Generalized Autoregressive Conditional Heteroskedasticity Model Research Articles

Matrix GARCH model: Inference and application*

Matrix-variate time series data are largely available in applications. However, no attempt has been made to study their conditional heteroskedasticity that is often observed in economic and financial data. To address this gap, we propose a novel matrix generalized autoregressive conditional heteroskedasticity (GARCH) model to capture the dynamics of conditional row and column covariance matrices of matrix time series. The key innovation of the matrix GARCH model is the use of a univariate GARCH specification for the trace of conditional row or column covariance matrix, which allows for the model identification. Moreover, we introduce a quasi-maximum likelihood estimator (QMLE) for model estimation and develop a portmanteau test for model diagnostic checking. Simulation studies are conducted to assess the finite-sample performance of the QMLE and portmanteau test. To handle large dimensional matrix time series, we also propose a matrix factor GARCH model, and establish its theoretical properties. Finally, we demonstrate the superiority of the matrix GARCH and matrix factor GARCH models over existing multivariate GARCH-type models in volatility forecasting and portfolio allocations using three applications on credit default swap prices, global stock sector indices, and future prices.

Time Series Analysis of Onion and Potato Prices using Parametric Models

This study presents a detailed analysis of onion and potato price series using parametric models, specifically Auto- regressive Integrated Moving Average (ARIMA) and Generalized Autoregressive Conditional Heteroscedasticity (GARCH). Data from various Indian markets were used to identify trends, correlations, and volatility. The findings indicate significant volatility in onion prices compared to potato prices, with both series demonstrating non- stationarity and non-normality. The ARIMA models were effective in capturing the seasonality and trends in the data, while GARCH models provided insights into the conditional variance.

Safe Havens Characteristics of Islamic Stock Indices During Bearish and Bullish Markets: A Case of Developed and Developing Countries

Purpose — This study examines the performance of Islamic and conventional indices during bullish (positive) and bearish (negative) markets while the economic and financial conditions are different. Design/Methodology/Approach — This study uses GARCH (1,1) and wavelet-assisted cross-spectral, cross-correlation analyses to investigate the performance of Islamic and conventional stock indices and study the presence of safe havens, which reflect low volatility in the indices during times of market turbulence, for investors during bearish and bullish periods from 2019–2021. Findings — The paper finds that there is high and persistent volatility in Islamic indices in bullish markets, while in bearish periods the Islamic indices show low volatility persistence; and conventional indices were highly volatile during crisis periods. Also, the impact of news shocks on Islamic indices was intense before COVID-19, while after that news, shocks had a high impact on the conventional indices. The wavelet coherence analysis shows that there is high coherency between Islamic and conventional indices in developed countries but in developing countries, the opposite prevails. Originality/Value — The study is the first to use both Generalised Autoregressive Conditional Heteroskedasticity (GARCH) and wavelet coherence models for examining Islamic and conventional stock indices during bearish and bullish markets, especially in the COVID-19 pandemic era. Practical Implications — The results conclude that there are safe havens present in Islamic indices in both developing and developed countries, especially the United States (US), the United Kingdom (UK), India, and Indonesia, while in other countries the presence is weak. Both GARCH and wavelet coherence models show the same results regarding the presence of volatility and safe havens for investors in different countries.

Multiscale neural dynamics in sleep transition volatility across age scales: a multimodal EEG-EMG-EOG analysis of temazepam effects.

Recent advances in computational modeling techniques have facilitated a more nuanced understanding of sleep neural dynamics across the lifespan. In this study, we tensorize multiscale multimodal electroencephalogram (EEG), electromyogram (EMG), and electrooculogram (EOG) signals and apply Generalized Autoregressive Conditional Heteroskedasticity (GARCH) modeling to quantify interactions between age scales and the use of pharmacological sleep aids on sleep stage transitions. Our cohort consists of 22 subjects in a crossover design study, where each subject received both a sleep aid and a placebo in different sessions. To understand these effects across the lifespan, three evenly distributed age groups were formed: 18-29, 30-49, and 50-66years. The methodological framework implemented here employs tensor-based machine learning techniques to compute continuous wavelet transform time-frequency features and utilizes a GARCH model to quantify sleep signal volatility across age scales. Support Vector Machines are used for feature ranking, and our analysis captures interactions between signal entropy, age, and sleep aid status across frequency bands, sleep transitions, and sleep stages. GARCH model results reveal statistically significant volatility clustering in EEG, EMG, and EOG signals, particularly during transitions between REM and non-REM sleep. Notably, volatility was higher in the 50-66 age group compared to the 18-29 age group, with marked fluctuations during transitions from deep sleep to REM sleep (standard deviation of 0.35 in the older group vs. 0.30 in the 18-29 age group, p < 0.05). Statistical comparisons of volatility across frequency bands, age scales, and sleep stages highlight significant differences attributable to sleep aid use. Mean conditional volatility parameterization of the GARCH model reveals directional influences, with a causality index of 0.75 from frontal to occipital regions during REM sleep transition periods. Our methodological framework identifies distinct neural behavior patterns across age groups associated with each sleep stage and transition, offering insights into the development of targeted interventions for sleep regularity across the lifespan.

The Effect of Exchange Rate Volatility on Foreign Tourist Visits in Indonesia: GARCH Analysis

In this era of globalization, currency exchange rates are one of the crucial factors affecting a country's economy. The exchange rate of a country's currency plays an important role in economic stability and international trade, especially in the tourism sector. This study aims to conduct an in-depth analysis to identify patterns of rupiah exchange rate volatility using the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model, and then examine whether these patterns have a significant correlation with the number of foreign tourist arrivals in Indonesia. In analyzing the data, researchers conducted several classical assumption tests, namely the normality test and the heteroscedasticity test. The test results show that the data used, namely the rupiah exchange rate data and the number of foreign tourist visits, are not normally distributed and heteroscedastic. In this case, it resulted in researchers not use the Pearson correlation test so they used the Kendall-Tau correlation test and Granger Causality test which resulted in the two variables being correlated. Furthermore, GARCH modeling is carried out which provides forecasting on both data to increase in 2024. The results of this study are expected to provide a basis for consideration for stakeholders to determine policies, especially in the financial and tourism sectors.

Applications of GARCH Models in Forecasting Financial Market Volatility: Insights from Leading Global Stock Indexes

This study investigates the volatility dynamics of major global stock indexes, including the FTSE 100, Hang Seng Index, NIKKEI 225, and S&amp;P 500, using a range of Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models. The analysis spans a comprehensive 20-year period from January 1, 2004, to December 31, 2023, encompassing diverse market conditions such as bull and bear markets, the 2008 financial crisis, and the COVID-19 pandemic. The methodology includes preprocessing steps such as calculating daily log returns, performing descriptive statistics, and conducting stationarity and ARCH effect tests to ensure data suitability for volatility modelling. The study evaluates several GARCH models, including GARCH, EGARCH, NGARCH, APARCH, GJR-GARCH, and TGARCH, to forecast volatility and address both symmetric and asymmetric effects. The TGARCH model exhibits strong performance in capturing leverage effects and asymmetries, particularly for the FTSE 100 and Hang Seng Index. The APARCH model performs well for the S&amp;P 500, demonstrating sensitivity to past shocks. Overall, the findings underscore the importance of advanced GARCH models in accurately predicting volatility in global financial markets, highlighting the TGARCH model's effectiveness in addressing asymmetries and providing insights into selecting appropriate models for enhanced financial analysis and risk management.

Estimating and forecasting bitcoin daily prices using ARIMA-GARCH models

PurposeThe goal of the study is to offer important insights into the dynamics of the cryptocurrency market by analyzing pricing data for Bitcoin. Using quantitative analytic methods, the study makes use of a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model and an Autoregressive Integrated Moving Average (ARIMA). The study looks at how predictable Bitcoin price swings and market volatility will be between 2021 and 2023.Design/methodology/approachThe data used in this study are the daily closing prices of Bitcoin from Jan 17th, 2021 to Dec 17th, 2023, which corresponds to a total of 1065 observations. The estimation process is run using 3 years of data (2021–2023), while the remaining (Jan 1st 2024 to Jan 17th 2024) is used for forecasting. The ARIMA-GARCH method is a robust framework for forecasting time series data with non-seasonal components. The model was selected based on the Akaike Information Criteria corrected (AICc) minimum values and maximum log-likelihood. Model adequacy was checked using plots of residuals and the Ljung–Box test.FindingsUsing the Box–Jenkins method, various AR and MA lags were tested to determine the most optimal lags. ARIMA (12,1,12) is the most appropriate model obtained from the various models using AIC. As financial time series, such as Bitcoin returns, can be volatile, an attempt is made to model this volatility using GARCH (1,1).Originality/valueThe study used partially processed secondary data to fit for time series analysis using the ARIMA (12,1,12)-GARCH(1,1) model and hence reliable and conclusive results.

Comparative Analysis of GARCH-Based Volatility Models of Financial Market Volatility: A Case of Nairobi Security Market PLC, Kenya

This paper conducted a comprehensive comparative analysis of various GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models to forecast financial market volatility, with a specific focus on the Nairobi Stock Exchange Market. The examined models include symmetric and asymmetric GARCH types, such as sGARCH, GJR-GARCH, AR (1) GJG-GARCH, among others. The primary objective is to identify the most suitable model for capturing the complex dynamics of financial market volatility. The study employs rigorous evaluation criteria, including the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Mean Error (ME), and Root Mean Absolute Error (RMAE), to assess the performance of each model. These criteria facilitate the selection of the optimal model for volatility forecasting. The analysis reveals that the GJR-GARCH (1,1) model emerges as the best-fit model, with AIC and BIC values of −5.5008 and −5.4902, respectively. This selection aligns with the consensus in the literature, highlighting the superiority of asymmetric GARCH models in capturing volatility dynamics. The comparison also involves symmetric GARCH models, such as sGARCH (1,1), and other asymmetric models like AR (1) GJG-GARCH. While these models were considered, the GJR-GARCH (1,1) model demonstrated superior forecasting capabilities. The study emphasizes the importance of accurate model selection and the incorporation of asymmetry in volatility modeling. The research provides essential insights into financial market volatility modeling and forecasting using both asymmetric and symmetric GARCH models. These findings have significant implications for government policymakers, financial institutions, and investors, offering improved tools for risk assessment and decision-making during periods of market turbulence.

Utilizing the GARCH Model for Analysis and Prediction of Stock Market Trends

This study utilizes the Generalized Auto-regressive Conditional Heteroskedasticity (GARCH) model to analyze the Shanghai Stock Exchange Composite Index and the Shenzhen Composite Index. It computes the returns and logarithmic returns, fits the GARCH model with a student-t distribution, and calculates the standardized residuals of the stock. Tests for normality and heteroskedasticity of residuals are conducted, follows by the creation of auto-correlation and partial auto-correlation plots to aid in volatility forecasting. The findings indicate that both returns of stock distributions exhibit multiple peaks and significant fluctuations, revealing that there exists pronounced volatility clustering and strong abrupt changes. Historical fluctuations significantly affect current fluctuations, showing a clustering effect. Volatility in returns has been very persistent, reflecting strong speculation and elevated overall risk. Histograms and skewness assessments show a left-skewed distribution for both indices, with a kurtosis greater than three, indicating a "fat-tailed" distribution. The positive mean values imply that China's economy is growing constantly, while the low standard deviation implies the immaturity of China's stock market. These insights highlight the dynamic behaviors and risk profiles in Chinas major stock indices, offering a deeper understanding of their speculative nature and economic implications. The Advantages and disadvantages of GARCH models and limitations of a single model are acknowledged in the report.

The COVID-19 Outbreak and Volatility Spillover Between Stock Exchange’s Overall and Small- and Medium-enterprise Indices: Evidence from India

The research examines the dynamic correlation between the volatility of the Bombay Stock Exchange (BSE) and the Small- and Medium-enterprises (SMEs) Index of BSE in India. It focuses on analysing the volatility spillover between SME and BSE by utilising the generalised autoregressive conditional heteroskedasticity (GARCH) model. The study extends its scope by examining the impact of the current pandemic, COVID-19, on the volatility of both SME and BSE by utilising the exponential GARCH and threshold GARCH models. Findings suggest that there is volatility spillover between the two indices. The results also highlight that the pandemic significantly affected the volatility of both indices. The findings of the study provide important implications for policymakers, portfolio managers, and investors, particularly when the Indian government has been emphasising on the growth and continuous support of the small and medium enterprises and increasing foreign investments in India’s SMEs in recent years.

Heteroscedasticity effects as component to future stock market predictions using RNN-based models.

Heteroscedasticity effects are useful for forecasting future stock return volatility. Stock volatility forecasting provides business insight into the stock market, making it valuable information for investors and traders. Predicting stock volatility is a crucial task and challenging. This study proposes a hybrid model that predicts future stock volatility values by considering the heteroscedasticity element of the stock price. The proposed model is a combination of Generalized Autoregressive Conditional Heteroskedasticity (GARCH) and a well-known Recurrent Neural Network (RNN) algorithm Long Short-Term Memory (LSTM). This proposed model is referred to as GARCH-LSTM model. The proposed model is expected to improve prediction accuracy by considering heteroscedasticity elements. First, the GARCH model is employed to estimate the model parameters. After that, the ARCH effect test is used to test the residuals obtained from the model. Any untrained heteroscedasticity element must be found using this step. The hypothesis of the ARCH test yielded a p-value less than 0.05 indicating there is valuable information remaining in the residual, known as heteroscedasticity element. Next, the dataset with heteroscedasticity is then modelled using an LSTM-based RNN algorithm. Experimental results revealed that hybrid GARCH-LSTM had the lowest MAE (7.961), RMSE (10.466), MAPE (0.516) and HMAE (0.005) values compared with a single LSTM. The accuracy of forecasting was also significantly improved by 15% and 13% with hybrid GARCH-LSTM in comparison to single LSTMs. Furthermore, the results reveal that hybrid GARCH-LSTM fully exploits the heteroscedasticity element, which is not captured by the GARCH model estimation, outperforming GARCH models on their own. This finding from this study confirmed that hybrid GARCH-LSTM models are effective forecasting tools for predicting stock price movements. In addition, the proposed model can assist investors in making informed decisions regarding stock prices since it is capable of closely predicting and imitating the observed pattern and trend of KLSE stock prices.

Support vector regression-based heteroscedastic models for cryptocurrency risk forecasting

In this study, we aimed to combine support vector regression (SVR) with an autoregressive (AR) model and a generalized autoregressive conditional heteroscedastic (GARCH) or asymmetric GARCH model. We compared the performances of these combined models with the maximum likelihood estimation (MLE)-based models under a normal, Student’s t, or skewed Student’s t distributional assumption. We employed such models to protect the heteroscedastic risks of four major cryptocurrencies, namely, Bitcoin, Ethereum, Tether, and Binance Coin, before and during COVID-19. We accomplished this task by forecasting their volatility, Value-at-Risk (VaR), and Expected Shortfall (ES). The empirical results revealed that in terms of the root-mean-square error (RMSE) and mean-absolute error (MAE) metrics, the SVR-based models with a polynomial or radial basis kernel function, which is nonlinear, resulted in a more accurate volatility forecast compared to the MLE-based models under any distributional assumption. The superiority of the former models was more apparent when applied to Tether exhibiting the most stable volatility. This supports evidence of nonlinear characteristics in the cryptocurrency return and volatility series. A combination between SVR and an asymmetric GARCH model also confirms the presence of an asymmetrically nonlinear relationship between the returns and volatility. In addition, the correct-VaR and Kupiec’s backtesting results showed that the SVR-based models tended to produce the VaR forecasts for the four cryptocurrencies with better accuracy at the 1%, 5%, and 10% significance levels compared to the MLE-based ones. The use of the correct-ES metric also leads to the superior ES forecasting performance of the former models at any level. However, McNeil and Frey’s backtesting results demonstrated their worse ES forecasting performance at the 1% level. As the COVID-19 pandemic progressed, we also found an increase in the Bitcoin and Binance Coin market risks as well as a decrease in the Ethereum and Tether market risks.

The impact of the COVID 19 pandemic on stock market volatility: evidence from a selection of developed and emerging stock markets

This study examines the impact of the COVID 19 pandemic on the stock markets of China, India, Pakistan, the UK and the US using Generalised Autoregressive Conditional Heteroscedasticity (GARCH) and Threshold GARCH models with COVID 19 as an exogenous dummy variable in the variance equation. The sample period of 2016–2021 is divided into two sub-periods: the pre-COVID 19 period and the COVID 19 period. The results of the study indicate that there was persistent volatility in these markets and that this volatility increased as a result of the pandemic. In addition, the Threshold GARCH results indicate that the asymmetric term was significant in all markets indicating that bad news, such as the pandemic, had a stronger impact on the conditional variance of the returns as compared to good news. In addition, the results further confirm that the US market had no significant impact on the volatility of the Chinese market during the pandemic. The results have important implications for (1) international investors regarding portfolio management and investment risk minimisation in situations like the COVID 19 pandemic; and (2) policy-makers in terms of how they respond to any future pandemic.

Forecasting tail risk of skewed financial returns having exponential‐polynomial tails

AbstractAggregated long and short trading risk positions of speculative assets over time are likely to be unequal. This may be because of irrational decisions of traders and investors as well as catastrophic events that lead to pronounce or salient market crashes. Returns of such assets are therefore more likely to have one polynomial tail and one exponential tail. The generalized hyperbolic (GH) skewed Student‐t distribution is known to handle such situations quite well. In this paper, we use generalized autoregressive conditional heteroscedasticity (GARCH) models to empirically show the superiority of the GH skewed Student‐t distribution in forecasting the extreme tail risks of cryptocurrency returns in the presence of substantial skewness in comparison with some competing distributions. Furthermore, we show the practical significance of the GH skewed Student‐t distribution‐based risk forecasts in computing daily capital requirements. Evidence from the study suggests that the GH skewed Student‐t distribution model tends to be superior in forecasting volatility and expected shortfall (ES) but not value‐at‐risk. In addition, the distribution yields higher value‐at‐risk (VaR) exceptions but surprisingly avoids the red zone of the Basel II accord penalty zones and produces lower but optimal daily capital requirements. Therefore, in the presence of substantially skewed returns having exponential‐polynomial tails, we recommend the use of the GH skewed Student‐t distribution for parametric GARCH models in forecasting extreme tail risk.

Comparative Analysis of Bilinear Time Series Models with Time-Varying and Symmetric GARCH Coefficients: Estimation and Simulation

This paper makes a significant contribution by focusing on estimating the coefficients of a sample of non-linear time series, a subject well-established in the statistical literature, using bilinear time series. Specifically, this study delves into a subset of bilinear models where Generalized Autoregressive Conditional Heteroscedastic (GARCH) models serve as the white noise component. The methodology involves applying the Klimko–Nilsen theorem, which plays a crucial role in extracting the asymptotic behavior of the estimators. In this context, the Generalized Autoregressive Conditional Heteroscedastic model of order (1,1) noted that the GARCH (1,1) model is defined as the white noise for the coefficients of the example models. Notably, this GARCH model satisfies the condition of having time-varying coefficients. This study meticulously outlines the essential stationarity conditions required for these models. The estimation of coefficients is accomplished by applying the least squares method. One of the key contributions lies in utilizing the fundamental theorem of Klimko and Nilsen, to prove the asymptotic behavior of the estimators, particularly how they vary with changes in the sample size. This paper illuminates the impact of estimators and their approximations based on varying sample sizes. Extending our study to include the estimation of bilinear models alongside GARCH and GARCH symmetric coefficients adds depth to our analysis and provides valuable insights into modeling financial time series data. Furthermore, this study sheds light on the influence of the GARCH white noise trace on the estimation of model coefficients. The results establish a clear connection between the model characteristics and the nature of the white noise, contributing to a more profound understanding of the relationship between these elements.

Online Muscle Activation Onset Detection Using Likelihood of Conditional Heteroskedasticity of Electromyography Signals.

The real-time, accurate detection of muscle activation onset (MAO) is significant for EMG-triggered control strategies in embedded applications like prostheses and exoskeletons. This paper investigates sEMG signals using the generalized autoregressive conditional heteroskedasticity (GARCH) model, focusing on variance. A novel feature, the likelihood of conditional heteroskedasticity (LCH) extracted from the maximum likelihood estimation of GARCH parameters, is proposed. This feature effectively distinguishes signal from noise based on heteroskedasticity, allowing for the detection of MAO through the LCH feature and a basic threshold classifier. For online calculation, the model parameter estimation is simplified, enabling direct calculation of the LCH value using fixed parameters. The proposed method was validated on two open-source datasets and demonstrated superior performance over existing methods. The mean absolute error of onset detection, compared with visual detection results, is approximately 65 ms under online conditions, showcasing high accuracy, universality, and noise insensitivity. The results indicate that the proposed method using the LCH feature from the GARCH model is highly effective for real-time detection of muscle activation onset in sEMG signals. This novel approach shows great potential and possibility for real-world applications, reflecting its superior performance in accuracy, universality, and insensitivity to noise.

Quantitative Assessment and Impact Analysis of Land Surface Deformation in Wuxi Based on PS-InSAR and GARCH Model

Land surface deformation, including subsidence and uplift, has significant impacts on human life and the natural environment. In recent years, the city of Wuxi, China has experienced large-scale surface deformation following the implementation of a groundwater abstraction ban policy in 2005. To accurately measure the regional impacts and understand the underlying mechanisms, we investigated the spatiotemporal characteristics of surface deformation in Wuxi from 2015 to 2023 using 100 Sentinel-1A SAR images and the Persistent Scatterer InSAR (PS-InSAR) technique. The results revealed that surface deformation in Wuxi exhibited significant spatial and temporal variations, with some areas experiencing alternating trends of subsidence and uplift rather than consistent unidirectional change. To uncover the factors influencing this volatility, we conducted a comprehensive analysis focusing on groundwater, precipitation, and soil geology. This study found strong correlations between the groundwater level changes and surface deformation, with the soft soil geology of the area, characterized by alternating layers of sand and clay, further increasing the surface volatility. Moreover, we innovatively applied the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model, typically used in financial analyses, to analyze the subsidence displacement time series in Wuxi. Based on this model, we propose a new “Amplitude Factor” index to evaluate overall surface deformation volatility in the city. Our qualitative assessment of surface stability based on the Amplitude Factor was consistent with research findings, demonstrating the accuracy and effectiveness of the proposed model. These results provide valuable insights for urban planning, construction, and safety control, highlighting the importance of continuous monitoring and analysis of surface deformation volatility for the city’s future development and safety.

Crude Oil Prices Forecast Based on Mixed-Frequency Deep Learning Approach and Intelligent Optimization Algorithm.

Precisely forecasting the price of crude oil is challenging due to its fundamental properties of nonlinearity, volatility, and stochasticity. This paper introduces a novel hybrid model, namely, the KV-MFSCBA-G model, within the decomposition-integration paradigm. It combines the mixed-frequency convolutional neural network-bidirectional long short-term memory network-attention mechanism (MFCBA) and generalized autoregressive conditional heteroskedasticity (GARCH) models. The MFCBA and GARCH models are employed to respectively forecast the low-frequency and high-frequency components decomposed through variational mode decomposition optimized by Kullback-Leibler divergence (KL-VMD). The classification of these components is performed using the fuzzy entropy (FE) algorithm. Therefore, this model can fully exploit the advantages of deep learning networks in fitting nonlinearities and traditional econometric models in capturing volatilities. Furthermore, the intelligent optimization algorithm and the low-frequency economic variable are introduced to improve forecasting performance. Specifically, the sparrow search algorithm (SSA) is employed to determine the optimal parameter combination of the MFCBA model, which is incorporated with monthly global economic conditions (GECON) data. The empirical findings of West Texas Intermediate (WTI) and Brent crude oil indicate that the proposed approach outperforms other models in evaluation indicators and statistical tests and has good robustness. This model can assist investors and market regulators in making decisions.

Decoding herding dynamics in the generative AI investment amid key technological advancements: A timeline perspective

This study, using two herding dynamics metrics and Glosten–Jagannathan–Runkle Generalized Autoregressive Conditional Heteroskedasticity (GJR-GARCH) model, forecasts market trends, captures asymmetric volatility, and reveals the generative AI (GenAI) ecosystem's impact on individual assets’ returns. Results of this study highlight distinctive traits of each GenAI equity, crucial for strategic positioning, especially for investors in tech stocks tied to GenAI. Herding behavior exhibits greater strength in the initial four months post-announcement of ChatGPT, gradually diminishing. GJR-GARCH reports that most of GenAI stocks do not exhibit statistically significant leverage effects. These findings provide valuable insights to navigate the dynamic landscape of GenAI investments.

VIX option pricing through nonaffine GARCH dynamics and semianalytical formula

AbstractThis paper develops analytical approximations for volatility index (VIX) option pricing under nonaffine generalized autoregressive conditional heteroskedasticity (GARCH) models as advocated by Christoffersen et al. We obtain the approximation formulae for pricing VIX options and then evaluate their performance with three expansions under four empirically well‐tested models. Our numerical experiments find that the weighted expansion generated by the fat‐tailed weighting kernel can significantly reduce approximation error over the Gram‐Charlier expansion; the Taylor expansion of conditional moments can lead to divergence for parameters with certain high persistence in the affine GARCH, nonlinear asymmetric GARCH, and Glosten‐Jagannathan‐Runkle GARCH models, while surviving during high persistence in the exponential GARCH.