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Quantitative Finance. Edition No. 1. Statistics in Practice

  • Book

  • 496 Pages
  • January 2020
  • John Wiley and Sons Ltd
  • ID: 5840987

Presents a multitude of topics relevant to the quantitative finance community by combining the best of the theory with the usefulness of applications

Written by accomplished teachers and researchers in the field, this book presents quantitative finance theory through applications to specific practical problems and comes with accompanying coding techniques in R and MATLAB, and some generic pseudo-algorithms to modern finance. It also offers over 300 examples and exercises that are appropriate for the beginning student as well as the practitioner in the field.

The Quantitative Finance book is divided into four parts. Part One begins by providing readers with the theoretical backdrop needed from probability and stochastic processes. We also present some useful finance concepts used throughout the book. In part two of the book we present the classical Black-Scholes-Merton model in a uniquely accessible and understandable way. Implied volatility as well as local volatility surfaces are also discussed. Next, solutions to Partial Differential Equations (PDE), wavelets and Fourier transforms are presented. Several methodologies for pricing options namely, tree methods, finite difference method and Monte Carlo simulation methods are also discussed. We conclude this part with a discussion on stochastic differential equations (SDE’s). In the third part of this book, several new and advanced models from current literature such as general Lvy processes, nonlinear PDE's for stochastic volatility models in a transaction fee market, PDE's in a jump-diffusion with stochastic volatility models and factor and copulas models are discussed. In part four of the book, we conclude with a solid presentation of the typical topics in fixed income securities and derivatives. We discuss models for pricing bonds market, marketable securities, credit default swaps (CDS) and securitizations.

  • Classroom-tested over a three-year period with the input of students and experienced practitioners
  • Emphasizes the volatility of financial analyses and interpretations
  • Weaves theory with application throughout the book
  • Utilizes R and MATLAB software programs
  • Presents pseudo-algorithms for readers who do not have access to any particular programming system
  • Supplemented with extensive author-maintained web site that includes helpful teaching hints, data sets, software programs, and additional content 

Quantitative Finance is an ideal textbook for upper-undergraduate and beginning graduate students in statistics, financial engineering, quantitative finance, and mathematical finance programs. It will also appeal to practitioners in the same fields.

Table of Contents

List of Figures xv

List of Tables xvii

Part I Stochastic Processes and Finance 1

1 Stochastic Processes 3

1.1 Introduction 3

1.2 General Characteristics of Stochastic Processes 4

1.2.1 The Index Set I 4

1.2.2 The State Space S 4

1.2.3 Adaptiveness, Filtration, and Standard Filtration 5

1.2.4 Pathwise Realizations 7

1.2.5 The Finite Dimensional Distribution of Stochastic Processes 8

1.2.6 Independent Components 9

1.2.7 Stationary Process 9

1.2.8 Stationary and Independent Increments 10

1.3 Variation and Quadratic Variation of Stochastic Processes 11

1.4 Other More Specific Properties 13

1.5 Examples of Stochastic Processes 14

1.5.1 The Bernoulli Process (Simple Random Walk) 14

1.5.2 The Brownian Motion (Wiener Process) 17

1.6 Borel - Cantelli Lemmas 19

1.7 Central Limit Theorem 20

1.8 Stochastic Differential Equation 20

1.9 Stochastic Integral 21

1.9.1 Properties of the Stochastic Integral 22

1.10 Maximization and Parameter Calibration of Stochastic Processes 22

1.10.1 Approximation of the Likelihood Function (Pseudo Maximum Likelihood Estimation) 24

1.10.2 Ozaki Method 24

1.10.3 Shoji-Ozaki Method 25

1.10.4 Kessler Method 25

1.11 Quadrature Methods 26

1.11.1 Rectangle Rule: (n = 1) (Darboux Sums) 27

1.11.2 Midpoint Rule 28

1.11.3 Trapezoid Rule 28

1.11.4 Simpson’s Rule 28

1.12 Problems 29

2 Basics of Finance 33

2.1 Introduction 33

2.2 Arbitrage 33

2.3 Options 35

2.3.1 Vanilla Options 35

2.3.2 Put-Call Parity 36

2.4 Hedging 39

2.5 Modeling Return of Stocks 40

2.6 Continuous Time Model 41

2.6.1 Itô’s Lemma 42

2.7 Problems 45

Part II Quantitative Finance in Practice 47

3 Some Models Used in Quantitative Finance 49

3.1 Introduction 49

3.2 Assumptions for the Black-Scholes-Merton Derivation 49

3.3 The B-S Model 50

3.4 Some Remarks on the B-S Model 58

3.4.1 Remark 1 58

3.4.2 Remark 2 58

3.5 Heston Model 60

3.5.1 Heston PDE Derivation 61

3.6 The Cox-Ingersoll-Ross (CIR) Model 63

3.7 Stochastic 𝛼, 𝛽, 𝜌 (SABR) Model 64

3.7.1 SABR Implied Volatility 64

3.8 Methods for Finding Roots of Functions: Implied Volatility 65

3.8.1 Introduction 65

3.8.2 The Bisection Method 65

3.8.3 The Newton’s Method 66

3.8.4 Secant Method 67

3.8.5 Computation of Implied Volatility Using the Newton’s Method 68

3.9 Some Remarks of Implied Volatility (Put-Call Parity) 69

3.10 Hedging Using Volatility 70

3.11 Functional Approximation Methods 73

3.11.1 Local Volatility Model 74

3.11.2 Dupire’s Equation 74

3.11.3 Spline Approximation 77

3.11.4 Numerical Solution Techniques 78

3.11.5 Pricing Surface 79

3.12 Problems 79

4 Solving Partial Differential Equations 83

4.1 Introduction 83

4.2 Useful Definitions and Types of PDEs 83

4.2.1 Types of PDEs (2-D) 83

4.2.2 Boundary Conditions (BC) for PDEs 84

4.3 Functional Spaces Useful for PDEs 85

4.4 Separation of Variables 88

4.5 Moment-Generating Laplace Transform 91

4.5.1 Numeric Inversion for Laplace Transform 92

4.5.2 Fourier Series Approximation Method 93

4.6 Application of the Laplace Transform to the Black-Scholes PDE 96

4.7 Problems 99

5 Wavelets and Fourier Transforms 101

5.1 Introduction 101

5.2 Dynamic Fourier Analysis 101

5.2.1 Tapering 102

5.2.2 Estimation of Spectral Density with Daniell Kernel 103

5.2.3 Discrete Fourier Transform 104

5.2.4 The Fast Fourier Transform (FFT) Method 106

5.3 Wavelets Theory 109

5.3.1 Definition 109

5.3.2 Wavelets and Time Series 110

5.4 Examples of Discrete Wavelets Transforms (DWT) 112

5.4.1 Haar Wavelets 112

5.4.2 Daubechies Wavelets 115

5.5 Application of Wavelets Transform 116

5.5.1 Finance 116

5.5.2 Modeling and Forecasting 117

5.5.3 Image Compression 117

5.5.4 Seismic Signals 117

5.5.5 Damage Detection in Frame Structures 118

5.6 Problems 118

6 Tree Methods 121

6.1 Introduction 121

6.2 Tree Methods: the Binomial Tree 122

6.2.1 One-Step Binomial Tree 122

6.2.2 Using the Tree to Price a European Option 125

6.2.3 Using the Tree to Price an American Option 126

6.2.4 Using the Tree to Price Any Path-Dependent Option 127

6.2.5 Using the Tree for Computing Hedge Sensitivities: the Greeks 128

6.2.6 Further Discussion on the American Option Pricing 128

6.2.7 A Parenthesis: the Brownian Motion as a Limit of Simple Random Walk 132

6.3 Tree Methods for Dividend-Paying Assets 135

6.3.1 Options on Assets Paying a Continuous Dividend 135

6.3.2 Options on Assets Paying a Known Discrete Proportional Dividend 136

6.3.3 Options on Assets Paying a Known Discrete Cash Dividend 136

6.3.4 Tree for Known (Deterministic) Time-Varying Volatility 137

6.4 Pricing Path-Dependent Options: Barrier Options 139

6.5 Trinomial Tree Method and Other Considerations 140

6.6 Markov Process 143

6.6.1 Transition Function 143

6.7 Basic Elements of Operators and Semigroup Theory 146

6.7.1 Infinitesimal Operator of Semigroup 150

6.7.2 Feller Semigroup 151

6.8 General Diffusion Process 152

6.8.1 Example: Derivation of Option Pricing PDE 155

6.9 A General Diffusion Approximation Method 156

6.10 Particle Filter Construction 159

6.11 Quadrinomial Tree Approximation 163

6.11.1 Construction of the One-Period Model 164

6.11.2 Construction of the Multiperiod Model: Option Valuation 170

6.12 Problems 173

7 Approximating PDEs 177

7.1 Introduction 177

7.2 The Explicit Finite Difference Method 179

7.2.1 Stability and Convergence 180

7.3 The Implicit Finite Difference Method 180

7.3.1 Stability and Convergence 182

7.4 The Crank-Nicolson Finite Difference Method 183

7.4.1 Stability and Convergence 183

7.5 A Discussion About the Necessary Number of Nodes in the Schemes 184

7.5.1 Explicit Finite Difference Method 184

7.5.2 Implicit Finite Difference Method 185

7.5.3 Crank-Nicolson Finite Difference Method 185

7.6 Solution of a Tridiagonal System 186

7.6.1 Inverting the Tridiagonal Matrix 186

7.6.2 Algorithm for Solving a Tridiagonal System 187

7.7 Heston PDE 188

7.7.1 Boundary Conditions 189

7.7.2 Derivative Approximation for Nonuniform Grid 190

7.8 Methods for Free Boundary Problems 191

7.8.1 American Option Valuations 192

7.8.2 Free Boundary Problem 192

7.8.3 Linear Complementarity Problem (LCP) 193

7.8.4 The Obstacle Problem 196

7.9 Methods for Pricing American Options 199

7.10 Problems 201

8 Approximating Stochastic Processes 203

8.1 Introduction 203

8.2 Plain Vanilla Monte Carlo Method 203

8.3 Approximation of Integrals Using the Monte Carlo Method 205

8.4 Variance Reduction 205

8.4.1 Antithetic Variates 205

8.4.2 Control Variates 206

8.5 American Option Pricing with Monte Carlo Simulation 208

8.5.1 Introduction 209

8.5.2 Martingale Optimization 210

8.5.3 Least Squares Monte Carlo (LSM) 210

8.6 Nonstandard Monte Carlo Methods 216

8.6.1 Sequential Monte Carlo (SMC) Method 216

8.6.2 Markov Chain Monte Carlo (MCMC) Method 217

8.7 Generating One-Dimensional Random Variables by Inverting the cdf 218

8.8 Generating One-Dimensional Normal Random Variables 220

8.8.1 The Box-Muller Method 221

8.8.2 The Polar Rejection Method 222

8.9 Generating Random Variables: Rejection Sampling Method 224

8.9.1 Marsaglia’s Ziggurat Method 226

8.10 Generating Random Variables: Importance Sampling 236

8.10.1 Sampling Importance Resampling 240

8.10.2 Adaptive Importance Sampling 241

8.11 Problems 242

9 Stochastic Differential Equations 245

9.1 Introduction 245

9.2 The Construction of the Stochastic Integral 246

9.2.1 Itô Integral Construction 249

9.2.2 An Illustrative Example 251

9.3 Properties of the Stochastic Integral 253

9.4 Itô Lemma 254

9.5 Stochastic Differential Equations (SDEs) 257

9.5.1 Solution Methods for SDEs 259

9.6 Examples of Stochastic Differential Equations 260

9.6.1 An Analysis of Cox-Ingersoll-Ross (CIR)-Type Models 263

9.6.2 Moments Calculation for the CIR Model 265

9.6.3 Interpretation of the Formulas for Moments 267

9.6.4 Parameter Estimation for the CIR Model 267

9.7 Linear Systems of SDEs 268

9.8 Some Relationship Between SDEs and Partial Differential Equations (PDEs) 271

9.9 Euler Method for Approximating SDEs 273

9.10 Random Vectors: Moments and Distributions 277

9.10.1 The Dirichlet Distribution 279

9.10.2 Multivariate Normal Distribution 280

9.11 Generating Multivariate (Gaussian) Distributions with Prescribed Covariance Structure 281

9.11.1 Generating Gaussian Vectors 281

9.12 Problems 283

Part III Advanced Models for Underlying Assets 287

10 Stochastic Volatility Models 289

10.1 Introduction 289

10.2 Stochastic Volatility 289

10.3 Types of Continuous Time SV Models 290

10.3.1 Constant Elasticity of Variance (CEV) Models 291

10.3.2 Hull-White Model 292

10.3.3 The Stochastic Alpha Beta Rho (SABR) Model 293

10.3.4 Scott Model 294

10.3.5 Stein and Stein Model 295

10.3.6 Heston Model 295

10.4 Derivation of Formulae Used: Mean-Reverting Processes 296

10.4.1 Moment Analysis for CIR Type Processes 299

10.5 Problems 301

11 Jump Diffusion Models 303

11.1 Introduction 303

11.2 The Poisson Process (Jumps) 303

11.3 The Compound Poisson Process 304

11.4 The Black-Scholes Models with Jumps 305

11.5 Solutions to Partial-Integral Differential Systems 310

11.5.1 Suitability of the Stochastic Model Postulated 311

11.5.2 Regime-Switching Jump Diffusion Model 312

11.5.3 The Option Pricing Problem 313

11.5.4 The General PIDE System 314

11.6 Problems 322

12 General Lévy Processes 325

12.1 Introduction and Definitions 325

12.2 Lévy Processes 325

12.3 Examples of Lévy Processes 329

12.3.1 The Gamma Process 329

12.3.2 Inverse Gaussian Process 330

12.3.3 Exponential Lévy Models 330

12.4 Subordination of Lévy Processes 331

12.5 Rescaled Range Analysis (Hurst Analysis) and Detrended Fluctuation Analysis (DFA) 332

12.5.1 Rescaled Range Analysis (Hurst Analysis) 332

12.5.2 Detrended Fluctuation Analysis 334

12.5.3 Stationarity and Unit Root Test 335

12.6 Problems 336

13 Generalized Lévy Processes, Long Range Correlations, and Memory Effects 337

13.1 Introduction 337

13.1.1 Stable Distributions 337

13.2 The Lévy Flight Models 339

13.2.1 Background 339

13.2.2 Kurtosis 343

13.2.3 Self-Similarity 345

13.2.4 The H - 𝛼 Relationship for the Truncated Lévy Flight 346

13.3 Sum of Lévy Stochastic Variables with Different Parameters 347

13.3.1 Sum of Exponential Random Variables with Different Parameters 348

13.3.2 Sum of Lévy Random Variables with Different Parameters 351

13.4 Examples and Applications 352

13.4.1 Truncated Lévy Models Applied to Financial Indices 352

13.4.2 Detrended Fluctuation Analysis (DFA) and Rescaled Range Analysis Applied to Financial Indices 357

13.5 Problems 362

14 Approximating General Derivative Prices 365

14.1 Introduction 365

14.2 Statement of the Problem 368

14.3 A General Parabolic Integro-Differential Problem 370

14.3.1 Schaefer’s Fixed Point Theorem 371

14.4 Solutions in Bounded Domains 372

14.5 Construction of the Solution in the Whole Domain 385

14.6 Problems 386

15 Solutions to Complex Models Arising in the Pricing of Financial Options 389

15.1 Introduction 389

15.2 Option Pricing with Transaction Costs and Stochastic Volatility 389

15.3 Option Price Valuation in the Geometric Brownian Motion Case with Transaction Costs 390

15.4 Stochastic Volatility Model with Transaction Costs 392

15.5 The PDE Derivation When the Volatility is a Traded Asset 393

15.5.1 The Nonlinear PDE 395

15.5.2 Derivation of the Option Value PDEs in Arbitrage Free and Complete Markets 397

15.6 Problems 400

16 Factor and Copulas Models 403

16.1 Introduction 403

16.2 Factor Models 403

16.2.1 Cross-Sectional Regression 404

16.2.2 Expected Return 406

16.2.3 Macroeconomic Factor Models 407

16.2.4 Fundamental Factor Models 408

16.2.5 Statistical Factor Models 408

16.3 Copula Models 409

16.3.1 Families of Copulas 411

16.4 Problems 412

Part IV Fixed Income Securities and Derivatives 413

17 Models for the Bond Market 415

17.1 Introduction and Notations 415

17.2 Notations 415

17.3 Caps and Swaps 417

17.4 Valuation of Basic Instruments: Zero Coupon and Vanilla Options on Zero Coupon 419

17.4.1 Black Model 419

17.4.2 Short Rate Models 420

17.5 Term Structure Consistent Models 422

17.6 Inverting the Yield Curve 426

17.6.1 Affine Term Structure 427

17.7 Problems 428

18 Exchange Traded Funds (ETFs), Credit Default Swap (CDS), and Securitization 431

18.1 Introduction 431

18.2 Exchange Traded Funds (ETFs) 431

18.2.1 Index ETFs 432

18.2.2 Stock ETFs 433

18.2.3 Bond ETFs 433

18.2.4 Commodity ETFs 433

18.2.5 Currency ETFs 434

18.2.6 Inverse ETFs 435

18.2.7 Leverage ETFs 435

18.3 Credit Default Swap (CDS) 436

18.3.1 Example of Credit Default Swap 437

18.3.2 Valuation 437

18.3.3 Recovery Rate Estimates 439

18.3.4 Binary Credit Default Swaps 439

18.3.5 Basket Credit Default Swaps 439

18.4 Mortgage Backed Securities (MBS) 440

18.5 Collateralized Debt Obligation (CDO) 441

18.5.1 Collateralized Mortgage Obligations (CMO) 441

18.5.2 Collateralized Loan Obligations (CLO) 442

18.5.3 Collateralized Bond Obligations (CBO) 442

18.6 Problems 443

Bibliography 445

Index 459

Authors

Maria Cristina Mariani University of Texas at El Paso. Ionut Florescu Stevens Institute of Technology.