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Abstract Abstract Dedication. Acknowledgements 1 Introduction 1.1 Approaches to evaluating Greeks 1.2 Background on the Malliavin calculus 1.3 Aims and structure of the thesis 2 Basic Properties of the Malliavin calculus 2.1 Wiener Space 2.2 Malliavin derivative 2.3 Skorohod integral 2.4 SDEs and Malliavin calculus. 2.5 The integration by parts formula 2.6 Iterated Wiener-It? integrals . 2.7 Malliavin derivative via chaos expansion 2.8 Skorohod integral via chaos expansion 2.9 The Clark-Haussmann-Ocone formula 3 Application of Malliavin calculus to the Calculations of Greeks for Continuous Processes 3.1 Generalized Greeks 3.2 Greeks for European Options 3.3 Greeks for Exotic options 3.4 Greeks for Barriers and Look-back options. 3.5 Greeks for the Heston model 4 Application of white noise calculus for Gaussian Processes to the Calculs tion of Greeks 4.1 Basic concepts of Gaussian white noise analysis 4.2 Stochastic test functions and stochastic distribution functions 4.3 The Wick product 4.4 The Hermite Transform 4.5 Hida-Malliavin derivative 4.6 Conditional expectation on (S)*4.7 The Donsker delta function 4.8 Financial Application: Calculating Greeks 5 Malliavin calculus for Pure Jump L¨¦vy SDEs 5.1 Basic definitions and results for L¨¦vy processes 5.2 Chaos expansion 5.3 Skorohod integral 5.4 Stochastic derivative 5.5 Differentiability of pure jump L¨¦vy stochastic differential equation 5.6 The necessary and sufficient condition for a function to serve as a weighting function 6 Calculations of Greeks for Jump Diffusion Processes 6.1 Basic elements of a L¨¦vy chaotic calculus 6.2 Chaos expansion 6.2.1 Directional derivative 6.2.2 Wiener-Poisson space 6.3 Skorohod integral 6.4 Greeks for jump diffusion models 6.5 Greeks for the Heston model with jumps 6.6 Greeks for L¨¦vy process 7 White noise calculus for L¨¦vy Processes and its Application to the Calcu-lations of Greeks 7.1 Basic concepts of L¨¦vy white noise analysis 7.2 Chaos expansion 7.3 The Hida/Kondratiev spaces 7.4 L¨¦vy Wick product 7.5 L¨¦vy Hermite transform 7.6 L¨¦vy stochastic derivative 7.7 Donsker delta function of a L¨¦vy process 7.8 Application: Computing Greeks References ±à¼­ÊÖ¼Ç