114 research outputs found
Estimation of integrated volatility of volatility with applications to goodness-of-fit testing
In this paper, we are concerned with nonparametric inference on the
volatility of volatility process in stochastic volatility models. We construct
several estimators for its integrated version in a high-frequency setting, all
based on increments of spot volatility estimators. Some of those are positive
by construction, others are bias corrected in order to attain the optimal rate
. Associated central limit theorems are proven which can be widely
used in practice, as they are the key to essentially all tools in model
validation for stochastic volatility models. As an illustration we give a brief
idea on a goodness-of-fit test in order to check for a certain parametric form
of volatility of volatility.Comment: Published at http://dx.doi.org/10.3150/14-BEJ648 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
A note on central limit theorems for quadratic variation in case of endogenous observation times
This paper is concerned with a central limit theorem for quadratic variation
when observations come as exit times from a regular grid. We discuss the
special case of a semimartingale with deterministic characteristics and finite
activity jumps in detail and illustrate technical issues in more general
situations.Comment: 16 pages, 1 figur
Nonparametric inference on L\'evy measures and copulas
In this paper nonparametric methods to assess the multivariate L\'{e}vy
measure are introduced. Starting from high-frequency observations of a L\'{e}vy
process , we construct estimators for its tail integrals and the
Pareto-L\'{e}vy copula and prove weak convergence of these estimators in
certain function spaces. Given n observations of increments over intervals of
length , the rate of convergence is for
which is natural concerning inference on the L\'{e}vy measure. Besides
extensions to nonequidistant sampling schemes analytic properties of the
Pareto-L\'{e}vy copula which, to the best of our knowledge, have not been
mentioned before in the literature are provided as well. We conclude with a
short simulation study on the performance of our estimators and apply them to
real data.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1116 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Estimation of Volatility Functionals in the Simultaneous Presence of Microstructure Noise and Jumps
We propose a new concept of modulated bipower variation for diffusion models with microstructure noise. We show that this method provides simple estimates for such important quantities as integrated volatility or integrated quarticity. Under mild conditions the consistency of modulated bipower variation is proven. Under further assumptions we prove stable convergence of our estimates with the optimal rate n^(-1/4). Moreover, we construct estimates which are robust to finite activity jumps. --Bipower Variation,Central Limit Theorem,Finite Activity Jumps,High-Frequency Data,Integrated Volatility,Microstructure Noise
Bipower-type estimation in a noisy diffusion setting
We consider a new class of estimators for volatility functionals in the setting of frequently observed It¯o diffusions which are disturbed by i.i.d. noise. These statistics extend the approach of pre-averaging as a general method for the estimation of the integrated volatility in the presence of microstructure noise and are closely related to the original concept of bipower variation in the no-noise case. We show that this approach provides efficient estimators for a large class of integrated powers of volatility and prove the associated (stable) central limit theorems. In a more general It¯o semimartingale framework this method can be used to define both estimators for the entire quadratic variation of the underlying process and jump-robust estimators which are consistent for various functionals of volatility. As a by-product we obtain a simple test for the presence of jumps in the underlying semimartingale. --Bipower Variation,Central Limit Theorem,High-Frequency Data,Microstructure Noise,Quadratic Variation,Semimartingale Theory,Test for Jumps
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