Option calibration of exponential Lévy models: confidence intervals and empirical results

Abstract

Observing prices of European put and call options, we calibrate exponential Lévy models nonparametrically. We discuss the efficient implementation of the spectral estimation procedures for Lévy models of finite jump activity as well as for self-decomposable Lévy models. Based on finite sample variances, confidence intervals are constructed for the volatility, for the drift and, pointwise, for the jump density. As demonstrated by simulations, these intervals perform well in terms of size and coverage probabilities. We compare the performance of the procedures for finite and infinite jump activity based on options on the German DAX index and find that both methods achieve good calibration results. The stability of the finite activity model is studied when the option prices are observed in a sequence of trading days.

Confidence sets in nonparametric calibration of exponential Lévy models

“…The choice of the weight functions is discussed in [28], where also possible weight functions are given. The weight functions for all U > 0 can be obtained from w 1 σ , w 1 γ , w 1 λ and w 1 µ by rescaling:…”

Testing the characteristics of a Lévy process

“…In the related setting of estimation from financial option data the weight function approach has been successfully analysed and applied by Belomestny and Reiß [4]. Söhl [23] addresses testing and confidence regions for this situation in detail and Söhl and Trabs [22] apply this to financial data. Generalisations to other processes defined by their Fourier transform are possible, e.g.…”

Quantile estimation for Lévy measures

“…Söhl [25] have derived confidence sets. An empirical study of the calibration methods can be found in Söhl and Trabs [26]. By arbitrage arguments the price process of an asset should be a martingale implying E[S t ] = S 0 for all t > 0.…”

“…Let us finally apply the estimation method to prices of DAX options from May 29, 2008. This data set 1 has already been studied by Söhl and Trabs [26]. Fig.…”

“…The corresponding Hölder norm of the latter is of order t −s−4 . With a standard Taylor expansion argument and applying the order of the kernel, the approximation error is of the order (|t| −s−4 ∨ 1)h s+1 by (26).…”