Forecasting volatility: does continuous time do better than discrete time?

In this paper we compare the forecast performance of continuous and discrete-time volatility models. In discrete time, we consider more than ten GARCH-type models and an asymmetric autoregressive stochastic volatility model. In continuous-time, a stochastic volatility model with mean reversion, volatility feedback and leverage. We estimate each model by maximum likelihood and evaluate their ability to forecast the two scales realized volatility, a nonparametric estimate of volatility based on highfrequency data that minimizes the biases present in realized volatility caused by microstructure errors. We find that volatility forecasts based on continuous-time models may outperform those of GARCH-type discrete-time models so that, besides other merits of continuous-time models, they may be used as a tool for generating reasonable volatility forecasts. However, within the stochastic volatility family, we do not find such evidence. We show that volatility feedback may have serious drawbacks in terms of forecasting and that an asymmetric disturbance distribution (possibly with heavy tails) might improve forecasting.Asymmetry, Continuous and discrete-time stochastic volatility models, GARCH-type models, Maximum likelihood via iterated filtering, Particle filter, Volatility forecasting

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 model \emph{stochvol} as well as the Gaussian process volatility model \emph{GPVol}, on average negative log-likelihood

Localizing Volatilities

We propose two main applications of Gy\"{o}ngy (1986)'s construction of inhomogeneous Markovian stochastic differential equations that mimick the one-dimensional marginals of continuous It\^{o} processes. Firstly, we prove Dupire (1994) and Derman and Kani (1994)'s result. We then present Bessel-based stochastic volatility models in which this relation is used to compute analytical formulas for the local volatility. Secondly, we use these mimicking techniques to extend the well-known local volatility results to a stochastic interest rates framework

The Heston stochastic volatility model in Hilbert space

We extend the Heston stochastic volatility model to a Hilbert space framework. The tensor Heston stochastic variance process is defined as a tensor product of a Hilbert-valued Ornstein-Uhlenbeck process with itself. The volatility process is then defined by a Cholesky decomposition of the variance process. We define a Hilbert-valued Ornstein-Uhlenbeck process with Wiener noise perturbed by this stochastic volatility, and compute the characteristic functional and covariance operator of this process. This process is then applied to the modelling of forward curves in energy markets. Finally, we compute the dynamics of the tensor Heston volatility model when the generator is bounded, and study its projection down to the real line for comparison with the classical Heston dynamics