Option Pricing and Estimation of Financial Models With R

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ISBN: 9780470745847 | 0470745843
Cover: Hardcover
Copyright: 4/4/2011

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Presents inference and simulation of stochastic process in the field of model calibration for financial times series modeled with continuous time processes and numerical option pricing. I ntroduces the basis of probability theory and goes on to explain how to model financial times series with continuous models, how to calibrate them and covers option pricing with one or more underlying assets based on these models.Analysis and implementation of models based on switching models or models with jumps are featured along with new models (Levy and telegraph process modeling) and topics such as; volatilty, covariation, p-variation and regime switching analysis, attention is focused on the calibration of these topics 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	p. xiii
A synthetic view	p. 1
The world of derivatives	p. 2
Different kinds of contracts	p. 2
Vanilla options	p. 3
Why options?	p. 6
A variety of options	p. 7
How to model asset prices	p. 8
One step beyond	p. 9
Bibliographical notes	p. 10
References	p. 10
Probability, random variables and statistics	p. 13
Probability	p. 13
Conditional probability	p. 15
Bayes' rule	p. 16
Random variables	p. 18
Characteristic function	p. 23
Moment generating function	p. 24
Examples of random variables	p. 24
Sum of random variables	p. 35
Infinitely divisible distributions	p. 37
Stable laws	p. 38
Fast Fourier Transform	p. 42
Inequalities	p. 46
Asymptotics	p. 48
Types of convergences	p. 48
Law of large numbers	p. 50
Central limit theorem	p. 52
Conditional expectation	p. 54
Statistics	p. 57
Properties of estimators	p. 57
The likelihood function	p. 61
Efficiency of estimators	p. 63
Maximum likelihood estimation	p. 64
Moment type estimators	p. 65
Least squares method	p. 65
Estimating functions	p. 66
Confidence intervals	p. 66
Numerical maximization of the likelihood	p. 68
The ¿-method	p. 70
Solution to exercises	p. 71
Bibliographical notes	p. 77
References	p. 77
Stochastic processes	p. 79
Definition and first properties	p. 79
Measurability and filtrations	p. 81
Simple and quadratic variation of a process	p. 83
Moments, covariance, and increments of stochastic processes	p. 84
Martingales	p. 84
Examples of martingales	p. 85
Inequalities for martingales	p. 88
Stopping times	p. 89
Markov property	p. 91
Discrete time Markov chains	p. 91
Continuous time Markov processes	p. 98
Continuous time Markov chains	p. 99
Mixing property	p. 101
Stable convergence	p. 103
Brownian motion	p. 104
Brownian motion and random walks	p. 106
Brownian motion is a martingale	p. 107
Brownian motion and partial differential equations	p. 107
Counting and marked processes	p. 108
Poisson process	p. 109
Compound Poisson process	p. 110
Compensated Poisson processes	p. 113
Telegraph process	p. 113
Telegraph process and partial differential equations	p. 115
Moments of the telegraph process	p. 117
Telegraph process and Brownian motion	p. 118
Stochastic integrals	p. 118
Properties of the stochastic integral	p. 122
Itô formula	p. 124
More properties and inequalities for the Itô integral	p. 127
Stochastic differential equations	p. 128
Existence and uniqueness of solutions	p. 128
Girsanov's theorem for diffusion processes	p. 130
Local martingales and semimartingales	p. 131
Lévy processes	p. 132
Lévy-Khintchine formula	p. 134
Lévy jumps and random measures	p. 135
Itô-Lévy decomposition of a Lévy process	p. 137
More on the Lévy measure	p. 138
The Itô formula for Lévy processes	p. 139
Lévy processes and martingales	p. 140
Stochastic differential equations with jumps	p. 143
Itô formula for Lévy driven stochastic differential equations	p. 144
Stochastic differential equations in Rⁿ	p. 145
Markov switching diffusions	p. 147
Solution to exercises	p. 148
Bibliographical notes	p. 155
References	p. 155
Numerical methods	p. 159
Monte Carlo method	p. 159
An application	p. 160
Numerical differentiation	p. 162
Root finding	p. 165
Numerical optimization	p. 167
Simulation of stochastic processes	p. 169
Poisson processes	p. 169
Telegraph process	p. 172
One-dimensional diffusion processes	p. 174
Multidimensional diffusion processes	p. 177
Lévy processes	p. 178
Simulation of stochastic differential equations with jumps	p. 181
Simulation of Markov switching diffusion processes	p. 183
Solution to exercises	p. 187
Bibliographical notes	p. 187
References	p. 187
Estimation of stochastic models for finance	p. 191
Geometric Brownian motion	p. 191
Properties of the increments	p. 193
Estimation of the parameters	p. 194
Quasi-maximum likelihood estimation	p. 195
Short-term interest rates models	p. 199
The special case of the CIR model	p. 201
Ahn-Gao model	p. 202
Aït-Sahalia model	p. 202
Exponential Lévy model	p. 205
Examples of Lévy models in finance	p. 205
Telegraph and geometric telegraph process	p. 210
Filtering of the geometric telegraph process	p. 216
Solution to exercises	p. 217
Bibliographical notes	p. 217
References	p. 218
European option pricing	p. 221
Contingent claims	p. 221
The main ingredients of option pricing	p. 223
One period market	p. 224
The Black and Scholes market	p. 227
Portfolio strategies	p. 228
Arbitrage and completeness	p. 229
Derivation of the Black and Scholes equation	p. 229
Solution of the Black and Scholes equation	p. 232
European call and put prices	p. 236
Put-call parity	p. 238
Option pricing with R	p. 239
The Monte Carlo approach	p. 242
Sensitivity of price to parameters	p. 246
The ¿-hedging and the Greeks	p. 249
The hedge ratio as a function of time	p. 251
Hedging of generic options	p. 252
The density method	p. 253
The numerical approximation	p. 254
The Monte Carlo approach	p. 255
Mixing Monte Carlo and numerical approximation	p. 256
Other Greeks of options	p. 258
Put and call Greeks with Rmetrics	p. 260
Pricing under the equivalent martingale measure	p. 261
Pricing of generic claims under the risk neutral measure	p. 264
Arbitrage and equivalent martingale measure	p. 264
More on numerical option pricing	p. 265
Pricing of path-dependent options	p. 266
Asian option pricing via asymptotic expansion	p. 269
Exotic option pricing with Rmetrics	p. 272
Implied volatility and volatility smiles	p. 273
Volatility smiles	p. 276
Pricing of basket options	p. 278
Numerical implementation	p. 280
Completeness and arbitrage	p. 280
An example with two assets	p. 280
Numerical pricing	p. 282
Solution to exercises	p. 282
Bibliographical notes	p. 283
References	p. 284
American options	p. 285
Finite difference methods	p. 285
Explicit finite-difference method	p. 286
Numerical stability	p. 292
Implicit finite-difference method	p. 293
The quadratic approximation	p. 297
Geske and Johnson and other approximations	p. 300
Monte Carlo methods	p. 300
Broadie and Glasserman simulation method	p. 300
Longstaff and Schwartz Least Squares Method	p. 307
Bibliographical notes	p. 311
References	p. 311
Pricing outside the standard Black and Scholes model	p. 313
The Lévy market model	p. 313
Why the Lévy market is incomplete?	p. 314
The Esscher transform	p. 315
The mean-correcting martingale measure	p. 317
Pricing of European options	p. 318
Option pricing using Fast Fourier Transform method	p. 318
The numerical implementation of the FFT pricing	p. 320
Pricing under the jump telegraph process	p. 325
Markov switching diffusions	p. 327
Monte Carlo pricing	p. 335
Semi-Monte Carlo method	p. 337
Pricing with the Fast Fourier Transform	p. 339
Other applications of Markov switching diffusion models	p. 341
The benchmark approach	p. 341
Benchmarking of the savings account	p. 344
Benchmarking of the risky asset	p. 344
Benchmarking the option price	p. 344
Martingale representation of the option price process	p. 345
Bibliographical notes	p. 346
References	p. 346
Miscellanea	p. 349
Monitoring of the volatility	p. 349
The least squares approach	p. 350
Analysis of multiple change points	p. 352
An example of real-time analysis	p. 354
More general quasi maximum likelihood approach	p. 355
Construction of the quasi-MLE	p. 356
A modified quasi-MLE	p. 357
First- and second-stage estimators	p. 358
Numerical example	p. 359
Asynchronous covariation estimation	p. 362
Numerical example	p. 364
LASSO model selection	p. 367
Modified LASSO objective function	p. 369
Adaptiveness of the method	p. 370
LASSO identification of the model for term structure of interest rates	p. 370
Clustering of financial time series	p. 374
The Markov operator distance	p. 375
Application to real data	p. 376
Sensitivity to misspecification	p. 383
Bibliographical notes	p. 387
References	p. 387
Appendices
'How to' guide to R	p. 393
Something to know first about R	p. 393
The workspace	p. 394
Graphics	p. 394
Getting help	p. 394
Installing packages	p. 395
Objects	p. 395
Assignments	p. 395
Basic object types	p. 398
Accessing objects and subsetting	p. 401
Coercion between data types	p. 405
S4 objects	p. 405
Functions	p. 408
Vectorization	p. 409
Parallel computing in R	p. 411
The foreach approach	p. 413
A note of warning on the multicore package	p. 416
Bibliographical notes	p. 416
References	p. 417
R in finance	p. 419
Overview of existing R frameworks	p. 419
Rmetrics	p. 420
RQuantLib	p. 420
The quantmod package	p. 421
Summary of main time series objects in R	p. 422
The ts class	p. 423
The zoo class	p. 424
The xts class	p. 426
The irts class	p. 427
The timeSeries class	p. 428
Dates and time handling	p. 428
Dates manipulation	p. 431
Using date objects to index time series	p. 433
Binding of time series	p. 434
Subsetting of time series	p. 440
Loading data from financial data servers	p. 442
Bibliographical notes	p. 445
References	p. 445
Index	p. 447
Table of Contents provided by Ingram. All Rights Reserved.

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Option Pricing and Estimation of Financial Models With R

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Option Pricing and Estimation of Financial Models With R

Summary

Author Biography

Table of Contents

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