Asymptotic equivalence for inference on the volatility from noisy observations

We consider discrete-time observations of a continuous martingale under measurement error. This serves as a fundamental model for high-frequency data in finance, where an efficient price process is observed under microstructure noise. It is shown that this nonparametric model is in Le Cam's sense asymptotically equivalent to a Gaussian shift experiment in terms of the square root of the volatility function $\sigma$ and a nonstandard noise level. As an application, new rate-optimal estimators of the volatility function and simple efficient estimators of the integrated volatility are constructed.Comment: Published in at http://dx.doi.org/10.1214/10-AOS855 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Functional estimation and hypothesis testing in nonparametric boundary models

Consider a Poisson point process with unknown support boundary curve $g$, which forms a prototype of an irregular statistical model. We address the problem of estimating non-linear functionals of the form $\int \Phi(g(x))\,dx$. Following a nonparametric maximum-likelihood approach, we construct an estimator which is UMVU over H\"older balls and achieves the (local) minimax rate of convergence. These results hold under weak assumptions on $\Phi$ which are satisfied for $\Phi(u)=|u|^p$, $p\ge 1$. As an application, we consider the problem of estimating the $L^p$-norm and derive the minimax separation rates in the corresponding nonparametric hypothesis testing problem. Structural differences to results for regular nonparametric models are discussed.Comment: 21 pages, 1 figur

Asymptotic equivalence and sufficiency for volatility estimation under microstructure noise

The basic model for high-frequency data in finance is considered, where an efficient price process is observed under microstructure noise. It is shown that this nonparametric model is in Le Cam's sense asymptotically equivalent to a Gaussian shift experiment in terms of the square root of the volatility function σ. As an application, simple rateoptimal estimators of the volatility and efficient estimators of the integrated volatility are constructed.High-frequency data, integrated volatility, spot volatility estimation, Le Cam deficiency, equivalence of experiments, Gaussian shift

Spectral calibration of exponential Lévy Models [1]

We investigate the problem of calibrating an exponential Lévy model based on market prices of vanilla options. We show that this inverse problem is in general severely ill-posed and we derive exact minimax rates of convergence. The estimation procedure we propose is based on the explicit inversion of the option price formula in the spectral domain and a cut-off scheme for high frequencies as regularisation.European option, jump diffusion, minimax rates, severely ill-posed, nonlinear inverse problem, spectral cut-off

Height of Right and Left Ethmoid Roofs: Aspects of Laterality in 644 Patients

Objective. The goal of the study was to determine the asymmetric distribution of the height of the ethmoid roof (fovea ethmoidalis). Method. We retrospectively reviewed 644 coronal sinus computer tomography (CT) scans. The height of the ethmoid roof was examined for possible lateral differences between the right and left sides. Results. In 221 CT scans (31%), there was an asymmetry between the height of the fovea ethmoidalis on the right and left side. Of these 221, 160 (72.4%) were lower on the right side, whereas 61 (27.6%) were lower on the left. The height of the ethmoid roof of the remaining 433 patients (66%) was symmetric. There were statistically significantly more asymmetric cases in men than in women (38% versus 29%). Conclusions. The present paper underlines the asymmetry, variability of the ethmoid roof, and the possible practical implications arising from that fact. The asymmetry of the roof of one side presents an additional point of consideration for careful preoperative and perioperative review of paranasal sinus CT scans in patients undergoing endonasal sinus surgery

Spectral calibration of exponential Lévy Models [2]

The calibration of financial models has become rather important topic in recent years mainly because of the need to price increasingly complex options in a consistent way. The choice of the underlying model is crucial for the good performance of any calibration procedure. Recent empirical evidences suggest that more complex models taking into account such phenomenons as jumps in the stock prices, smiles in implied volatilities and so on should be considered. Among most popular such models are Levy ones which are on the one hand able to produce complex behavior of the stock time series including jumps, heavy tails and on other hand remain tractable with respect to option pricing. The work on calibration methods for financial models based on Lévy processes has mainly focused on certain parametrisations of the underlying Lévy process with the notable exception of Cont and Tankov (2004). Since the characteristic triplet of a Lévy process is a priori an infinite-dimensional object, the parametric approach is always exposed to the problem of misspecification, in particular when there is no inherent economic foundation of the parameters and they are only used to generate different shapes of possible jump distributions. In this work we propose and test a non-parametric calibration algorithm which is based on the inversion of the explicit pricing formula via Fourier transforms and a regularisation in the spectral domain. Using the Fast Fourier Transformation, the procedure is fast, easy to implement and yields good results in simulations in view of the severe ill-posedness of the underlying inverse problem.European option, jump diffusion, minimax rates, severely ill-posed, nonlinear inverse problem, spectral cut-off

Estimation of the characteristics of a Lévy process observed at arbitrary frequency

A Lévy process is observed at time points of distance delta until time T. We construct an estimator of the Lévy-Khinchine characteristics of the process and derive optimal rates of convergence simultaneously in T and delta. Thereby, we encompass the usual low- and high-frequency assumptions and obtain also asymptotics in the mid-frequency regime.Lévy process, Lévy-Khinchine characteristics, Nonparametric estimation, Inverse problem, Optimal rates of convergence