Option Pricing and Estimation of Financial Models with R

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.

Table of Contents:
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 δ-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 References 155 4 Numerical methods 159 4.1 Monte Carlo method 159 4.1.1 An application 160 4.2 Numerical differentiation 162 4.3 Root finding 165 4.4 Numerical optimization 167 4.5 Simulation of stochastic processes 169 4.5.1 Poisson processes 169 4.5.2 Telegraph process 172 4.5.3 One-dimensional diffusion processes 174 4.5.4 Multidimensional diffusion processes 177 4.5.5 Lévy processes 178 4.5.6 Simulation of stochastic differential equations with jumps 181 4.5.7 Simulation of Markov switching diffusion processes 183 4.6 Solution to exercises 187 4.7 Bibliographical notes 187 References 187 5 Estimation of stochastic models for finance 191 5.1 Geometric Brownian motion 191 5.1.1 Properties of the increments 193 5.1.2 Estimation of the parameters 194 5.2 Quasi-maximum likelihood estimation 195 5.3 Short-term interest rates models 199 5.3.1 The special case of the CIR model 201 5.3.2 Ahn-Gao model 202 5.3.3 Aït-Sahalia model 202 5.4 Exponential Lévy model 205 5.4.1 Examples of Lévy models in finance 205 5.5 Telegraph and geometric telegraph process 210 5.5.1 Filtering of the geometric telegraph process 216 5.6 Solution to exercises 217 5.7 Bibliographical notes 217 References 218 6 European option pricing 221 6.1 Contingent claims 221 6.1.1 The main ingredients of option pricing 223 6.1.2 One period market 224 6.1.3 The Black and Scholes market 227 6.1.4 Portfolio strategies 228 6.1.5 Arbitrage and completeness 229 6.1.6 Derivation of the Black and Scholes equation 229 6.2 Solution of the Black and Scholes equation 232 6.2.1 European call and put prices 236 6.2.2 Put-call parity 238 6.2.3 Option pricing with R 239 6.2.4 The Monte Carlo approach 242 6.2.5 Sensitivity of price to parameters 246 6.3 The δ-hedging and the Greeks 249 6.3.1 The hedge ratio as a function of time 251 6.3.2 Hedging of generic options 252 6.3.3 The density method 253 6.3.4 The numerical approximation 254 6.3.5 The Monte Carlo approach 255 6.3.6 Mixing Monte Carlo and numerical approximation 256 6.3.7 Other Greeks of options 258 6.3.8 Put and call Greeks with Rmetrics 260 6.4 Pricing under the equivalent martingale measure 261 6.4.1 Pricing of generic claims under the risk neutral measure 264 6.4.2 Arbitrage and equivalent martingale measure 264 6.5 More on numerical option pricing 265 6.5.1 Pricing of path-dependent options 266 6.5.2 Asian option pricing via asymptotic expansion 269 6.5.3 Exotic option pricing with Rmetrics 272 6.6 Implied volatility and volatility smiles 273 6.6.1 Volatility smiles 276 6.7 Pricing of basket options 278 6.7.1 Numerical implementation 280 6.7.2 Completeness and arbitrage 280 6.7.3 An example with two assets 280 6.7.4 Numerical pricing 282 6.8 Solution to exercises 282 6.9 Bibliographical notes 283 References 284 7 American options 285 7.1 Finite difference methods 285 7.2 Explicit finite-difference method 286 7.2.1 Numerical stability 292 7.3 Implicit finite-difference method 293 7.4 The quadratic approximation 297 7.5 Geske and Johnson and other approximations 300 7.6 Monte Carlo methods 300 7.6.1 Broadie and Glasserman simulation method 300 7.6.2 Longstaff and Schwartz Least Squares Method 307 7.7 Bibliographical notes 311 References 311 8 Pricing outside the standard Black and Scholes model 313 8.1 The Lévy market model 313 8.1.1 Why the Lévy market is incomplete? 314 8.1.2 The Esscher transform 315 8.1.3 The mean-correcting martingale measure 317 8.1.4 Pricing of European options 318 8.1.5 Option pricing using Fast Fourier Transform method 318 8.1.6 The numerical implementation of the FFT pricing 320 8.2 Pricing under the jump telegraph process 325 8.3 Markov switching diffusions 327 8.3.1 Monte Carlo pricing 335 8.3.2 Semi-Monte Carlo method 337 8.3.3 Pricing with the Fast Fourier Transform 339 8.3.4 Other applications of Markov switching diffusion models 341 8.4 The benchmark approach 341 8.4.1 Benchmarking of the savings account 344 8.4.2 Benchmarking of the risky asset 344 8.4.3 Benchmarking the option price 344 8.4.4 Martingale representation of the option price process 345 8.5 Bibliographical notes 346 References 346 9 Miscellanea 349 9.1 Monitoring of the volatility 349 9.1.1 The least squares approach 350 9.1.2 Analysis of multiple change points 352 9.1.3 An example of real-time analysis 354 9.1.4 More general quasi maximum likelihood approach 355 9.1.5 Construction of the quasi-MLE 356 9.1.6 A modified quasi-MLE 357 9.1.7 First- and second-stage estimators 358 9.1.8 Numerical example 359 9.2 Asynchronous covariation estimation 362 9.2.1 Numerical example 364 9.3 LASSO model selection 367 9.3.1 Modified LASSO objective function 369 9.3.2 Adaptiveness of the method 370 9.3.3 LASSO identification of the model for term structure of interest rates 370 9.4 Clustering of financial time series 374 9.4.1 The Markov operator distance 375 9.4.2 Application to real data 376 9.4.3 Sensitivity to misspecification 383 9.5 Bibliographical notes 387 References 387 Appendices A ‘How to’ guide to R 393 A.1 Something to know first about R 393 A.1.1 The workspace 394 A.1.2 Graphics 394 A.1.3 Getting help 394 A.1.4 Installing packages 395 A.  Objects 395 A.2.1 Assignments 395 A.2.2 Basic object types 398 A.2.3 Accessing objects and subsetting 401 A.2.4 Coercion between data types 405 A.3 S4 objects 405 A.4 Functions 408 A.5 Vectorization 409 A.6 Parallel computing in R 411 A.6.1 The foreach approach 413 A.6.2 A note of warning on the multicore package 416 A.7 Bibliographical notes 416 References 417 B R in finance 419 B.1 Overview of existing R frameworks 419 B.1.1 Rmetrics 420 B.1.2 RQuantLib 420 B.1.3 The quantmod package 421 B.2 Summary of main time series objects in R 422 B.2.1 The ts class 423 B.2.2 The zoo class 424 B.2.3 The xts class 426 B.2.4 The irts class 427 B.2.5 The timeSeries class 428 B.3 Dates and time handling 428 B.3.1 Dates manipulation 431 B.3.2 Using date objects to index time series 433 B.4 Binding of time series 434 B.4.1 Subsetting of time series 440 B.5 Loading data from financial data servers 442 B.6 Bibliographical notes 445 References 445 Index 447

About the Author :
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.