[eBook Code] Fourier Transform Methods in Finance (eBook Code, 1st)

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 to Fourier pricing.

1.5 Hilbert transform.

1.6 Pricing via FFT.

1.7 Related literature.

2 The Dynamics of Asset Prices.

2.1 Introduction.

2.2 Efficient markets and Lévy processes.

2.3 Construction of Lévy markets.

2.4 Properties of Lévy processes.

3 Non-stationary Market Dynamics.

3.1 Non-stationary processes.

3.2 Time changes.

3.3 Simulation of Lévy processes.

4 Arbitrage-Free Pricing.

4.1 Introduction.

4.2 Equilibrium and arbitrage.

4.3 Arbitrage-free pricing.

4.4 Derivatives.

4.5 Lévy martingale processes.

4.6 Lévy markets.

5 Generalized Functions.

5.1 Introduction.

5.2 The vector space of test functions.

5.3 Distributions.

5.4 The calculus of distributions.

5.5 Slow growth distributions.

5.6 Function convolution.

5.7 Distributional convolution.

5.8 The convolution of distributions in S.

6 The Fourier Transform.

6.1 Introduction.

6.2 The Fourier transformation of functions.

6.3 Fourier transform and option pricing.

6.4 Fourier transform for generalized functions.

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.

7.4 Infinite activity models.

7.5 Stochastic volatility.

7.6 FFT at work.

Appendices.

A Elements of probability.

A.1 Elements of measure theory.

A.2 Elements of the theory of stochastic processes.

B Elements of Complex Analysis.

B.1 Complex numbers.

B.2 Functions of complex variables.

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.

E The Fast Fourier Transform.

E.1 Discrete Fourier transform.

E.2 Fast Fourier transform.

F The Fractional Fourier Transform.

F.1 Circular matrix.

F.2 Toepliz matrix.

F.3 Some numerical results.

G Affine Models: The Path Integral Approach.

G.1 The problem.

G.2 Solution of the Riccati equations.

Bibliogrsphy.

Index.