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Spectral calibration of exponential Lévy models. (English) Zbl 1126.91022
The goal of this paper is to investigate the problem of nonparametric inference for the Lévy triplet when the asset price follows an exponential Lévy model. It is supposed that at time \(t=0\) one disposes of prices for vanilla European call and put option on this asset with different strike prices and possibly different maturities. By basing the estimates on option data, the authors draw inference on the underlying risk neutral price process, which in general cannot be determined from the historical price data due to the incompleteness of the Lévy market. It is unrealistic to determine the triplet correctly, so they try to provide an estimator which is as good as possible for the given accuracy of the data. This optimality property is assessed by the minimax paradigm which quantifies the error in the worst case scenario. The lower bound is established and it is demonstrated that already in the simple exponential Lévy model the estimation problem is in general severely ill-posed. It means that the estimation error as a function of the accuracy of observations converges with a logarithmic rate. An explicit construction of an estimator is proposed that attains this optimal minimax rate.

MSC:
91B28 Finance etc. (MSC2000)
60G51 Processes with independent increments; Lévy processes
62G20 Asymptotic properties of nonparametric inference
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