The Threshold log GARCH Stochastic Volatility Model

In this article, we introduce a new class of volatility models, referred to asThreshold log GARCH-SV (T-log GARCH-SV) models, designed tocapture asymmetry and leverage effects in the volatility of financial timeseries. Our model extends the symmetric log GARCH-SV framework byincorporating a threshold effect into the log-volatility equation, makingit particularly well-suited for modeling time-varying volatility withregime-dependent dynamics. This new formulation enhances the flexibility ofexisting SV-type models. We establish a sufficient condition to ensurestrict stationarity, geometric ergodicity, and the existence of higher-orderlog-moments. For parameter estimation, we propose a quasi-maximumlikelihood approach that combines an Expectation Maximization algorithm withparticle filtering and smoothing techniques. The performance of the proposedmethod is evaluated through a Monte Carlo simulation study. Finally, we applyour formulation to the S&P 500 stock index. The empirical resultsdemonstrate that the T-log GARCH-SV model outperforms competing$SV$-type models in volatility forecasting, highlighting its effectiveness incapturing asymmetries in financial return dynamics.