Spectral estimation of the fractional order of a Lévy process

Abstract

We consider the problem of estimating the fractional order of a Lévy process from low frequency historical and options data. An estimation methodology is developed which allows us to treat both estimation and calibration problems in a unified way. The corresponding procedure consists of two steps: the estimation of a conditional characteristic function and the weighted least squares estimation of the fractional order in spectral domain. While the second step is identical for both calibration and estimation, the first one depends on the problem at hand. Minimax rates of convergence for the fractional order estimate are derived, the asymptotic normality is proved and a data-driven algorithm based on aggregation is proposed. The performance of the estimator in both estimation and calibration setups is illustrated by a simulation study.

Published in at http://dx.doi.org/10.1214/09-AOS715 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Estimating the degree of activity of jumps in high frequency data
• Spatial aggregation of local likelihood estimates with applications to classification
• Nonparametric estimation for Lévy processes from low-frequency observations

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• Asymptotically optimal discretization of hedging strategies with jumps
• Confidence sets in nonparametric calibration of exponential Lévy models
• Identifying the successive Blumenthal-Getoor indices of a discretely observed process
• Calibration of self-decomposable Lévy models
• High-frequency Donsker theorems for Lévy measures
• Quantile estimation for Lévy measures
• Adaptive nonparametric estimation for Lévy processes observed at low frequency
• High-order short-time expansions for ATM option prices of exponential Lévy models
• The Normal-Generalised Gamma-Pareto process: A novel pure-jump Lévy process with flexible tail and jump-activity properties
• Testing the characteristics of a Lévy process