Confidence sets in nonparametric calibration of exponential Lévy models

Abstract

Confidence intervals and joint confidence sets are constructed for the nonparametric calibration of exponential Lévy models based on prices of European options. To this end, we show joint asymptotic normality in the spectral calibration method for the estimators of the volatility, the drift, the jump intensity and the Lévy density at finitely many points.

Efficient nonparametric inference for discretely observed compound Poisson processes

“…Chen, Delaigle and Hall [7] generalise the noise component to a symmetric stable process, also estimate the distribution function under the sup-norm and obtain minimax rates of the problem complementing those in [28]. Using prices of financial options and assuming the logarithm of the prices of financial assets is a compound Poisson process with a diffusion component, [2,12,33,35] perform density estimation and develop confidence sets. Also using the spectral approach, Kappus [27] adaptively estimates the jump density of a Lévy process.…”

Quantile estimation for Lévy measures

“…This problem is similar to the one considered by Söhl [25] who has determined the asymptotic distribution of the finite activity estimators by Belomestny and Reiß [3] and has derived confidence sets. In fact, we will only estimate an upper bound of the standard deviation.…”

“…The equivalence to more general error distributions follows from Grama and Nussbaum [13]. More details on this equivalence can be found in Söhl [25] and Trabs [28,Supplement].…”

“…In the following, t is fixed and thus omitted in the constants. Using the error decomposition (25), Lemma 13, Proposition 14, we obtain…”

“…Using |F g t (u)| (1 + |u|) −1 , Plancherel's identity and Lemma 12 yield for the stochastic error from (25) and the linearized stochastic error term L ∆,n defined in (27) …”