An Efficient Markov-Chain Monte Carlo Method for the State Space Models with Stochastic Volatility

ABSTRACTWe introduce a novel class of stochastic volatility models, which can utilize and relate many high‐frequency realized volatility (RV) measures to latent volatility. Instrumental variable methods provide a unified framework for estimation and testing. We study parameter inference problems in the proposed framework with nonstationary stochastic volatility and exogenous predictors in the latent volatility process. Identification‐robust methods are developed for a joint hypothesis involving the volatility persistence parameter and the autocorrelation parameter of the composite error (or the noise ratio). For inference about the volatility persistence parameter, projection techniques are applied. The proposed tests include Anderson‐Rubin‐type tests and their point‐optimal versions. For distributional theory, we provide finite‐sample tests and confidence sets for Gaussian errors, establish exact Monte Carlo test procedures for non‐Gaussian errors (possibly heavy‐tailed), and show asymptotic validity under weaker assumptions. Simulation results show that the proposed tests outperform the asymptotic test regarding size and exhibit excellent power in empirically realistic settings. The proposed inference methods are applied to IBM's price and option data (2009–2013). We consider 175 different instruments (IVs) spanning 22 classes and analyze their ability to describe the low‐frequency volatility. IVs are compared based on the average length of the proposed identification‐robust confidence intervals. The superior instrument set mostly comprises 5‐min HF realized measures, and these IVs produce confidence sets which show that the volatility process is nearly unit‐root. In addition, we find RVs with higher frequency yield wider confidence intervals than RVs with slightly lower frequency, indicating that these confidence intervals adjust to absorb market microstructure noise. Furthermore, when we consider irrelevant or weak IVs (jumps and signed jumps), the proposed tests produce unbounded confidence intervals. We also find that both RV and BV measures produce almost identical confidence intervals across all 14 subclasses, confirming that our methodology is robust in the presence of jumps. Finally, although jumps contain little information regarding the low‐frequency volatility, we find evidence that there may be a nonlinear relationship between jumps and low‐frequency volatility.

This paper studies the portfolio optimization of mean-variance utility with state-dependent risk aversion, where the stock asset is driven by a stochastic process. The sub-game perfect Nash equilibrium strategies and the extended Hamilton-Jacobi-Bellman equations have been used to derive the system of non-linear partial differential equations. From the economic point of view, we demonstrate the numerical evaluation of the suggested solution for a special case where the risk aversion rate is proportional to the wealth value. Our results show that the asset driven by the stochastic volatility process is more general and reasonable than the process with a constant volatility.

We introduce and derive the asymptotic behavior of a new measure constructed from high-frequency data which we call the realized Laplace transform of volatility. The statistic provides a nonparametric estimate for the empirical Laplace transform function of the latent stochastic volatility process over a given interval of time and is robust to the presence of jumps in the price process. With a long span of data, that is, under joint long-span and infill asymptotics, the statistic can be used to construct a nonparametric estimate of the volatility Laplace transform as well as of the integrated joint Laplace transform of volatility over different points of time. We derive feasible functional limit theorems for our statistic both under fixed-span and infill asymptotics as well as under joint long-span and infill asymptotics which allow us to quantify the precision in estimation under both sampling schemes.

Finite sample accuracy and choice of sampling frequency in integrated volatility estimation

ABSTRACTThis paper develops Bayesian inference of extreme value models with a flexible time-dependent latent structure. The generalized extreme value distribution is utilized to incorporate state variables that follow an autoregressive moving average (ARMA) process with Gumbel-distributed innovations. The time-dependent extreme value distribution is combined with heavy-tailed error terms. An efficient Markov chain Monte Carlo algorithm is proposed using a state-space representation with a finite mixture of normal distributions to approximate the Gumbel distribution. The methodology is illustrated by simulated data and two different sets of real data. Monthly minima of daily returns of stock price index, and monthly maxima of hourly electricity demand are fit to the proposed model and used for model comparison. Estimation results show the usefulness of the proposed model and methodology, and provide evidence that the latent autoregressive process and heavy-tailed errors play an important role to describe the monthly series of minimum stock returns and maximum electricity demand.

This paper uses an extensive set of market data of forward swap rates and swaptions covering 3 July 2002 to 21 May 2009 to identify a two-dimensional stochastic volatility process for the level of rates. The process is identified step by step by increasing the requirement of the model and introducing appropriate adjustments. The first part of the paper investigates the smile dynamics of forward swap rates at their setting dates. Comparing the SABR (with different ) and Heston stochastic volatility models informs us what different specifications of the driving SDEs have to offer in terms of reflecting the dynamics of the smile across dates. The outcome of the analysis is that a normal SABR model () satisfactorily passes all tests and seems to provide a good match to the market. In contrast, we find that the Heston model does not. The next step is to seek a model of the forward swap rates (in their own swaption measure) based on only two Brownian motions that enables a specification with common parameters. It turns out that this can be done by extending the SABR model with a time-dependent volatility function and a mean-reverting volatility process. The performance of the extended (SABR with mean-reversion) model is analysed over several historical dates and is shown to be a stable and flexible choice that allows for good calibration across expiries and strikes. Finally, a time-homogeneous candidate stochastic volatility process that can be used as a driver for all swap rates is identified. This candidate process may in future work be used as a building block for a separable stochastic volatility LIBOR market model or a stochastic volatility Markov-functional model.

This paper proposes the new concept of stochastic leverage in stochastic volatility models. Stochastic leverage refers to a stochastic process which replaces the classical constant correlation parameter between the asset return and the stochastic volatility process. We provide a systematic treatment of stochastic leverage and propose to model the stochastic leverage effect explicitly, e.g. by means of a linear transformation of a Jacobi process. Such models are both analytically tractable and allow for a direct economic interpretation. In particular, we propose two new stochastic volatility models which allow for a stochastic leverage effect: the generalised Heston model and the generalised Barndorff-Nielsen & Shephard model. We investigate the impact of a stochastic leverage effect in the risk neutral world by focusing on implied volatilities generated by option prices derived from our new models. Furthermore, we give a detailed account on statistical properties of the new models.

Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations

In this paper we show that in the lognormal discrete-time stochastic volatility model with predictable conditional expected returns, the conditional expected value of the discounted payo of a European call option is infinite. Our empirical illustration shows that the characteristics of the predictive distributions of the discounted payos, obtained using Monte Carlo methods, do not indicate directly that the expected discounted payos are infinite.

Astract- This paper presents a new energy-based continuous time series probability model that has stochastic process in its latent dynamics. Because the model assuming its output is used for connecting a vector space such as embedding space and latent dynamics, one of the possible application of the present study is to predict sequence of segments such as in the human motion behavior recognition. The present model gives joint distribution of all variable, therefore, it has the property to generate learned sequence without conditioning any values including initial state. This property overcomes existing problem in initial value condition for frequently applied time series model such as recurrent neural network. Furthermore, energy-based continuous formulation gives us consistent method for both learning and estimating the state by the gradient descent and its derivative methods. Such consistent method is not proposed in existing study. In this paper, the model formulation, gradient estimator, and its learning method is shown. Experiments on synthetic data is given, and the validity of the proposed model and method discussed.

International audience

ABSTRACTWe consider a borderline case: The central limit theorem for a strictly stationary time series with infinite variance but a Gaussian limit. In the i.i.d. case, a well‐known sufficient condition for this central limit theorem is regular variation of the marginal distribution with tail index . In the dependent case, we assume the stronger condition of sequential regular variation of the time series with tail index . We assume that a sample of size from this time series can be split into blocks of size such that as and that the block sums are asymptotically independent. Then we apply classical central limit theory for row‐wise i.i.d. triangular arrays. The necessary and sufficient conditions for such independent block sums will be verified by using large deviation results for the time series. We derive the central limit theorem for ‐dependent sequences, linear processes, stochastic volatility processes and solutions to affine stochastic recurrence equations whose marginal distributions have infinite variance and are regularly varying with tail index .

Reconsidering the continuous time limit of the GARCH(1, 1) process

Time Deformation and Term Structure of Interest Rates

We develop a flexible business cycle indicator that accounts for potential time variation in macroeconomic variables. The coincident economic indicator is based on a multivariate trend cycle decomposition model and is constructed from a moderate set of US macroeconomic time series. In particular, we consider an unobserved components time series model with a common cycle that is shared across different time series but adjusted for phase shift and amplitude. The extracted cycle can be interpreted as a model‐based bandpass filter and is designed to emphasize the business cycle frequencies that are of interest to applied researchers and policymakers. Stochastic volatility processes and mixture distributions for the irregular components and the common cycle disturbances enable us to account for the heteroskedasticity present in the data. Forecasting results are presented for a set of different specifications. Point forecasts from the preferred model indicate a future recession with the uncertainty over the business cycle growing quickly as the forecast horizon increases. Copyright © 2010 John Wiley & Sons, Ltd.