First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The paper introduces a GP-Vol model to flexibly capture the time-dependent changes in variance, and develops a new online algorithm for fully Bayesian inference under the model. The paper is clearly written, the developed inference method seems technically sound, and the presented results look promising. My opinion on the model itself, using a non-parametric approach such as using the GP prior on the transition function (as in the paper), seems, though, a bit an obvious way of extending the prior work developed in the finance area. So, I wouldn't put too high grade on the paper in terms of its originality.

This study evaluates deep neural networks for forecasting probability distributions of financial returns. 1D convolutional neural networks (CNN) and Long Short-Term Memory (LSTM) architectures are used to forecast parameters of three probability distributions: Normal, Student's t, and skewed Student's t. Using custom negative log-likelihood loss functions, distribution parameters are optimized directly. The models are tested on six major equity indices (S\&P 500, BOVESPA, DAX, WIG, Nikkei 225, and KOSPI) using probabilistic evaluation metrics including Log Predictive Score (LPS), Continuous Ranked Probability Score (CRPS), and Probability Integral Transform (PIT). Results show that deep learning models provide accurate distributional forecasts and perform competitively with classical GARCH models for Value-at-Risk estimation. The LSTM with skewed Student's t distribution performs best across multiple evaluation criteria, capturing both heavy tails and asymmetry in financial returns. This work shows that deep neural networks are viable alternatives to traditional econometric models for financial risk assessment and portfolio management.

This study seeks to advance the understanding and prediction of stock market return uncertainty through the application of advanced deep learning techniques. We introduce a novel deep learning model that utilizes a Gaussian mixture distribution to capture the complex, time-varying nature of asset return distributions in the Chinese stock market. By incorporating the Gaussian mixture distribution, our approach effectively characterizes short-term fluctuations and non-traditional features of stock returns, such as skewness and heavy tails, that are often overlooked by traditional models. Compared to GARCH models and their variants, our method demonstrates superior performance in volatility estimation, particularly during periods of heightened market volatility. It provides more accurate volatility forecasts and offers unique risk insights for different assets, thereby deepening the understanding of return uncertainty. Additionally, we propose a novel use of Code embedding which utilizes a bag-of-words approach to train hidden representations of stock codes and transforms the uncertainty attributes of stocks into high-dimensional vectors. These vectors are subsequently reduced to two dimensions, allowing the observation of similarity among different stocks. This visualization facilitates the identification of asset clusters with similar risk profiles, offering valuable insights for portfolio management and risk mitigation. Since we predict the uncertainty of returns by estimating their latent distribution, it is challenging to evaluate the return distribution when the true distribution is unobservable. However, we can measure it through the CRPS to assess how well the predicted distribution matches the true returns, and through MSE and QLIKE metrics to evaluate the error between the volatility level of the predicted distribution and proxy measures of true volatility.

The Pairwise Markov Chain (PMC) is a probabilistic graphical model extending the well-known Hidden Markov Model. This model, although highly effective for many tasks, has been scarcely utilized for continuous value prediction. This is mainly due to the issue of modeling observations inherent in generative probabilistic models. In this paper, we introduce a new algorithm for prediction with the PMC. On the one hand, this algorithm allows circumventing the feature problem, thus fully exploiting the capabilities of the PMC. On the other hand, it enables the PMC to extend any predictive model by introducing hidden states, updated at each time step, and allowing the introduction of non-stationarity for any model. We apply the PMC with its new algorithm for volatility forecasting, which we compare to the highly popular GARCH(1,1) and feedforward neural models across numerous pairs. This is particularly relevant given the regime changes that we can observe in volatility. For each scenario, our algorithm enhances the performance of the extended model, demonstrating the value of our approach.

Volatility, which indicates the dispersion of returns, is a crucial measure of risk and is hence used extensively for pricing and discriminating between different financial investments. As a result, accurate volatility prediction receives extensive attention. The Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model and its succeeding variants are well established models for stock volatility forecasting. More recently, deep learning models have gained popularity in volatility prediction as they demonstrated promising accuracy in certain time series prediction tasks. Inspired by Physics-Informed Neural Networks (PINN), we constructed a new, hybrid Deep Learning model that combines the strengths of GARCH with the flexibility of a Long Short-Term Memory (LSTM) Deep Neural Network (DNN), thus capturing and forecasting market volatility more accurately than either class of models are capable of on their own. We refer to this novel model as a GARCH-Informed Neural Network (GINN). When compared to other time series models, GINN showed superior out-of-sample prediction performance in terms of the Coefficient of Determination ($R^2$), Mean Squared Error (MSE), and Mean Absolute Error (MAE).

The AI traders in financial markets have sparked significant interest in their effects on price formation mechanisms and market volatility, raising important questions for market stability and regulation. Despite this interest, a comprehensive model to quantitatively assess the specific impacts of AI traders remains undeveloped. This study aims to address this gap by modeling the influence of AI traders on market price formation and volatility within a multi-agent framework, leveraging the concept of microfoundations. Microfoundations involve understanding macroeconomic phenomena, such as market price formation, through the decision-making and interactions of individual economic agents. While widely acknowledged in macroeconomics, microfoundational approaches remain unexplored in empirical finance, particularly for models like the GARCH model, which captures key financial statistical properties such as volatility clustering and fat tails. This study proposes a multi-agent market model to derive the microfoundations of the GARCH model, incorporating three types of agents: noise traders, fundamental traders, and AI traders. By mathematically aggregating the micro-structure of these agents, we establish the microfoundations of the GARCH model. We validate this model through multi-agent simulations, confirming its ability to reproduce the stylized facts of financial markets. Finally, we analyze the impact of AI traders using parameters derived from these microfoundations, contributing to a deeper understanding of their role in market dynamics.

This paper investigates the forecasting performance of COMEX copper futures realized volatility across various high-frequency intervals using both econometric volatility models and deep learning recurrent neural network models. The econometric models considered are GARCH and HAR, while the deep learning models include RNN (Recurrent Neural Network), LSTM (Long Short-Term Memory), and GRU (Gated Recurrent Unit). In forecasting daily realized volatility for COMEX copper futures with a rolling window approach, the econometric models, particularly HAR, outperform recurrent neural networks overall, with HAR achieving the lowest QLIKE loss function value. However, when the data is replaced with hourly high-frequency realized volatility, the deep learning models outperform the GARCH model, and HAR attains a comparable QLIKE loss function value. Despite the black-box nature of machine learning models, the deep learning models demonstrate superior forecasting performance, surpassing the fixed QLIKE value of HAR in the experiment. Moreover, as the forecast horizon extends for daily realized volatility, deep learning models gradually close the performance gap with the GARCH model in certain loss function metrics. Nonetheless, HAR remains the most effective model overall for daily realized volatility forecasting in copper futures.

In this study, we examine the fluctuation in the value of the Great Britain Pound (GBP). We focus particularly on its relationship with the United States Dollar (USD) and the Euro (EUR) currency pairs. Utilizing data from June 15, 2018, to June 15, 2023, we apply various mathematical models to assess their effectiveness in predicting the 20-day variation in the pairs' daily returns. Our analysis involves the implementation of Exponentially Weighted Moving Average (EWMA), Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models, and Implied Volatility (IV) models. To evaluate their performance, we compare the accuracy of their predictions using Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) metrics. We delve into the intricacies of GARCH models, examining their statistical characteristics when applied to the provided dataset. Our findings suggest the existence of asymmetric returns in the EUR/GBP pair, while such evidence is inconclusive for the GBP/USD pair. Additionally, we observe that GARCH-type models better fit the data when assuming residuals follow a standard t-distribution rather than a standard normal distribution. Furthermore, we investigate the efficacy of different forecasting techniques within GARCH-type models. Comparing rolling window forecasts to expanding window forecasts, we find no definitive superiority in either approach across the tested scenarios. Our experiments reveal that for the GBP/USD pair, the most accurate volatility forecasts stem from the utilization of GARCH models employing a rolling window methodology. Conversely, for the EUR/GBP pair, optimal forecasts are derived from GARCH models and Ordinary Least Squares (OLS) models incorporating the annualized implied volatility of the exchange rate as an independent variable. Keywords: Euro, Exponentially Weighted Moving Average, Generalized Autoregressive Conditional Heteroskedasticity, Great Britain Pound, Implied Volatility, United States Dollar 1. Introduction As defined by the [1], a volatility of a financial asset is simply how much its price changes over time.

Volatility, as a measure of uncertainty, plays a crucial role in numerous financial activities such as risk management. The Econometrics and Machine Learning communities have developed two distinct approaches for financial volatility forecasting: the stochastic approach and the neural network (NN) approach. Despite their individual strengths, these methodologies have conventionally evolved in separate research trajectories with little interaction between them. This study endeavors to bridge this gap by establishing an equivalence relationship between models of the GARCH family and their corresponding NN counterparts. With the equivalence relationship established, we introduce an innovative approach, named GARCH-NN, for constructing NN-based volatility models. It obtains the NN counterparts of GARCH models and integrates them as components into an established NN architecture, thereby seamlessly infusing volatility stylized facts (SFs) inherent in the GARCH models into the neural network. We develop the GARCH-LSTM model to showcase the power of the GARCH-NN approach. Experiment results validate that amalgamating the NN counterparts of the GARCH family models into established NN models leads to enhanced outcomes compared to employing the stochastic and NN models in isolation.

Time series forecasting represents a significant and challenging task across various fields. Recently, methods based on mode decomposition have dominated the forecasting of complex time series because of the advantages of capturing local characteristics and extracting intrinsic modes from data. Unfortunately, most models fail to capture the implied volatilities that contain significant information. To enhance the prediction of contemporary diverse and complex time series, we propose a novel time series forecasting paradigm that integrates decomposition with the capability to capture the underlying fluctuation information of the series. In our methodology, we implement the Variational Mode Decomposition algorithm to decompose the time series into K distinct sub-modes. Following this decomposition, we apply the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model to extract the volatility information in these sub-modes. Subsequently, both the numerical data and the volatility information for each sub-mode are harnessed to train a neural network. This network is adept at predicting the information of the sub-modes, and we aggregate the predictions of all sub-modes to generate the final output. By integrating econometric and artificial intelligence methods, and taking into account both the numerical and volatility information of the time series, our proposed framework demonstrates superior performance in time series forecasting, as evidenced by the significant decrease in MSE, RMSE, and MAPE in our comparative experimental results.