Realized Laplace transforms for pure-jump semimartingales

We consider specification and inference for the stochastic scale of discretely-observed pure-jump semimartingales with locally stable L\'{e}vy densities in the setting where both the time span of the data set increases, and the mesh of the observation grid decreases. The estimation is based on constructing a nonparametric estimate for the empirical Laplace transform of the stochastic scale over a given interval of time by aggregating high-frequency increments of the observed process on that time interval into a statistic we call realized Laplace transform. The realized Laplace transform depends on the activity of the driving pure-jump martingale, and we consider both cases when the latter is known or has to be inferred from the data.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1006 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Limit theorems for power variations of pure-jump processes with application to activity estimation

This paper derives the asymptotic behavior of realized power variation of pure-jump It\^{o} semimartingales as the sampling frequency within a fixed interval increases to infinity. We prove convergence in probability and an associated central limit theorem for the realized power variation as a function of its power. We apply the limit theorems to propose an efficient adaptive estimator for the activity of discretely-sampled It\^{o} semimartingale over a fixed interval.Comment: Published in at http://dx.doi.org/10.1214/10-AAP700 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Volatility occupation times

We propose nonparametric estimators of the occupation measure and the occupation density of the diffusion coefficient (stochastic volatility) of a discretely observed It\^{o} semimartingale on a fixed interval when the mesh of the observation grid shrinks to zero asymptotically. In a first step we estimate the volatility locally over blocks of shrinking length, and then in a second step we use these estimates to construct a sample analogue of the volatility occupation time and a kernel-based estimator of its density. We prove the consistency of our estimators and further derive bounds for their rates of convergence. We use these results to estimate nonparametrically the quantiles associated with the volatility occupation measure.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1135 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Rational Pessimism, Rational Exuberance, and Asset Pricing Models

The paper estimates and examines the empirical plausibiltiy of asset pricing models that attempt to explain features of financial markets such as the size of the equity premium and the volatility of the stock market. In one model, the long run risks model of Bansal and Yaron (2004), low frequency movements and time varying uncertainty in aggregate consumption growth are the key channels for understanding asset prices. In another, as typified by Campbell and Cochrane (1999), habit formation, which generates time-varying risk-aversion and consequently time-variation in risk-premia, is the key channel. These models are fitted to data using simulation estimators. Both models are found to fit the data equally well at conventional significance levels, and they can track quite closely a new measure of realized annual volatility. Further scrutiny using a rich array of diagnostics suggests that the long run risk model is preferred.

Pricing of the Time-Change Risks

We develop an equilibrium endowment economy with Epstein–Zin recursive utility and a Lévy time-change subordinator, which represents a clock that connects business and calendar time. Our setup provides a tractable equilibrium framework for pricing non-Gaussian jump-like risks induced by the time-change, with closed-form solutions for asset prices. Persistence of the time-change shocks leads to predictability of consumption and dividends and time-variation in asset prices and risk premia in calendar time. In numerical calibrations, we show that the risk compensation for Lévy risks accounts for about one-third of the overall equity premium

A New Class of Stochastic Volatility Models with Jumps: Theory and Estimation

The purpose of this paper is to propose a new class of jump diffusions which feature both stochastic volatility and random intensity jumps. Previous studies have focussed primarily on pure jump processes with constant intensity and log-normal jumps or constant jump intensity combined with a one factor stochastic volatility model. We introduce several generalizations which can better accommodate several empirical features of returns data. In their most general form we introduce a class of processes which nests jump-diffusions previously considered in empirical work and includes the affine class of random intensity models studied by Bates (1998) and Duffie, Pan and Singleton (1998) but also allows for non-affine random intensity jump components. We attain the generality of our specification through a generic Lévy process characterization of the jump component. The processes we introduce share the desirable feature with the affine class that they yield analytically tractable and explicit option pricing formula. The non-affine class of processes we study include specifications where the random intensity jump component depends on the size of the previous jump which represent an alternative to affine random intensity jump processes which feature correlation between the stochastic volatility and jump component. We also allow for and experiment with different empirical specifications of the jump size distributions. We use two types of data sets. One involves the S&P500 and the other comprises of 100 years of daily Dow Jones index. The former is a return series often used in the literature and allows us to compare our results with previous studies. The latter has the advantage to provide a long time series and enhances the possibility of estimating the jump component more precisely. The non-affine random intensity jump processes are more parsimonious than the affine class and appear to fit the data much better. Nous présentons une nouvelle classe de processus à sauts avec volatilité stochastique. Cette nouvelle classe généralise les modèles affinés proposés par Duffie, Pan et Singleton (1998). La généralité se manifeste par une représentation générique des sauts par un processus de Lévy. La classe des processus que nous présentons nous fournit également des prix d'options. Une application empirique démontre la présence de sauts dans des séries financières telles le S&P500 et le Dow Jones. De plus, les processus n'ont pas une intensité constante. Nous analysons plusieurs spécifications empiriques.Efficient method of moments, Poisson processes, jump processes, stochastic volatility models, filtering, Processus à sauts, mesures de Lévy, modèles à volatilité stochastique

Simulation methods for Lévy-driven continuous-time autoregressive moving average (CARMA) stochastic volatility models

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . We develop simulation schemes for the new classes of non-Gaussian pure jump Levy processes for sto chastic volatility. We write the price and volatility processes as integrals against a vector Levy process, which makes series approximation methods directly applicable. These methods entail simulation of the Levy increments and formation of weighted sums of the increments; they do not require a closed-form expression for a tail mass function or specification of a copula function. We also present a new, and apparently quite flexible, bivariate mixture-of-gammas model for the driving Levy process. Within this setup, it is quite straightforward to generate simulations from a L?vy-driven continuous-time autoregres sive moving average stochastic volatility model augmented by a pure-jump price component. Simulations reveal the wide range of different types of financial price processes that can be generated in this manner, including processes with persistent stochastic volatility, dynamic leverage, and jumps. American Statistical Associatio