The accurate prediction of time-changing variances is an important task in the modeling of financial data. Standard econometric models are often limited as they assume rigid functional relationships for the variances. Moreover, function parameters are usually learned using maximum likelihood, which can lead to overfitting. To address these problems we introduce a novel model for time-changing variances using Gaussian Processes. A Gaussian Process (GP) defines a distribution over functions, which allows us to capture highly flexible functional relationships for the variances. In addition, we develop an online algorithm to perform inference. The algorithm has two main advantages. First, it takes a Bayesian approach, thereby avoiding overfitting. Second, it is much quicker than current offline inference procedures. Finally, our new model was evaluated on financial data and showed significant improvement in predictive performance over current standard models.
Financial studies require volatility based models which provides useful insights on risks related to investments. Stochastic volatility models are one of the most popular approaches to model volatility in such studies. The asset returns under study may come in multiple clusters which are not captured well assuming standard distributions. Mixture distributions are more appropriate in such situations. In this work, an algorithm is demonstrated which is capable of studying finite mixtures but with unknown number of components. This algorithm uses a Birth-Death process to adjust the number of comp…
A broad class of stochastic volatility models are defined by systems of stochastic differential equations. While these models have seen widespread success in domains such as finance and statistical climatology, they typically lack an ability to condition on historical data to produce a true posterior distribution. To address this fundamental limitation, we show how to re-cast a class of stochastic volatility models as a hierarchical Gaussian process (GP) model with specialized covariance functions. This GP model retains the inductive biases of the stochastic volatility model while providing th…
In this work, we propose a model for estimating volatility from financial time series, extending the non-Gaussian family of space-state models with exact marginal likelihood proposed by Gamerman, Santos and Franco (2013). On the literature there are models focused on estimating financial assets risk, however, most of them rely on MCMC methods based on Metropolis algorithms, since full conditional posterior distributions are not known. We present an alternative model capable of estimating the volatility, in an automatic way, since all full conditional posterior distributions are known, and it i…
In this paper we use Gaussian Process (GP) regression to propose a novel approach for predicting volatility of financial returns by forecasting the envelopes of the time series. We provide a direct comparison of their performance to traditional approaches such as GARCH. We compare the forecasting power of three approaches: GP regression on the absolute and squared returns; regression on the envelope of the returns and the absolute returns; and regression on the envelope of the negative and positive returns separately. We use a maximum a posteriori estimate with a Gaussian prior to determine ou…
We introduce a new regression framework, Gaussian process regression networks (GPRN), which combines the structural properties of Bayesian neural networks with the non-parametric flexibility of Gaussian processes. This model accommodates input dependent signal and noise correlations between multiple response variables, input dependent length-scales and amplitudes, and heavy-tailed predictive distributions. We derive both efficient Markov chain Monte Carlo and variational Bayes inference procedures for this model. We apply GPRN as a multiple output regression and multivariate volatility model, …
The stochastic volatility model is a popular tool for modeling the volatility of assets. The model is a nonlinear and non-Gaussian state space model, and consequently is difficult to fit. Many approaches, both classical and Bayesian, have been developed that rely on numerically intensive techniques such as quasi-maximum likelihood estimation and Markov chain Monte Carlo (MCMC). Convergence and mixing problems still plague MCMC algorithms when drawing samples sequentially from the posterior distributions. While particle Gibbs methods have been successful when applied to nonlinear or non-Gaussia…
We propose a factor state-space approach with stochastic volatility to model and forecast the term structure of future contracts on commodities. Our approach builds upon the dynamic 3-factor Nelson-Siegel model and its 4-factor Svensson extension and assumes for the latent level, slope and curvature factors a Gaussian vector autoregression with a multivariate Wishart stochastic volatility process. Exploiting the conjugacy of the Wishart and the Gaussian distribution, we develop a computationally fast and easy to implement MCMC algorithm for the Bayesian posterior analysis. An empirical applica…
Robust volatility updates for Hierarchical Gaussian Filtering
Interacting Minds Centre, Aarhus University（阿arhus大学交互心智中心）；Centre for Humanities Computing, Aarhus University（阿arhus大学人文计算中心）；Translational Neuromodeling Unit, University of Zurich & ETH Zurich（苏黎世大学转化神经建模单元及苏黎世联邦理工学院）；Institute for Cognitive Science, University of Osnabrück（奥斯纳布吕克大学认知科学研究所）
Local volatility is a versatile option pricing model due to its state dependent diffusion coefficient. Calibration is, however, non-trivial as it involves both proposing a hypothesis model of the latent function and a method for fitting it to data. In this paper we present novel Bayesian inference with Gaussian process priors. We obtain a rich representation of the local volatility function with a probabilistic notion of uncertainty attached to the calibrate. We propose an inference algorithm and apply our approach to S&P 500 market data.
This paper proposes factor stochastic volatility models with skew error distributions. The generalized hyperbolic skew t-distribution is employed for common-factor processes and idiosyncratic shocks. Using a Bayesian sparsity modeling strategy for the skewness parameter provides a parsimonious skew structure for possibly high-dimensional stochastic volatility models. Analyses of daily stock returns are provided. Empirical results show that the skewness is important for common-factor processes but less for idiosyncratic shocks. The sparse skew structure improves prediction and portfolio perform…
We explore the abilities of two machine learning approaches for no-arbitrage interpolation of European vanilla option prices, which jointly yield the corresponding local volatility surface: a finite dimensional Gaussian process (GP) regression approach under no-arbitrage constraints based on prices, and a neural net (NN) approach with penalization of arbitrages based on implied volatilities. We demonstrate the performance of these approaches relative to the SSVI industry standard. The GP approach is proven arbitrage-free, whereas arbitrages are only penalized under the SSVI and NN approaches. …
University College London（伦敦大学学院）；Shanghai Jiao Tong University（上海交通大学）
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…
Jump diffusion processes are widely used to model asset prices over time, mainly for their ability to capture complex discontinuous behavior, but inference on the model parameters remains a challenge. Here our goal is posterior inference on the volatility coefficient of the diffusion part of the process based on discrete samples. A Bayesian approach requires specification of a model for the jump part of the process, prior distributions for the corresponding parameters, and computation of the joint posterior. Since the volatility coefficient is our only interest, it would be desirable to avoid …
Several studies explore inferences based on stochastic volatility (SV) models, taking into account the stylized facts of return data. The common problem is that the latent parameters of many volatility models are high-dimensional and analytically intractable, which means inferences require approximations using, for example, the Markov Chain Monte Carlo or Laplace methods. Some SV models are expressed as a linear Gaussian state-space model that leads to a marginal likelihood, reducing the dimensionality of the problem. Others are not linearized, and the latent parameters are integrated out. How…
Local volatility is an important quantity in option pricing, portfolio hedging, and risk management. It is not directly observable from the market; hence calibrations of local volatility models are necessary using observable market data. Unlike most existing point-estimate methods, we cast the large-scale nonlinear inverse problem into the Bayesian framework, yielding a posterior distribution of the local volatility, which naturally quantifies its uncertainty. This extra uncertainty information enables traders and risk managers to make better decisions. To alleviate the computational cost, we …
The Gaussian Graphical Model (GGM) is a popular tool for incorporating sparsity into joint multivariate distributions. The G-Wishart distribution, a conjugate prior for precision matrices satisfying general GGM constraints, has now been in existence for over a decade. However, due to the lack of a direct sampler, its use has been limited in hierarchical Bayesian contexts, relegating mixing over the class of GGMs mostly to situations involving standard Gaussian likelihoods. Recent work, however, has developed methods that couple model and parameter moves, first through reversible jump methods a…
Although there are many methods for functional data analysis (FDA), little emphasis is put on characterizing variability among volatilities of individual functions. In particular, certain individuals exhibit erratic swings in their trajectory while other individuals have more stable trajectories. There is evidence of such volatility heterogeneity in blood pressure trajectories during pregnancy, for example, and reason to suspect that volatility is a biologically important feature. Most FDA models implicitly assume similar or identical smoothness of the individual functions, and hence can lead …
We consider the problem of testing the parametric form of the volatility for high frequency data. It is demonstrated that in the presence of microstructure noise commonly used tests do not keep the preassigned level and are inconsistent. The concept of preaveraging is used to construct new tests, which do not suffer from these drawbacks. These tests are based on a Kolmogorov-Smirnov or Cramer-von-Mises functional of an integrated stochastic process, for which weak convergence to a (conditional) Gaussian process is established. The finite sample properties of a bootstrap version of the test are…
Prediction of linear fractional stable motions using codifference, with application to non-Gaussian rough volatility
De Vinci Higher Education, De Vinci Research Center（德芬奇高等教育，德芬奇研究中心）；ESILV；Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院数学研究所）；Ecole Polytechnique（巴黎综合理工学院）；LadHyX, UMR CNRS 7646, Ecole Polytechnique, Institut Polytechnique de Paris（拉迪力学实验室，CNRS联合研究单位7646，巴黎综合理工学院，巴黎理工学院）
This paper considers an augmented double autoregressive (DAR) model, which allows null volatility coefficients to circumvent the over-parameterization problem in the DAR model. Since the volatility coefficients might be on the boundary, the statistical inference methods based on the Gaussian quasi-maximum likelihood estimation (GQMLE) become non-standard, and their asymptotics require the data to have a finite sixth moment, which narrows applicable scope in studying heavy-tailed data. To overcome this deficiency, this paper develops a systematic statistical inference procedure based on the sel…
This paper introduces a novel Ito diffusion process to model high-frequency financial data, which can accommodate low-frequency volatility dynamics by embedding the discrete-time non-linear exponential GARCH structure with log-integrated volatility in a continuous instantaneous volatility process. The key feature of the proposed model is that, unlike existing GARCH-Ito models, the instantaneous volatility process has a non-linear structure, which ensures that the log-integrated volatilities have the realized GARCH structure. We call this the exponential realized GARCH-Ito (ERGI) model. Given t…
This article studies the finite sample behaviour of a number of estimators for the integrated power volatility process of a Brownian semistationary process in the non semi-martingale setting. We establish three consistent feasible estimators for the integrated volatility, two derived from parametric methods and one non-parametrically. We then use a simulation study to compare the convergence properties of the estimators to one another, and to a benchmark of an infeasible estimator. We further establish bounds for the asymptotic variance of the infeasible estimator and assess whether a central …
We propose a family of models that enable predictive estimation of time-varying extreme event probabilities in heavy-tailed and nonlinearly dependent time series. The models are a white noise process with conditionally log-Laplace stochastic volatility. In contrast to other, similar stochastic volatility formalisms, this process has analytic expressions for its conditional probabilistic structure that enable straightforward estimation of dynamically changing extreme event probabilities. The process and volatility are conditionally Pareto-tailed, with tail exponent given by the reciprocal of th…
