Presents inference and simulation of stochastic process in the field of model calibration for financial times series modelled by continuous time processes and numerical option pricing. Introduces the bases of probability theory and goes on to explain how to model financial times series with continuous models, how to calibrate them from discrete data and further covers option pricing with one or more underlying assets based on these models. Analysis and implementation of models goes beyond the standard Black and Scholes framework and includes Markov switching models, Levy models and other models with jumps (e.g. the telegraph process); Topics other than option pricing include: volatility and covariation estimation, change point analysis, asymptotic expansion and classification of financial time series from a statistical viewpoint. The book features problems with solutions and examples. All the examples and R code are available as an additional R package, therefore all the examples can be reproduced
Preface. 1. A Synthetic View. 1.1 The World of Derivatives. 1.2 Bibliographic Notes. References. 2. Probability, Random Variables and Statistics. 2.1 Probability. 2.2 Bayes' Rule. 2.3 Random Variables. 2.4 Asymptotics. 2.5 Conditional Expectation. 2.6 Statistics. 2.7 Solution to Exercises. 2.8 Bibliographic Notes. References. 3. Stochastic Processes. 3.1 Definition and First Properties. 3.3 Stopping Times. 3.4 Markov Property. 3.5 Mixing Property. 3.6 Stable Convergence. 3.7 Brownian Motion. 3.8 Counting and Marked Processes. 3.9 Poisson Process. 3.10 Compound Poisson process. 3.11 Compensated Poisson processes. 3.12 Telegraph Process. 3.13 Stochastic Integrals. 3.14 More Properties and Inequalities for the Ito Integral. 3.15 Stochastic Differential Equations. 3.16 Girsanov's theorem for diffusion processes. 3.17 Local Martingales and Semimartingales. 3.18 Levy Processes. 3.19 Stochastic Differential Equations in Rn. 3.20 Markov Switching Diffusions. 3.21 Solution to Exercises. 3.22 Bibliographic Notes. References. 4. Numerical Methods. 4.1 Monte Carlo Method. 4.2 Numerical Differentiation. 4.3 Root Finding. 4.4 Numerical Optimization. 4.5 Simulation of Stochastic Processes. 4.6 Solution to Exercises. 4.7 Bibliographic Notes. References. 5. Estimation of Stochastic Models for Finance. 5.1 Geometric Brownian Motion. 5.2 Quasi-Maximum Likelihood Estimation. 5.3 Short-Term Interest Rates Models. 5.4 Exponential Levy Model. 5.5 Telegraph and Geometric Telegraph Process. 5.6 Solution to Exercises. 5.7 Bibliographic Notes. References. 6. European Option Pricing. 6.1 Contingent Claims. 6.2 Solution of the Black & Scholes Equation. 6.3 The Hedging and the Greeks. 6.4 Pricing Under the Equivalent Martingale Measure. 6.5 More on Numerical Option Pricing. 6.6 Implied Volatility and Volatility Smiles. 6.7 Pricing of Basket Options. 6.8 Solution to Exercises. 6.9 Bibliographic Notes. References. 7. American Options. 7.1 Finite Difference Methods. 7.2 Explicit Finite-Difference Method. 7.3 Implicit Finite-Difference Method. 7.4 The Quadratic Approximation. 7.5 Geske & Johnson and Other Approximations. 7.6 Monte Carlo Methods. 7.7 Bibliographic Notes. References. 8. Pricing Outside the Standard Black & Scholes Model. 8.1 The Levy Market Model. 8.2 Pricing Under the Jump Telegraph Process. 8.3 Markov Switching Diffusions. 8.4 The Benchmark approach. 8.5 Bibliographic Notes. References. 9. Miscellanea. 9.1 Monitoring of the Volatility. 9.2 Asynchronous Covariation Estimation. 9.3 LASSO Model Selection. 9.4 Clustering of Financial Time Series. 9.5 Bibliographic Notes. References. A. How to Guide to R. A.1 Something to Know Soon About R. A.2 Objects. A.3 S4 Objects. A.4 Functions. A.5 Vectorization. A.6 Parallel Computing in R. A.7 Bibliographic Notes. References. B. R in Finance. B.1 Overview of Existing R Frameworks. B.2 Summary of Main Time Series Objects in R. B.3 Dates and Time Handling. B.4 Binding of Time Series. B.5 Loading Data From Financial Data Servers. B.6 Bibliographic Notes. References. Index.
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