Fourier Transform Methods in Finance -- Contents -- Preface -- List of Symbols -- 1 Fourier Pricing Methods -- 1.1 Introduction -- 1.2 A General Representation of Option Prices -- 1.3 The Dynamics of Asset Prices -- 1.4 A Generalized Function Approach -- 1.4 A generalized function approach to Fourier pricing -- 1.4.1 Digital payoffs and the Dirac delta function -- 1.4.2 The Fourier transform of digital payoffs -- 1.4.3 The cash-or-nothing option -- 1.4.4 The asset-or-nothing option -- 1.4.5 European options: the general pricing formula -- 1.5 Hilbert Transform -- 1.6 Pricing Via FFT -- 1.6.1 The sampling theorem -- 1.6.2 The truncated sampling theorem -- 1.6.3 Why bother? -- 1.6.4 The pricing formula -- 1.6.5 Application of the FFT -- 1.7 Related Literature -- 2 The Dynamics of Asset Prices -- 2.1 Introduction -- 2.2 Efficient Markets and Levy ́Processes -- 2.2.1 Random walks and Brownian motions -- 2.2.2 Geometric Brownian motion -- 2.2.3 Stable processes -- 2.2.4 Characteristic functions -- 2.2.5 Levy ́processes -- 2.2.6 Infinite divisibility -- 2.3 Construction of Levy ́Markets -- 2.3.1 The compound Poisson process -- 2.3.2 The Poisson point process -- 2.3.3 Sums over Poisson point processes -- 2.3.4 The decomposition theorem -- 2.4 Properties of Levy ́Processes -- 2.4.1 Pathwise properties of Levy ́processes -- 2.4.2 Completely monotone Levy ́densities -- 2.4.3 Moments of a Levy ́process -- 3 Non-stationary Market Dynamics -- 3.1 Non-stationary Processes -- 3.1.1 Self-similar processes -- 3.1.2 Self-decomposable distributions -- 3.1.3 Additive processes -- 3.1.4 Sato processes -- 3.2 Time Changes -- 3.2.1 Stochastic clocks -- 3.2.2 Subordinators -- 3.2.3 Stochastic volatility -- 3.2.4 The time-change technique.

3.3 Simulation of Levy ́Processes -- 3.3.1 Simulation via embedded random walks -- 3.3.2 Simulation via truncated Poisson point processes -- 4 Arbitrage-Free Pricing -- 4.1 Introduction -- 4.2 Equilibrium And Arbitrage -- 4.3 Arbitrage-free Pricing -- 4.3.1 Arbitrage pricing theory -- 4.3.2 Martingale pricing theory -- 4.3.3 Radon-Nikodym derivative -- 4.4 Derivatives -- 4.4.1 The replicating portfolio -- 4.4.2 Options and pricing kernels -- 4.4.3 Plain vanilla options and digital options -- 4.4.4 The Black-Scholes model -- 4.5 Levy ́Martingale Processes -- 4.5.1 Construction of martingales through Levy ́processes -- 4.5.2 Change of equivalent measures for Levy ́processes -- 4.5.3 The Esscher transform -- 4.6 Levy ́Markets -- 5 Generalized Functions -- 5.1 Introduction -- 5.2 The Vector Space of Test Functions -- 5.3 Distributions -- 5.3.1 Dirac delta and other singular distributions -- 5.4 The Calculus of Distributions -- 5.4.1 Distribution derivative -- 5.4.2 Special examples of distributions -- 5.5 Slow Growth Distributions -- 5.6 Function Convolution -- 5.6.1 Definitions -- 5.6.2 Some properties of convolution -- 5.7 Distributional Convolution -- 5.7.1 The direct product of distribution -- 5.7.2 The convolution of distributions -- 5.8 The Convolution of Distributions In S -- 6 The Fourier Transform -- 6.1 Introduction -- 6.2 The Fourier Transformation of Functions -- 6.2.1 Fourier series -- 6.2.2 Fourier transform -- 6.2.3 Parseval theorem -- 6.3 Fourier Transform And Option Pricing -- 6.3.1 The Carr-Madan approach -- 6.3.2 The Lewis approach -- 6.4 Fourier Transform For Generalized Functions -- 6.4.1 The Fourier transforms of testing functions of rapid descent -- 6.4.2 The Fourier transforms of distribution of slow growth -- 6.5 Exercises -- 6.6 Fourier option pricing with generalized functions -- 7 Fourier Transforms at Work -- 7.1 Introduction.

7.2 The Black-scholes Model -- 7.3 Finite Activity Models -- 7.3.1 Discrete jumps -- 7.3.2 The Merton model -- 7.4 Infinite Activity Models -- 7.4.1 The Variance Gamma model -- 7.4.2 The CGMY model -- 7.5 Stochastic Volatility -- 7.5.1 The Heston model -- 7.5.2 Vanilla options in the Heston model -- 7.6 Fft At Work -- 7.6.1 Market calibration -- 7.6.2 Pricing exotics -- Appendices -- A Elements of Probability -- A.1 Elements of Measure Theory -- A.1.1 Integration -- A.1.2 Lebesgue integral -- A.1.3 The characteristic function -- A.1.4 Relevant probability distributions -- A.1.5 Convergence of sequences of random variables -- A.1.6 The Radon-Nikodym derivative -- A.1.7 Conditional expectation -- A.2 Elements of The Theory of Stochastic Processes -- A.2.1 Stochastic processes -- A.2.2 Martingales -- B Elements of Complex Analysis -- B.1 Complex Numbers -- B.1.1 Why complex numbers? -- B.1.2 Imaginary numbers -- B.1.3 The complex plane -- B.1.4 Elementary operations -- B.1.5 Polar form -- B.2 Functions of Complex Variables -- B.2.1 De.nitions -- B.2.2 Analytic functions -- B.2.3 Cauchy-Riemann conditions -- B.2.4 Multi-valued functions -- C Complex Integration -- C.1 Definitions -- C.2 The Cauchy-Goursat Theorem -- C.3 Consequences of Cauchy's Theorem -- C.4 Principal value -- C.5 Laurent series -- C.6 Complex residue -- C.7 Residue theorem -- C.8 Jordan's Lemma -- D Vector Spaces and Function Spaces -- D.1 Definitions -- D.2 Inner product space -- D.3 Topological vector spaces -- D.4 Functionals and dual space -- D.4.1 Algebraic dual space -- D.4.2 Continuous dual space -- E The Fast Fourier Transform -- E.1 Discrete Fourier transform -- E.2 Fast Fourier transform -- F The Fractional Fast Fourier Transform -- F.1 Circular matrix -- F.1.1 Matrix vector multiplication -- F.2 Toepliz Matrix -- F.2.1 Embedding in a circular matrix.

F.2.2 Applications to pricing -- F.3 Some Numerical Results -- F.3.1 The Variance Gamma model -- F.3.2 The Heston model -- G Affine Models: The Path Integral Approach -- G.1 The problem -- G.2 Solution of the Riccati equations -- Bibliography -- Index.