmetry and the leverage effect observed in the volatility of financial time series. Extending the classical threshold stochastic volatility model, our approach expresses the parameters governing log-volatility as a function of a homogeneous Markov chain with a finite state space. The primary goal of our proposed model is to capture the dynamic behavior of volatility driven by a Markov chain, enabling the accommodation of both gradual shifts due to economic forces and sudden changes caused by abnormal events. Following the model's definition, we derive several probabilistic properties of the MS-TSV models, including strict (or second-order) stationarity, causality, ergodicity, and the computation of higher-order moments. Additionally, we provide the expression for the covariance function of the squared (or powered) process. Furthermore, we establish the limit theory for the Quasi-Maximum Likelihood Estimator (QMLE) and demonstrate the strong consistency of this estimator. Finally, a simulation study is presented to assess the performance of the proposed estimation method.

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