Malliavin calculus in the Canonical Lévy process: White noise theory and financial applications
Abstract
We constructed a white noise theory for the Canonical Lévy process by Solé, Utzet, and Vives. The construction is based on the alternative construction of the chaos expansion of square integrable random variable. Then, we showed a Clark-Ocone theorem in L 2(P) and under the change of measure. The result from the Clark-Ocone theorem was used for the mean-variance hedging problem and applied it to stochastic volatility models such as the Barndorff-Nielsen and Shepard model model and the Bates model. A Donsker Delta approach is employed on a Binary option to solve the mean-variance hedging problem. Finally, we are able to derive the Delta and Gamma for a barrier and lookback options for an exp-Lévy process using the methodology of Bernis, Gobet, and Kohatsu-Higa by employing a dominating process.
Degree
Ph.D.
Advisors
Viens, Purdue University.
Subject Area
Mathematics|Statistics|Finance
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