Option Pricing and Estimation of Financial Models with R

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.

• Preface xiii1 A synthetic view 11.1 The world of derivatives 21.1.1 Different kinds of contracts 21.1.2 Vanilla options 31.1.3 Why options? 61.1.4 A variety of options 71.1.5 How to model asset prices 81.1.6 One step beyond 91.2 Bibliographical notes 10References 102 Probability, random variables and statistics 132.1 Probability 132.1.1 Conditional probability 152.2 Bayes’ rule 162.3 Random variables 182.3.1 Characteristic function 232.3.2 Moment generating function 242.3.3 Examples of random variables 242.3.4 Sum of random variables 352.3.5 Infinitely divisible distributions 372.3.6 Stable laws 382.3.7 Fast Fourier Transform 422.3.8 Inequalities 462.4 Asymptotics 482.4.1 Types of convergences 482.4.2 Law of large numbers 502.4.3 Central limit theorem 522.5 Conditional expectation 542.6 Statistics 572.6.1 Properties of estimators 572.6.2 The likelihood function 612.6.3 Efficiency of estimators 632.6.4 Maximum likelihood estimation 642.6.5 Moment type estimators 652.6.6 Least squares method 652.6.7 Estimating functions 662.6.8 Confidence intervals 662.6.9 Numerical maximization of the likelihood 682.6.10 The δ-method 702.7 Solution to exercises 712.8 Bibliographical notes 77References 773 Stochastic processes 793.1 Definition and first properties 793.1.1 Measurability and filtrations 813.1.2 Simple and quadratic variation of a process 833.1.3 Moments, covariance, and increments of stochastic processes 843.2 Martingales 843.2.1 Examples of martingales 853.2.2 Inequalities for martingales 883.3 Stopping times 893.4 Markov property 913.4.1 Discrete time Markov chains 913.4.2 Continuous time Markov processes 983.4.3 Continuous time Markov chains 993.5 Mixing property 1013.6 Stable convergence 1033.7 Brownian motion 1043.7.1 Brownian motion and random walks 1063.7.2 Brownian motion is a martingale 1073.7.3 Brownian motion and partial differential equations 1073.8 Counting and marked processes 1083.9 Poisson process 1093.10 Compound Poisson process 1103.11 Compensated Poisson processes 1133.12 Telegraph process 1133.12.1 Telegraph process and partial differential equations 1153.12.2 Moments of the telegraph process 1173.12.3 Telegraph process and Brownian motion 1183.13 Stochastic integrals 1183.13.1 Properties of the stochastic integral 1223.13.2 Itô formula 1243.14 More properties and inequalities for the Itô integral 1273.15 Stochastic differential equations 1283.15.1 Existence and uniqueness of solutions 1283.16 Girsanov’s theorem for diffusion processes 1303.17 Local martingales and semimartingales 1313.18 Lévy processes 1323.18.1 Lévy-Khintchine formula 1343.18.2 Lévy jumps and random measures 1353.18.3 Itô-Lévy decomposition of a Lévy process 1373.18.4 More on the Lévy measure 1383.18.5 The Itô formula for Lévy processes 1393.18.6 Lévy processes and martingales 1403.18.7 Stochastic differential equations with jumps 1433.18.8 Itô formula for Lévy driven stochastic differential equations 1443.19 Stochastic differential equations in R n 1453.20 Markov switching diffusions 1473.21 Solution to exercises 1483.22 Bibliographical notes 155References 1554 Numerical methods 1594.1 Monte Carlo method 1594.1.1 An application 1604.2 Numerical differentiation 1624.3 Root finding 1654.4 Numerical optimization 1674.5 Simulation of stochastic processes 1694.5.1 Poisson processes 1694.5.2 Telegraph process 1724.5.3 One-dimensional diffusion processes 1744.5.4 Multidimensional diffusion processes 1774.5.5 Lévy processes 1784.5.6 Simulation of stochastic differential equations with jumps 1814.5.7 Simulation of Markov switching diffusion processes 1834.6 Solution to exercises 1874.7 Bibliographical notes 187References 1875 Estimation of stochastic models for finance 1915.1 Geometric Brownian motion 1915.1.1 Properties of the increments 1935.1.2 Estimation of the parameters 1945.2 Quasi-maximum likelihood estimation 1955.3 Short-term interest rates models 1995.3.1 The special case of the CIR model 2015.3.2 Ahn-Gao model 2025.3.3 Aït-Sahalia model 2025.4 Exponential Lévy model 2055.4.1 Examples of Lévy models in finance 2055.5 Telegraph and geometric telegraph process 2105.5.1 Filtering of the geometric telegraph process 2165.6 Solution to exercises 2175.7 Bibliographical notes 217References 2186 European option pricing 2216.1 Contingent claims 2216.1.1 The main ingredients of option pricing 2236.1.2 One period market 2246.1.3 The Black and Scholes market 2276.1.4 Portfolio strategies 2286.1.5 Arbitrage and completeness 2296.1.6 Derivation of the Black and Scholes equation 2296.2 Solution of the Black and Scholes equation 2326.2.1 European call and put prices 2366.2.2 Put-call parity 2386.2.3 Option pricing with R 2396.2.4 The Monte Carlo approach 2426.2.5 Sensitivity of price to parameters 2466.3 The δ-hedging and the Greeks 2496.3.1 The hedge ratio as a function of time 2516.3.2 Hedging of generic options 2526.3.3 The density method 2536.3.4 The numerical approximation 2546.3.5 The Monte Carlo approach 2556.3.6 Mixing Monte Carlo and numerical approximation 2566.3.7 Other Greeks of options 2586.3.8 Put and call Greeks with Rmetrics 2606.4 Pricing under the equivalent martingale measure 2616.4.1 Pricing of generic claims under the risk neutral measure 2646.4.2 Arbitrage and equivalent martingale measure 2646.5 More on numerical option pricing 2656.5.1 Pricing of path-dependent options 2666.5.2 Asian option pricing via asymptotic expansion 2696.5.3 Exotic option pricing with Rmetrics 2726.6 Implied volatility and volatility smiles 2736.6.1 Volatility smiles 2766.7 Pricing of basket options 2786.7.1 Numerical implementation 2806.7.2 Completeness and arbitrage 2806.7.3 An example with two assets 2806.7.4 Numerical pricing 2826.8 Solution to exercises 2826.9 Bibliographical notes 283References 2847 American options 2857.1 Finite difference methods 2857.2 Explicit finite-difference method 2867.2.1 Numerical stability 2927.3 Implicit finite-difference method 2937.4 The quadratic approximation 2977.5 Geske and Johnson and other approximations 3007.6 Monte Carlo methods 3007.6.1 Broadie and Glasserman simulation method 3007.6.2 Longstaff and Schwartz Least Squares Method 3077.7 Bibliographical notes 311References 3118 Pricing outside the standard Black and Scholes model 3138.1 The Lévy market model 3138.1.1 Why the Lévy market is incomplete? 3148.1.2 The Esscher transform 3158.1.3 The mean-correcting martingale measure 3178.1.4 Pricing of European options 3188.1.5 Option pricing using Fast Fourier Transform method 3188.1.6 The numerical implementation of the FFT pricing 3208.2 Pricing under the jump telegraph process 3258.3 Markov switching diffusions 3278.3.1 Monte Carlo pricing 3358.3.2 Semi-Monte Carlo method 3378.3.3 Pricing with the Fast Fourier Transform 3398.3.4 Other applications of Markov switching diffusion models 3418.4 The benchmark approach 3418.4.1 Benchmarking of the savings account 3448.4.2 Benchmarking of the risky asset 3448.4.3 Benchmarking the option price 3448.4.4 Martingale representation of the option price process 3458.5 Bibliographical notes 346References 3469 Miscellanea 3499.1 Monitoring of the volatility 3499.1.1 The least squares approach 3509.1.2 Analysis of multiple change points 3529.1.3 An example of real-time analysis 3549.1.4 More general quasi maximum likelihood approach 3559.1.5 Construction of the quasi-MLE 3569.1.6 A modified quasi-MLE 3579.1.7 First- and second-stage estimators 3589.1.8 Numerical example 3599.2 Asynchronous covariation estimation 3629.2.1 Numerical example 3649.3 LASSO model selection 3679.3.1 Modified LASSO objective function 3699.3.2 Adaptiveness of the method 3709.3.3 LASSO identification of the model for term structure of interest rates 3709.4 Clustering of financial time series 3749.4.1 The Markov operator distance 3759.4.2 Application to real data 3769.4.3 Sensitivity to misspecification 3839.5 Bibliographical notes 387References 387AppendicesA ‘How to’ guide to R 393A.1 Something to know first about R 393A.1.1 The workspace 394A.1.2 Graphics 394A.1.3 Getting help 394A.1.4 Installing packages 395A.  Objects 395A.2.1 Assignments 395A.2.2 Basic object types 398A.2.3 Accessing objects and subsetting 401A.2.4 Coercion between data types 405A.3 S4 objects 405A.4 Functions 408A.5 Vectorization 409A.6 Parallel computing in R 411A.6.1 The foreach approach 413A.6.2 A note of warning on the multicore package 416A.7 Bibliographical notes 416References 417B R in finance 419B.1 Overview of existing R frameworks 419B.1.1 Rmetrics 420B.1.2 RQuantLib 420B.1.3 The quantmod package 421B.2 Summary of main time series objects in R 422B.2.1 The ts class 423B.2.2 The zoo class 424B.2.3 The xts class 426B.2.4 The irts class 427B.2.5 The timeSeries class 428B.3 Dates and time handling 428B.3.1 Dates manipulation 431B.3.2 Using date objects to index time series 433B.4 Binding of time series 434B.4.1 Subsetting of time series 440B.5 Loading data from financial data servers 442B.6 Bibliographical notes 445References 445Index 447