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Pricing European options and risk measurement under exponential Lévy models — a practical guide

Khaled Salhi
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Khaled Salhi: Institut Elie Cartan de Lorraine, Université de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France2CNRS, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France3Inria, Villers-lès-Nancy, F-54600, France

International Journal of Financial Engineering (IJFE), 2017, vol. 04, issue 02n03, 1-36

Abstract: This paper provides a thorough survey of the European option pricing, with new trends in the risk measurement, under exponential Lévy models. We develop all steps of pricing from equivalent martingale measures construction to numerical valuation of the option price under these measures. We then construct an algorithm, based on Rockafellar and Uryasev representation and fast Fourier transform, to compute Risk indicators, like the VaR and the CVaR of derivatives. The results are illustrated with an example of each exponential Lévy class. The main contribution of this paper is to build a comprehensive study from the theoretical point of view to practical numerical illustration and to give a complete characterization of the studied equivalent martingale measures by discussing their similarity and their applicability in practice. Furthermore, this work proposes applications to the Fourier inversion technique in risk measurement.

Keywords: Lévy process; incomplete market; Esscher martingale measure; minimal entropy martingale measure; fast Fourier transform; value-at-risk; conditional value-at-risk; Merton model; variance gamma model (search for similar items in EconPapers)
Date: 2017
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