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Quantitative Finance de Maria C. Mariani

Quantitative Finance

Autor Maria C. Mariani
en Limba Engleză Hardback – 22 noi 2019
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.
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Specificații

ISBN-13: 9781118629956
ISBN-10: 1118629957
Pagini: 492
Dimensiuni: 157 x 235 x 31 mm
Greutate: 0.86 kg
Editura: Wiley
Locul publicării:Hoboken, United States

Notă biografică

MARIA C. MARIANI, PHD, is Shigeko K. Chan Distinguished Professor and Chair in the Department of Mathematical Sciences at The University of Texas at El Paso. She currently focuses her research on mathematical finance, stochastic and non-linear differential equations, geophysics, and numerical methods. Dr. Mariani is co-organizer of the Conference on Modeling High-Frequency Data in Finance. IONUT FLORESCU, PHD, is Research Professor in Financial Engineering at Stevens Institute of Technology. He serves as Director of the Hanlon Laboratories as well as Director of the Financial Analytics program. His main research is in probability and stochastic processes and applications to domains such as finance, computer vision, robotics, earthquake studies, weather studies, and many more. Dr. Florescu is lead organizer of the Conference on Modeling High-Frequency Data in Finance.

Cuprins

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
1.3 Variation and Quadratic Variation of Stochastic Processes 11
1.4 Other More Specific Properties 13
1.5 Examples of Stochastic Processes 14
1.6 Borel-Cantelli Lemmas 19
1.7 Central Limit Theorem 20
1.8 Stochastic Differential Equation 20
1.9 Stochastic Integral 21
1.10 Maximization and Parameter Calibration of Stochastic Processes 22
1.11 Quadrature Methods 26
1.12 Problems 29
2 Basics of Finance 33
2.1 Introduction 33
2.2 Arbitrage 33
2.3 Options 35
2.4 Hedging 39
2.5 Modeling Return of Stocks 40
2.6 Continuous Time Model 41
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
3.5 Heston Model 60
3.6 The Cox-Ingersoll-Ross (CIR) Model 63
3.7 Stochastic ¿, ß, ¿ (SABR) Model 64
3.8 Methods for Finding Roots of Functions: Implied Volatility 65
3.9 Some Remarks of Implied Volatility (Put-Call Parity) 69
3.10 Hedging Using Volatility 70
3.11 Functional Approximation Methods 73
3.12 Problems 79
4 Solving Partial Differential Equations 83
4.1 Introduction 83
4.2 Useful Definitions and Types of PDEs 83
4.3 Functional Spaces Useful for PDEs 85
4.4 Separation of Variables 88
4.5 Moment-Generating Laplace Transform 91
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
5.3 Wavelets Theory 109
5.4 Examples of Discrete Wavelets Transforms (DWT) 112
5.5 Application of Wavelets Transform 116
5.6 Problems 118
6 Tree Methods 121
6.1 Introduction 121
6.2 Tree Methods: the Binomial Tree 122
6.3 Tree Methods for Dividend-Paying Assets 135
6.4 Pricing Path-Dependent Options: Barrier Options 139
6.5 Trinomial Tree Method and Other Considerations 140
6.6 Markov Process 143
6.7 Basic Elements of Operators and Semigroup Theory 146
6.8 General Diffusion Process 152
6.9 A General Diffusion Approximation Method 156
6.10 Particle Filter Construction 159
6.11 Quadrinomial Tree Approximation 163
6.12 Problems 173
7 Approximating PDEs 177
7.1 Introduction 177
7.2 The Explicit Finite Difference Method 179
7.3 The Implicit Finite Difference Method 180
7.4 The Crank-Nicolson Finite Difference Method 183
7.5 A Discussion About the Necessary Number of Nodes in the Schemes 184
7.6 Solution of a Tridiagonal System 186
7.7 Heston PDE 188
7.8 Methods for Free Boundary Problems 191
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
8.5 American Option Pricing with Monte Carlo Simulation 208
8.6 Nonstandard Monte Carlo Methods 216
8.7 Generating One-Dimensional Random Variables by Inverting the cdf 218
8.8 Generating One-Dimensional Normal Random Variables 220
8.9 Generating Random Variables: Rejection Sampling Method 224
8.10 Generating Random Variables: Importance Sampling 236
8.11 Problems 242
9 Stochastic Differential Equations 245
9.1 Introduction 245
9.2 The Construction of the Stochastic Integral 246
9.3 Properties of the Stochastic Integral 253
9.4 Itô Lemma 254
9.5 Stochastic Differential Equations (SDEs) 257
9.6 Examples of Stochastic Differential Equations 260
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
9.11 Generating Multivariate (Gaussian) Distributions with Prescribed Covariance Structure 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
10.4 Derivation of Formulae Used: Mean-Reverting Processes 296
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
11.6 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
12.4 Subordination of Lévy Processes 331
12.5 Rescaled Range Analysis (Hurst Analysis) and Detrended Fluctuation Analysis (DFA) 332
12.6 Problems 336
13 Generalized Lévy Processes, Long Range Correlations, and Memory Effects 337
13.1 Introduction 337
13.2 The Lévy Flight Models 339
13.3 Sum of Lévy Stochastic Variables with Different Parameters 347
13.4 Examples and Applications 352
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
14.4 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
15.6 Problems 400
16 Factor and Copulas Models 403
16.1 Introduction 403
16.2 Factor Models 403
16.3 Copula Models 409
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
17.5 Term Structure Consistent Models 422
17.6 Inverting the Yield Curve 426
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
18.3 Credit Default Swap (CDS) 436
18.4 Mortgage Backed Securities (MBS) 440
18.5 Collateralized Debt Obligation (CDO) 441
18.6 Problems 443
Bibliography 445
Index 459