The sensitivity analysis of options is as important as pricing in option theory since it is used for hedging strategies, hence for risk management purposes. This dissertation presents new sensitivities for options when the underlying follows an exponential Lévy process, specifically Variance Gamma and Normal Inverse Gaussian processes. The calculation of these sensitivities is based on a finite dimensional Malliavin calculus and the centered finite difference method via Monte-Carlo simulations. We give explicit formulas that are used directly in Monte-Carlo simulations. By using simulations, we show that a localized version of the Malliavin estimator outperforms others including the centered finite difference estimator for the call and digital options under Variance Gamma and Normal Inverse Gaussian processes driven option pricing models. In order to compare the performance of these methods we use an inverse Fourier transform method to calculate the exact values of the sensitivities of European call and digital options written on S&P 500 index. Our results show that a variation of localized Malliavin calculus approach gives a robust estimator while the convergence of centered finite difference method in Monte-Carlo simulations varies with different Greeks and new sensitivities that we introduce. We also discuss an approximation method for the Variance Gamma process. We introduce new random number generators for the path wise simulations of the approximating process. We improve convergence results for a type of sensitivity by using a mixed Malliavin calculus on the increments of the approximating process. / A Dissertation Submitted to the Department of Mathematics in Partial FulfiLlment of
the Requirements for the Degree of Doctor of Philosophy. / Summer Semester, 2010. / April 12, 2010. / Centered Finite Difference, Monte-Carlo simulations, FFT, Malliavin calculus, Inverse Fourier Transform method, Normal Inverse Gaussian process, Approximation of Lévy processes, Variance Gamma process, Greeks / Includes bibliographical references. / Craig A. Nolder, Professor Directing Thesis; Fred Huffer, University Representative; Bettye Anne Case, Committee Member; David Kopriva, Committee Member; Giray Okten, Committee Member; Jack Quine, Committee Member.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_175718 |
| Contributors | Bayazit, Dervis, 1979- (authoraut), Nolder, Craig A. (professor directing thesis), Huffer, Fred (university representative), Case, Bettye Anne (committee member), Kopriva, David (committee member), Okten, Giray (committee member), Quine, Jack (committee member), Department of Mathematics (degree granting department), Florida State University (degree granting institution) |
| Publisher | Florida State University, Florida State University |
| Source Sets | Florida State University |
| Language | English, English |
| Detected Language | English |
| Type | Text, text |
| Format | 1 online resource, computer, application/pdf |
| Rights | This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). The copyright in theses and dissertations completed at Florida State University is held by the students who author them. |
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