ATE - L ´ evy processes: Applications to Finance and...

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Unformatted text preview: L ´ evy processes: Applications to Finance and Dependence Structure Advanced Topic Examination | Spring 2010 Pierre Garreau Florida State University 1017 Academic Way, 208 Love Building Tallahassee, FL 32306 www.math.fsu.edu/~pgarreau Major Professor | Dr. C. Nolder Department of Mathematics | Florida State University Abstract The main definitions and properties of L´ evy processes are introduced in order to present mar- tingale transformations such as Girsanov Theorem and Essher transform methods to work with exponential L´ evy martingales. The problem of derivative pricing is addressed in the case of Itˆ o-L´ evy processes as well as portfolio replication for hedging purposes. The characterisation of multidimensional processes is presented and the simulation of L´ evy copula explained in the case a process of finite variation. Key words L´ evy processes, martingale measure, option pricing, L´ evy copula. CONTENTS Contents 1 Introduction 1 2 Introduction to L´ evy processes 3 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Characterisation of L´ evy processes . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Itˆ o-L´ evy processes and Stochastic Exponential . . . . . . . . . . . . . . . . . . . 10 3 L´ evy processes and martingale measures 13 3.1 Girsanov theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Stochastic exponential and martingale measures . . . . . . . . . . . . . . . . . . . 17 3.2.1 Stochastic exponential and exponential L´ evy . . . . . . . . . . . . . . . . 17 3.2.2 Esscher transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 Exponential L´ evy models in finance 21 4.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Delta-Gamma hedging in Stochastic exponential models . . . . . . . . . . . . . . 22 4.3 Option Pricing in Stochastic exponential models . . . . . . . . . . . . . . . . . . 25 4.3.1 FFT Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3.2 PIDE Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.3.3 Finite Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 Dependence structure of L´ evy processes 29 5.1 Introduction to dependence modelling for L´ evy processes . . . . . . . . . . . . . . 29 5.2 L´ evy copulae: Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 L´ evy Copulae: Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6 Summary and future research 36 ii 1 Introduction L´ evy processes are a class of stochastic processes with independent and stationnary increments named after the French Mathematician Paul L´ evy who worked on the characterization of in- finitely divisible distributions. The growing interest toward this class of processes in Physics, Biology and more recently Finance, is due to their rich properties - and notably the ability to...
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This note was uploaded on 01/15/2012 for the course MAT 5939 taught by Professor Garreau during the Fall '11 term at FSU.

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ATE - L ´ evy processes: Applications to Finance and...

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