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### High-dimensional Index Volatility Models via Stein's Identity

In this paper, we consider estimating the parametric components of index volatility models, whose variance function has semiparametric form with two common index structures: single index and multiple index. Our approach applies the first- and second-order Stein's identities on the empirical mean squared error (MSE) to extract the direction of true signals. We study both low-dimensional setting and high-dimensional setting under finite moment condition, which is weaker than existing literature and makes our estimators applicable even for some heavy-tailed data. From our theoretical analysis, we prove that the statistical rate of convergence has two components: parametric rate and nonparametric rate. For the parametric rate, we achieve $\sqrt{n}$-consistency for low-dimensional setting and optimal/sub-optimal rate for high-dimensional setting. For the nonparametric rate, we show it's asymptotically bounded by $n^{-4/5}$ under both settings when the mean function has bounded second derivative, so it only contributes high-order terms. Simulation results also back our theoretical conclusions.

### A Neural Stochastic Volatility Model

In this paper, we show that the recent integration of statistical models with deep recurrent neural networks provides a new way of formulating volatility (the degree of variation of time series) models that have been widely used in time series analysis and prediction in finance. The model comprises a pair of complementary stochastic recurrent neural networks: the generative network models the joint distribution of the stochastic volatility process; the inference network approximates the conditional distribution of the latent variables given the observables. Our focus here is on the formulation of temporal dynamics of volatility over time under a stochastic recurrent neural network framework. Experiments on real-world stock price datasets demonstrate that the proposed model generates a better volatility estimation and prediction that outperforms mainstream methods, e.g., deterministic models such as GARCH and its variants, and stochastic models namely the MCMC-based stochvol as well as the Gaussian-process-based, on average negative log-likelihood.

### A Neural Stochastic Volatility Model

In this paper, we show that the recent integration of statistical models with deep recurrent neural networks provides a new way of formulating volatility (the degree of variation of time series) models that have been widely used in time series analysis and prediction in finance. The model comprises a pair of complementary stochastic recurrent neural networks: the generative network models the joint distribution of the stochastic volatility process; the inference network approximates the conditional distribution of the latent variables given the observables. Our focus here is on the formulation of temporal dynamics of volatility over time under a stochastic recurrent neural network framework. Experiments on real-world stock price datasets demonstrate that the proposed model generates a better volatility estimation and prediction that outperforms stronge baseline methods, including the deterministic models, such as GARCH and its variants, and the stochastic MCMC-based models, and the Gaussian-process-based, on the average negative log-likelihood measure.

### Scalable inference for a full multivariate stochastic volatility model

We introduce a multivariate stochastic volatility model for asset returns that imposes no restrictions to the structure of the volatility matrix and treats all its elements as functions of latent stochastic processes. When the number of assets is prohibitively large, we propose a factor multivariate stochastic volatility model in which the variances and correlations of the factors evolve stochastically over time. Inference is achieved via a carefully designed feasible and scalable Markov chain Monte Carlo algorithm that combines two computationally important ingredients: it utilizes invariant to the prior Metropolis proposal densities for simultaneously updating all latent paths and has quadratic, rather than cubic, computational complexity when evaluating the multivariate normal densities required. We apply our modelling and computational methodology to $571$ stock daily returns of Euro STOXX index for data over a period of $10$ years. MATLAB software for this paper is available at http://www.aueb.gr/users/mtitsias/code/msv.zip.

### A long short-term memory stochastic volatility model

Stochastic Volatility (SV) models are widely used in the financial sector while Long Short-Term Memory (LSTM) models have been successfully used in many large-scale industrial applications of Deep Learning. Our article combines these two methods non trivially and proposes a model for capturing the dynamics of financial volatility process, which we call the LSTM-SV model. The proposed model overcomes the short-term memory problem in conventional SV models, is able to capture non-linear dependence in the latent volatility process, and often has a better out-of-sample forecast performance than SV models. The conclusions are illustrated through simulation studies and applications to three financial time series datasets: US stock market weekly index SP500, Australian stock weekly index ASX200 and Australian-US dollar daily exchange rates. We argue that there are significant differences in the underlying dynamics between the volatility process of SP500 and ASX200 datasets and that of the exchange rate dataset. For the stock index data, there is strong evidence of long-term memory and non-linear dependence in the volatility process, while this is not the case for the exchange rates. An user-friendly software package together with the examples reported in the paper are available at https://github.com/vbayeslab.