A jump-diffusion Libor model and its robust calibration

Abstract

In this paper we propose a jump-diffusion Libor model with jumps in a high-dimensional space (R m ) and test a stable non-parametric calibration algorithm which takes into account a given local covariance structure. The algorithm returns smooth and simply structured Lévy densities, and penalizes the deviation from the Libor market model. In practice, the procedure is FFT based, thus fast, easy to implement, and yields good results, particularly in view of the severe ill-posedness of the underlying inverse problem.

The Markov-switching jump diffusion LIBOR market model

“…The problem can, however, be circumvented by approximating (20) and (21) via freezing the L j 's at 0, as proposed by Glasserman and Merener (2003), and later adapted by Belomestny and Schoenmakers (2011). Given this approach, Wiener processes and compensator measures are approximated…”

“…Unfortunately, however, it has turned out that the assumption of simple forward rates following log-normal dynamics is not sufficient to account for complicated market movements or non-flat implied volatility surfaces (see Rebonato 2002). As a consequence, a large number of extensions have been brought forth over the course of years and, among others, the simple log-normal processes of the original model were replaced by displaced diffusions (see Joshi and Rebonato 2003), Lévy processes (see Eberlein and Özkan 2005), generalized jump diffusions (see Kou 2003 andSchoenmakers 2011), Markov-switching geometric Brownian motions (see Elliott and Valchev 2004), processes with stochastic volatility (see Brotherton-Ratcliffe 2005 andBelomestny et al 2010) and general semimartingales (see Jamshidian 1999).…”

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

“…The approach to minimize the calibration error was also applied by Belomestny and Schoenmakers (2011). Alternative data-driven choices of the cut-off value U are the "quasi-optimality" approach which was studied by Bauer and Reiß (2008) and which was applied by Belomestny (2011) or the use of a preestimator as proposed by Trabs (2012).…”

“…Whereas the above mentioned works focus on the asymptotic theory, we concentrate on the application of the method to realistic sample sizes. In a related framework of a jump-diffusion Libor model, Belomestny and Schoenmakers (2011) study the application of the spectral calibration method to finite sample data sets.…”

Approximate Option Pricing in the Lévy Libor Model

“…Efficient simulation algorithms suitable for pricing exotic options have been proposed in (Kohatsu-Higa and Tankov 2010;Papapantoleon, Schoenmakers, and Skovmand 2012), however, these Monte Carlo algorithms are probably not an option for the purposes of calibration because the computation is still too slow due to the presence of both discretization and statistical error. Eberlein andÖzkan (2005), Kluge (2005) and (Belomestny and Schoenmakers 2011) propose fast methods for computing caplet prices which are based on Fourier transform inversion and use the fact that the characteristic function of many parametric Lévy processes is known explicitly. Since in the Lévy Libor model, the Libor rate L k is not a geometric Lévy process under the corresponding probability measure Q T k , unless k = n (see Remark 3.1 below for details), using these methods for k < n requires an additional approximation (some random terms appearing in the compensator of the jump measure of L k are approximated by their values at time t = 0, a method known as freezing).…”