Finite-dimensional LDP

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Table of contents

1 Introduction 
1.1 Option pricing
1.1.1 The Black and Scholes model
1.1.2 Beyond the BS model
1.1.3 Non-equity derivatives
1.1.4 Pricing options with asymptotic methods
1.2 Summary of the thesis
1.3 The main results of the thesis
1.3.1 Ane stochastic volatility models, large deviations and optimal sampling (Chapters 2 and 3)
1.3.2 Options on realized variance and density expansion (Chapter 4)
1.3.3 Perturbation theory and interest rate derivatives pricing in the Levy Libor model (Chapter 5)
2 Pathwise large deviations for ane stochastic volatility models 
2.1 Introduction
2.2 Model description
2.3 Large deviations theory
2.4 Trajectorial large deviations for ane stochastic volatility model
2.4.1 Finite-dimensional LDP
2.4.2 Innite-dimensional LDP
2.5 Variance reduction
2.6 Numerical examples
2.6.1 European and Asian put options in the Heston model .
2.6.2 European put on the Heston model with negative exponential jumps
3 Large deviations for Wishart stochastic volatility model 
3.1 Introduction
3.2 The Wishart stochastic volatility model
3.3 Long-time large deviations for the Wishart volatility model
3.3.1 Reminder of large deviations theory
3.3.2 Long-time behaviour of the Laplace transform of the log-price
3.3.3 Long-time large deviation principle for the log-price process
3.4 Asymptotic implied volatility of basket options
3.4.1 Asymptotic price for the Wishart model
3.4.2 Implied volatility asymptotics
3.5 Variance reduction
3.5.1 The general variance reduction problem
3.5.2 Asymptotic variance reduction
3.6 Numerical results
3.6.1 Long-time implied volatility
3.6.2 Variance reduction
4 An asymptotic approach for the pricing of options on realized variance 
4.1 Introduction
4.2 Expansion of marginal densities
4.3 Denition and properties of the integrated variance process .
4.3.1 The integrated variance process
4.3.2 Hamiltonian equations and optimal control
4.3.3 Derivatives of the energy
4.4 Asymptotic expansion of the density
4.5 Pricing options on realized variance
4.5.1 Asymptotics for the price of options on realized variance
4.5.2 Asymptotics for the Black and Scholes implied volatility of options on realized variance
5 Approximate option pricing in the Levy Libor model 
5.1 Introduction
5.2 Presentation of the model
5.2.1 The driving process
5.2.2 The model
5.3 Option pricing via PIDEs
5.3.1 General payo
5.3.2 Caplet
5.3.3 Swaptions
5.4 Approximate pricing
5.4.1 Approximate pricing for general payos under the terminal measure
5.4.2 Approximate pricing of caplets
5.4.3 Approximate pricing of swaptions
5.5 Numerical examples