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A Workout in Computational Finance (with Website)

Aus der Reihe Wiley Finance Series

91,99 €

inkl. MwSt, Versandkostenfrei

Beschreibung

Details

Einband

Gebundene Ausgabe

Erscheinungsdatum

23.09.2013

Verlag

John Wiley & Sons Inc

Seitenzahl

336

Maße (L/B/H)

25,2/17,7/2,8 cm

Gewicht

745 g

Auflage

1. Auflage

Sprache

Englisch

ISBN

978-1-119-97191-7

Beschreibung

Details

Einband

Gebundene Ausgabe

Erscheinungsdatum

23.09.2013

Verlag

John Wiley & Sons Inc

Seitenzahl

336

Maße (L/B/H)

25,2/17,7/2,8 cm

Gewicht

745 g

Auflage

1. Auflage

Sprache

Englisch

ISBN

978-1-119-97191-7

Herstelleradresse

Libri GmbH
Europaallee 1
36244 Bad Hersfeld
DE

Email: gpsr@libri.de

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  • Produktbild: A Workout in Computational Finance
  • Produktbild: A Workout in Computational Finance

  • Acknowledgements xiii

    About the Authors xv

    1 Introduction and Reading Guide 1

    2 Binomial Trees 7

    2.1 Equities and Basic Options 7

    2.2 The One Period Model 8

    2.3 The Multiperiod Binomial Model 9

    2.4 Black-Scholes and Trees 10

    2.5 Strengths and Weaknesses of Binomial Trees 12

    2.5.1 Ease of Implementation 12

    2.5.2 Oscillations 12

    2.5.3 Non-recombining Trees 14

    2.5.4 Exotic Options and Trees 14

    2.5.5 Greeks and Binomial Trees 15

    2.5.6 Grid Adaptivity and Trees 15

    2.6 Conclusion 16

    3 Finite Differences and the Black-Scholes PDE 17

    3.1 A Continuous Time Model for Equity Prices 17

    3.2 Black-Scholes Model: From the SDE to the PDE 19

    3.3 Finite Differences 23

    3.4 Time Discretization 27

    3.5 Stability Considerations 30

    3.6 Finite Differences and the Heat Equation 30

    3.6.1 Numerical Results 34

    3.7 Appendix: Error Analysis 36

    4 Mean Reversion and Trinomial Trees 39

    4.1 Some Fixed Income Terms 39

    4.1.1 Interest Rates and Compounding 39

    4.1.2 Libor Rates and Vanilla Interest Rate Swaps 40

    4.2 Black76 for Caps and Swaptions 43

    4.3 One-Factor Short Rate Models 45

    4.3.1 Prominent Short Rate Models 45

    4.4 The Hull-White Model in More Detail 46

    4.5 Trinomial Trees 47

    5 Upwinding Techniques for Short Rate Models 55

    5.1 Derivation of a PDE for Short Rate Models 55

    5.2 Upwind Schemes 56

    5.2.1 Model Equation 57

    5.3 A Puttable Fixed Rate Bond under the Hull-White One Factor Model 63

    5.3.1 Bond Details 64

    5.3.2 Model Details 64

    5.3.3 Numerical Method 65

    5.3.4 An Algorithm in Pseudocode 68

    5.3.5 Results 69

    6 Boundary, Terminal and Interface Conditions and their Influence 71

    6.1 Terminal Conditions for Equity Options 71

    6.2 Terminal Conditions for Fixed Income Instruments 72

    6.3 Callability and Bermudan Options 74

    6.4 Dividends 74

    6.5 Snowballs and TARNs 75

    6.6 Boundary Conditions 77

    6.6.1 Double Barrier Options and Dirichlet Boundary Conditions 77

    6.6.2 Artificial Boundary Conditions and the Neumann Case 78

    7 Finite Element Methods 81

    7.1 Introduction 81

    7.1.1 Weighted Residual Methods 81

    7.1.2 Basic Steps 82

    7.2 Grid Generation 83

    7.3 Elements 85

    7.3.1 1D Elements 86

    7.3.2 2D Elements 88

    7.4 The Assembling Process 90

    7.4.1 Element Matrices 93

    7.4.2 Time Discretization 97

    7.4.3 Global Matrices 98

    7.4.4 Boundary Conditions 101

    7.4.5 Application of the Finite Element Method to Convection-Diffusion-Reaction Problems 103

    7.5 A Zero Coupon Bond Under the Two Factor Hull-White Model 105

    7.6 Appendix: Higher Order Elements 107

    7.6.1 3D Elements 109

    7.6.2 Local and Natural Coordinates 111

    8 Solving Systems of Linear Equations 117

    8.1 Direct Methods 118

    8.1.1 Gaussian Elimination 118

    8.1.2 Thomas Algorithm 119

    8.1.3 LU Decomposition 120

    8.1.4 Cholesky Decomposition 121

    8.2 Iterative Solvers 122

    8.2.1 Matrix Decomposition 123

    8.2.2 Krylov Methods 125

    8.2.3 Multigrid Solvers 126

    8.2.4 Preconditioning 129

    9 Monte Carlo Simulation 133

    9.1 The Principles of Monte Carlo Integration 133

    9.2 Pricing Derivatives with Monte Carlo Methods 134

    9.2.1 Discretizing the Stochastic Differential Equation 135

    9.2.2 Pricing Formalism 137

    9.2.3 Valuation of a Steepener under a Two Factor Hull-White Model 137

    9.3 An Introduction to the Libor Market Model 139

    9.4 Random Number Generation 146

    9.4.1 Properties of a Random Number Generator 147

    9.4.2 Uniform Variates 148

    9.4.3 Random Vectors 150

    9.4.4 Recent Developments in Random Number Generation 151

    9.4.5 Transforming Variables 152

    9.4.6 Random Number Generation for Commonly Used Distributions 155

    10 Advanced Monte Carlo Techniques 161

    10.1 Variance Reduction Techniques 161

    10.1.1 Antithetic Variates 161

    10.1.2 Control Variates 163

    10.1.3 Conditioning 166

    10.1.4 Additional Techniques for Variance Reduction 168

    10.2 Quasi Monte Carlo Method 169

    10.2.1 Low-Discrepancy Sequences 169

    10.2.2 Randomizing QMC 174

    10.3 Brownian Bridge Technique 175

    10.3.1 A Steepener under a Libor Market Model 177

    11 Valuation of Financial Instruments with Embedded American/Bermudan Options within Monte Carlo Frameworks 179

    11.1 Pricing American options using the Longstaff and Schwartz algorithm 179

    11.2 A Modified Least Squares Monte Carlo Algorithm for Bermudan Callable Interest Rate Instruments 181

    11.2.1 Algorithm: Extended LSMC Method for Bermudan Options 182

    11.2.2 Notes on Basis Functions and Regression 185

    11.3 Examples 186

    11.3.1 A Bermudan Callable Floater under Different Short-rate Models 186

    11.3.2 A Bermudan Callable Steepener Swap under a Two Factor Hull-White Model 188

    11.3.3 A Bermudan Callable Steepener Cross Currency Swap in a 3D IR/FX Model Framework 189

    12 Characteristic Function Methods for Option Pricing 193

    12.1 Equity Models 194

    12.1.1 Heston Model 196

    12.1.2 Jump Diffusion Models 198

    12.1.3 Infinite Activity Models 199

    12.1.4 Bates Model 200

    12.2 Fourier Techniques 201

    12.2.1 Fast Fourier Transform Methods 201

    12.2.2 Fourier-Cosine Expansion Methods 203

    13 Numerical Methods for the Solution of PIDEs 209

    13.1 A PIDE for Jump Models 209

    13.2 Numerical Solution of the PIDE 210

    13.2.1 Discretization of the Spatial Domain 211

    13.2.2 Discretization of the Time Domain 211

    13.2.3 A European Option under the Kou Jump Diffusion Model 212

    13.3 Appendix: Numerical Integration via Newton-Cotes Formulae 214

    14 Copulas and the Pitfalls of Correlation 217

    14.1 Correlation 218

    14.1.1 Pearson's ¿ 218

    14.1.2 Spearman's ¿ 218

    14.1.3 Kendall's ¿ 220

    14.1.4 Other Measures 221

    14.2 Copulas 221

    14.2.1 Basic Concepts 222

    14.2.2 Important Copula Functions 222

    14.2.3 Parameter estimation and sampling 229

    14.2.4 Default Probabilities for Credit Derivatives 234

    15 Parameter Calibration and Inverse Problems 239

    15.1 Implied Black-Scholes Volatilities 239

    15.2 Calibration Problems for Yield Curves 240

    15.3 Reversion Speed and Volatility 245

    15.4 Local Volatility 245

    15.4.1 Dupire's Inversion Formula 246

    15.4.2 Identifying Local Volatility 246

    15.4.3 Results 247

    15.5 Identifying Parameters in Volatility Models 248

    15.5.1 Model Calibration for the FTSE- 100 249

    16 Optimization Techniques 253

    16.1 Model Calibration and Optimization 255

    16.1.1 Gradient-Based Algorithms for Nonlinear Least Squares Problems 256

    16.2 Heuristically Inspired Algorithms 258

    16.2.1 Simulated Annealing 259

    16.2.2 Differential Evolution 260

    16.3 A Hybrid Algorithm for Heston Model Calibration 261

    16.4 Portfolio Optimization 265

    17 Risk Management 269

    17.1 Value at Risk and Expected Shortfall 269

    17.1.1 Parametric VaR 270

    17.1.2 Historical VaR 272

    17.1.3 Monte Carlo VaR 273

    17.1.4 Individual and Contribution VaR 274

    17.2 Principal Component Analysis 276

    17.2.1 Principal Component Analysis for Non-scalar Risk Factors 276

    17.2.2 Principal Components for Fast Valuation 277

    17.3 Extreme Value Theory 278

    18 Quantitative Finance on Parallel Architectures 285

    18.1 A Short Introduction to Parallel Computing 285

    18.2 Different Levels of Parallelization 288

    18.3 GPU Programming 288

    18.3.1 CUDA and OpenCL 289

    18.3.2 Memory 289

    18.4 Parallelization of Single Instrument Valuations using (Q)MC 290

    18.5 Parallelization of Hybrid Calibration Algorithms 291

    18.5.1 Implementation Details 292

    18.5.2 Results 295

    19 Building Large Software Systems for the Financial Industry 297

    Bibliography 301

    Index 307