This paper introduces a novel and enhanced class of Markov-switching threshold BLGARCH (MS-TBLG) models designed to better capture the intricate behavior of financial time series. By incorporating both a threshold mechanism and interaction effects between returns and their conditional volatility, the proposed framework significantly improves upon traditional Markov-switching BLGARCH (MS-BLG) models. This structure allows for a more flexible representation of regime-dependent volatility, particularly in modeling asymmetric effects such as the leverage effect—where negative and positive shocks have differing impacts across regimes. We established key conditions that guarantee the stationarity, causality, and ergodicity of the MS-TBLG process, and derived analytical expressions for its power covariance functions, taking the threshold effect into account. To estimate the model parameters efficiently, we developed a generalized method of moments (GMM) procedure specifically adapted to the complexity of Markov-switching dynamics. This approach utilizes moment conditions derived from the power-transformed squared process, ensuring consistent estimation despite the presence of unobserved regime shifts. The effectiveness and robustness of the estimation strategy were validated through extensive Monte Carlo simulations. Finally, the model's practical relevance was illustrated through an empirical application to oil price data, showcasing its effectiveness in capturing regime-switching behavior and complex volatility dynamics.

This paper introduces a novel logarithmic threshold stochastic volatility $GARCH$ model as an advanced extension of traditional $GARCH$ frameworks. The model combines a logarithmic transformation of volatility shocks with a dynamic threshold, allowing it to better capture asymmetric behavior and sudden regime shifts commonly observed in financial markets. We provide clear theoretical conditions for strict and second-order stationarity, and for the existence of higher-order moments, which fills an important gap in the literature on stochastic volatility models. Monte Carlo simulations demonstrate the model's efficiency in estimating parameters, yielding accurate results with minimal bias for a sample size of 5,000. When applied to Algerian Dinar/Euro exchange rate data from 2000 to 2011, the model successfully captures volatility clustering and leverage effects, revealing a 30% increase in volatility in response to negative shocks relative to positive ones. It also improves predictive accuracy by 15% over standard models, underscoring its strength in capturing volatility in emerging markets with complex and nonlinear patterns.

In this paper, we presented a mathematical model to analyze financial instruments that are sensitive to the difference between the highest and lowest prices of n independent stocks in a random volatility environment. This model relies on the multivariate distribution of the ranges of n independent Wiener processes, describing the difference between the highest and lowest stock prices for a known time period. In addition to deriving the statistical characteristics of this distribution and its truncated version, including reliability properties, moments, the stress–strength parameter, and order statistics; we considered Bonferroni and Lorenz curves and the Gini index of the proposed model, as well as assessed its robustness in turbulent market environments. The proposed distribution enhances the modeling of range-based financial products to enable the construction of more efficient risk management and hedging strategies. Simulations with real financial data also confirmed its effectiveness in modeling range-based products and reducing volatility in markets.