Spectral estimation ofthe fractional orderof a Lévy process

We consider the problem of estimating the fractional order of a L´evyprocess from low frequency historical and options data. An estimationmethodology is developed which allows us to treat both estimation andcalibration problems in a unified way. The corresponding procedureconsists of two steps: the estimation of a conditional characteristic functionand the weighted least squares estimation of the fractional order inspectral domain. While the second step is identical for both calibrationand estimation, the first one depends on the problem at hand. Minimaxrates of convergence for the fractional order estimate are derived,the asymptotic normality is proved and a data-driven algorithm basedon aggregation is proposed. The performance of the estimator in bothestimation and calibration setups is illustrated by a simulation study