Spectral calibration of exponential Lévy Models [1]

Spectral calibration of exponential Levy
models ^?

Denis Belomestny and Markus Reiβ

April 27, 2006

Abstract. We investigate the problem of calibrating an exponential Levy model
based on market prices of vanilla options. We show that this inverse problem is in
general severely ill-posed and we derive exact minimax rates of convergence. The
estimation procedure we propose is based on the explicit inversion of the option
price formula in the spectral domain and a cut-off scheme for high frequencies as
regularisation.

Key words. European option, jump diffusion, minimax rates, severely ill-
posed, nonlinear inverse problem, spectral cut-off

Mathematics Sub ject Classification (2000): 60G51; 62G20; 91B28
JEL Subject Classification: G13; C14

1. Introduction

The work on calibration methods for financial models based on Levy pro-
cesses has mainly focused on certain parametrisations of the underlying
Levy process with the notable exception of Cont and Tankov (2004b). Since
the characteristic triplet of a Levy process is a priori an infinite-dimensional
object, the parametric approach is always exposed to the problem of mis-
specification, in particular when there is no inherent economic foundation
of the parameters and they are only used to generate different shapes of
possible jump distributions. The goal of this paper is to investigate mathe-
matically the problem of nonparametric inference for the Levy triplet when
the asset price (St ) follows an exponential Levy model

St = Se^rt+Xt with a Levy process Xt for t > 0.           (1)

We suppose that at time t = 0 we dispose of prices for vanilla European
call and put options on this asset with different strike prices and possibly

^? This research was supported by the Deutsche Forschungsgemeinschaft through
the SFB 649 Economic Risk.

Weierstraβ Institute for Applied Analysis and Stochastics (WIAS), Mohrenstraβe
39, 10117 Berlin, Germany, e-mail: [email protected]

Institute of Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld

294, 69120 Heidelberg, Germany, e-mail: [email protected]

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