Stefano M. Iacus, Department of Economics, Business and Statistics, University of Milan, Italy
The aim of this book is twofold. The first goal is to summarize elementary and advanced topics on modern option pricing: from the basic models of the Black & Scholes theory to the more sophisticated approach based on Lévy processes and other jump processes.
At the same time, the other goal of the book is to identify, estimate and justify, with the use of statistically sound techniques, the choice of particular financial models starting from real financial data.
In the spirit of modern finance, this book considers only continuous time models like diffusion of Lévy processes. Therefore, the statistical techniques presented are those designed to work on real discrete time data obtained from these continuous time models.
- Provides a comprehensive and in–depth guide to financial modeling.
- Looks at basic and advanced option pricing with R.
- Explores simulation of multidimensional stochastic differential equations with jumps.
- Provides a comprehensive survey on empirical finance in the R statistical environment.
- Addresses model selection and identification of financial models from empirical financial data.
This book is an invaluable resource for post graduate students and researchers in economics, mathematics and statistics who want to approach mathematical finance from an applied point of view. Statisticians and data analysts working in a field related to finance will also benefit from this book.
1. A Synthetic View.
1.1 The World of Derivatives.
1.2 Bibliographic Notes.
2. Probability, Random Variables and Statistics.
2.2 Bayes′ Rule.
2.3 Random Variables.
2.5 Conditional Expectation.
2.7 Solution to Exercises.
2.8 Bibliographic Notes.
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 Itô Integral.
3.15 Stochastic Differential Equations.
3.16 Girsanov′s theorem for diffusion processes.
3.17 Local Martingales and Semimartingales.
3.18 Lévy Processes.
3.19 Stochastic Differential Equations in Rn.
3.20 Markov Switching Diffusions.
3.21 Solution to Exercises.
3.22 Bibliographic Notes.
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.
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 Lévy Model.
5.5 Telegraph and Geometric Telegraph Process.
5.6 Solution to Exercises.
5.7 Bibliographic Notes.
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.
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.
8. Pricing Outside the Standard Black & Scholes Model.
8.1 The Lévy Market Model.
8.2 Pricing Under the Jump Telegraph Process.
8.3 Markov Switching Diffusions.
8.4 The Benchmark approach.
8.5 Bibliographic Notes.
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.
A. ′How to′ Guide to R.
A.1 Something to Know Soon About R.
A.3 S4 Objects.
A.6 Parallel Computing in R.
A.7 Bibliographic Notes.
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.