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

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

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

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

Real-Time Detection of Local No-Arbitrage Violations

This paper focuses on the task of detecting local episodes involving violation of the standard It\^o semimartingale assumption for financial asset prices in real time that might induce arbitrage opportunities. Our proposed detectors, defined as stopping rules, are applied sequentially to continually incoming high-frequency data. We show that they are asymptotically exponentially distributed in the absence of Ito semimartingale violations. On the other hand, when a violation occurs, we can achieve immediate detection under infill asymptotics. A Monte Carlo study demonstrates that the asymptotic results provide a good approximation to the finite-sample behavior of the sequential detectors. An empirical application to S&P 500 index futures data corroborates the effectiveness of our detectors in swiftly identifying the emergence of an extreme return persistence episode in real time

Volatility Measurement with Pockets of Extreme Return Persistence

Increasing evidence points towards the episodic emergence of pockets with extreme return persistence. This notion refers to intraday periods of non-trivial duration, for which stock returns are highly positively autocorrelated. Such episodes include, but are not limited to, gradual jumps and prolonged bursts in the drift component. In this paper, we develop a family of integrated volatility estimators, labeled differenced-return volatility (DV) estimators, which provide robustness to these types of Itˆo semimartingale violations. Specifically, we show that, by using differences in consecutive high-frequency returns, our DV estimators can reduce the non-trivial bias that all commonly-used estimators exhibit during such periods of apparent short-term intraday return predictability. A Monte Carlo study demonstrates the reliability of the newly developed volatility estimators in finite samples. In our empirical volatility forecasting application to S&P 500 index futures and individual equities, our DV-based Heterogeneous Autoregressive (HAR) model performs well relative to existing procedures according to standard out-of-sample MSE and QLIKE criteria