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,
Lévy 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.
Autorentext
Stefano Maria Iacus, Professor (Professore Associato) of Probability and Mathematical Statistics at University of Milan, Department of Economics, Business and Statistics. Stefano is a member of the R development Core Team.
Klappentext
Option Pricing and Estimation of Financial Models with R
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.
Key Features:
- 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.
Zusammenfassung
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, Lévy 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.
Inhalt
Preface xiii
1 A synthetic view 1
1.1 The world of derivatives 2
1.1.1 Different kinds of contracts 2
1.1.2 Vanilla options 3
1.1.3 Why options? 6
1.1.4 A variety of options 7
1.1.5 How to model asset prices 8
1.1.6 One step beyond 9
1.2 Bibliographical notes 10
References 10
2 Probability, random variables and statistics 13
2.1 Probability 13
2.1.1 Conditional probability 15
2.2 Bayes' rule 16
2.3 Random variables 18
2.3.1 Characteristic function 23
2.3.2 Moment generating function 24
2.3.3 Examples of random variables 24
2.3.4 Sum of random variables 35
2.3.5 Infinitely divisible distributions 37
2.3.6 Stable laws 38
2.3.7 Fast Fourier Transform 42
2.3.8 Inequalities 46
2.4 Asymptotics 48
2.4.1 Types of convergences 48
2.4.2 Law of large numbers 50
2.4.3 Central limit theorem 52
2.5 Conditional expectation 54
2.6 Statistics 57
2.6.1 Properties of estimators 57
2.6.2 The likelihood function 61
2.6.3 Efficiency of estimators 63
2.6.4 Maximum likelihood estimation 64
2.6.5 Moment type estimators 65
2.6.6 Least squares method 65
2.6.7 Estimating functions 66
2.6.8 Confidence intervals 66
2.6.9 Numerical maximization of the likelihood 68
2.6.10 The -method 70
2.7 Solution to exercises 71
2.8 Bibliographical notes 77
References 77
3 Stochastic processes 79
3.1 Definition and first properties 79
3.1.1 Measurability and filtrations 81
3.1.2 Simple and quadratic variation of a process 83
3.1.3 Moments, covariance, and increments of stochastic processes 84
3.2 Martingales 84
3.2.1 Examples of martingales 85
3.2.2 Inequalities for martingales 88
3.3 Stopping times 89
3.4 Markov property 91
3.4.1 Discrete time Markov chains 91
3.4.2 Continuous time Markov processes 98
3.4.3 Continuous time Markov chains 99
3.5 Mixing property 101
3.6 Stable convergence 103
3.7 Brownian motion 104
3.7.1 Brownian motion and random walks 106
3.7.2 Brownian motion is a martingale 107
3.7.3 Brownian motion and partial differential equations 107
3.8 Counting and marked processes 108
3.9 Poisson process 109
3.10 Compound Poisson process 110
3.11 Compensated Poisson processes 113
3.12 Telegraph process 113
3.12.1 Telegraph process and partial differential equations 115
3.12.2 Moments of the telegraph process 117
3.12.3 Telegraph process and Brownian motion 118
3.13 Stochastic integrals 118
3.13.1 Properties of the stochastic integral 122
3.13.2 Itô formula 124
3.14 More properties and inequalities for the Itô integral 127
3.15 Stochastic differential equations 128
3.15.1 Existence and uniqueness of solutions 128
3.16 Girsanov's theorem for diffusion processes 130
3.17 Local martingales and semimartingales 131
3.18 Lévy processes 132
3.18.1 Lévy-Khintchine formula 134
3.18.2 Lévy jumps and random measures 135
3.18.3 Itô-Lévy decomposition of a Lévy process 137
3.18.4 More on the Lévy measure 138
3.18.5 The Itô formula for Lévy processes 139
3.18.6 Lévy processes and martingales 140
3.18.7 Stochastic differential equations with jumps 143
3.18.8 Itô formula for Lévy driven stochastic differential equations 144
3.19 Stochastic differential equations in R n 145
3.20 Markov switching diffusions 147
3.21 Solution to exercises 148
3.22 Bibliographical notes 155
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