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1 Introduction.
1.1 The need for better financial modeling of asset prices.
1.2 The family of stable distribution and its properties.
1.3 Option pricing with volatility clustering.
1.4 Model dependencies.
1.5 Monte Carlo.
1.6 Organization of the book.
2 Probability distributions.
2.1 Basic concepts.
2.2 Discrete probability distributions.
2.3 Continuous probability distributions.
2.4 Statistic moments and quantiles.
2.5 Characteristic function.
2.6 Joint probability distributions.
2.7 Summary.
3 Stable and tempered stable distributions.
3.1 α-Stable distribution.
3.2 Tempered stable distributions.
3.3 Infinitely divisible distributions.
3.3.1 Exponential Moments.
3.4 Summary.
3.5 Appendix.
4 Stochastic Processes in Continuous Time.
4.1 Some preliminaries.
4.2 Poisson Process.
4.3 Pure jump process.
4.4 Brownian motion.
4.5 Time-Changed Brownian motion.
4.6 L´evy process.
4.7 Summary.
5 Conditional Expectation and Change of Measure.
5.1 Events, σ-fields, and filtration.
5.2 Conditional expectation.
5.3 Change of measures.
5.4 Summary.
6 Exponential L´evy Models.
6.1 Exponential L´evy Models.
6.2 Fitting α-stable and tempered stable distributions.
6.3 Illustration: Parameter estimation for tempered stable distributions.
6.4 Summary.
6.5 Appendix : Numerical approximation of probability density and cumulative distribution functions.
7 Option Pricing in Exponential L´evy Models.
7.1 Option contract.
7.2 Boundary conditions for the price of an option.
7.3 No-arbitrage pricing and equivalent martingale measure.
7.4 Option pricing under the Black-Scholes model.
7.5 European option pricing under exponential tempered stable
Models.
7.6 The subordinated stock price model.
7.7 Summary.
8 Simulation.
8.1 Random number generators.
8.2 Simulation techniques for L´evy processes.
8.3 Tempered stable processes.
8.4 Tempered infinitely divisible processes.
8.5 Time-changed Brownian motion.
8.6 Monte Carlo methods.
Appendix.
9 Multi-tail t distribution.
9.1 Introduction.
9.2 Principal component analysis.
9.3 Estimating parameters.
9.4 Empirical results.
9.5 Conclusion.
10 Non-Gaussian portfolio allocation.
10.1 Introduction.
10.2 Multi-factor linear model.
10.3 Modeling dependencies.
10.4 Average value-at-risk.
10.5 Optimal portfolios.
10.6 The algorithm.
10.7 An empirical test.
10.8 Conclusions.
11 Normal GARCH models.
11.1 Introduction.
11.2 GARCH dynamics with normal innovation.
11.3 Market estimation.
11.4 Risk-neutral estimation.
11.5 Conclusions.
12 Smoothly truncated stable GARCH models.
12.1 Introduction.
12.2 A Generalized NGARCH Option Pricing Model.
12.3 Empirical Analysis.
12.4 Conclusion.
13 Infinitely divisible GARCH models.
13.1 Stock price dynamic.
13.2 Risk-neutral dynamic.
13.3 Non-normal infinitely divisible GARCH.
13.4 Simulate infinitely divisible GARCH.
Appendix.
14 Option Pricing with Monte Carlo.
14.1 Introduction.
14.2 Data set.
14.3 Performance of Option Pricing Models.
14.4 Conclusions.
15 American Option Pricing with Monte Carlo Methods.
15.1 American option pricing in discrete time.
15.2 The Least Squares Monte Carlo method.
15.3 LSM method in GARCH option pricing model.
15.4 Empirical illustration.
15.5 Summary.
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