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A comprehensive introduction to various numerical methods used in computational finance today
Quantitative skills are a prerequisite for anyone working in finance or beginning a career in the field, as well as risk managers. A thorough grounding in numerical methods is necessary, as is the ability to assess their quality, advantages, and limitations. This book offers a thorough introduction to each method, revealing the numerical traps that practitioners frequently fall into. Each method is referenced with practical, real-world examples in the areas of valuation, risk analysis, and…mehr
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A comprehensive introduction to various numerical methods used in computational finance today
Quantitative skills are a prerequisite for anyone working in finance or beginning a career in the field, as well as risk managers. A thorough grounding in numerical methods is necessary, as is the ability to assess their quality, advantages, and limitations. This book offers a thorough introduction to each method, revealing the numerical traps that practitioners frequently fall into. Each method is referenced with practical, real-world examples in the areas of valuation, risk analysis, and calibration of specific financial instruments and models. It features a strong emphasis on robust schemes for the numerical treatment of problems within computational finance. Methods covered include PDE/PIDE using finite differences or finite elements, fast and stable solvers for sparse grid systems, stabilization and regularization techniques for inverse problems resulting from the calibration of financial models to market data, Monte Carlo and Quasi Monte Carlo techniques for simulating high dimensional systems, and local and global optimization tools to solve the minimization problem.
Quantitative skills are a prerequisite for anyone working in finance or beginning a career in the field, as well as risk managers. A thorough grounding in numerical methods is necessary, as is the ability to assess their quality, advantages, and limitations. This book offers a thorough introduction to each method, revealing the numerical traps that practitioners frequently fall into. Each method is referenced with practical, real-world examples in the areas of valuation, risk analysis, and calibration of specific financial instruments and models. It features a strong emphasis on robust schemes for the numerical treatment of problems within computational finance. Methods covered include PDE/PIDE using finite differences or finite elements, fast and stable solvers for sparse grid systems, stabilization and regularization techniques for inverse problems resulting from the calibration of financial models to market data, Monte Carlo and Quasi Monte Carlo techniques for simulating high dimensional systems, and local and global optimization tools to solve the minimization problem.
Produktdetails
- Produktdetails
- Wiley Finance Series
- Verlag: Wiley & Sons
- Artikelnr. des Verlages: 1W119971910
- 1. Auflage
- Seitenzahl: 336
- Erscheinungstermin: 23. September 2013
- Englisch
- Abmessung: 252mm x 177mm x 28mm
- Gewicht: 746g
- ISBN-13: 9781119971917
- ISBN-10: 1119971918
- Artikelnr.: 35451080
- Wiley Finance Series
- Verlag: Wiley & Sons
- Artikelnr. des Verlages: 1W119971910
- 1. Auflage
- Seitenzahl: 336
- Erscheinungstermin: 23. September 2013
- Englisch
- Abmessung: 252mm x 177mm x 28mm
- Gewicht: 746g
- ISBN-13: 9781119971917
- ISBN-10: 1119971918
- Artikelnr.: 35451080
MICHAEL AICHINGER obtained his Ph.D. in Theoretical Physics from the Johannes Kepler University Linz with a thesis on numerical methods in density functional theory and their application to 2D finite electron systems. A mobility grant led him to the Texas A&M University (2003) and to the Helsinki University of Technology (2004). In 2007 Michael Aichinger joined the Industrial Mathematics Competence Center where he has been working as a senior researcher and consultant in the field of quantitative finance for the last five years. He also works for the Austrian Academy of Sciences at the Radon Institute for Computational and Applied Mathematics where he is involved in several industrial mathematics and computational physics projects. Michael has (co-) authored around 20 journal articles in the fields of computational physics and quantitative finance. ANDREAS BINDER obtained his Ph.D. in Industrial Mathematics from the Johannes Kepler University Linz with a thesis on continuous casting of steel. A research grant led him to the Oxford Center for Industrial and Applied Mathematics, UK, in 1991, where he got in touch with mathematical finance for the first time. After some years being an assistant professor at the Industrial Mathematics Institute, in 1996, he left university and became managing director of MathConsult GmbH, where he heads also the Computational Finance Group. Andreas has authored two introductory books on mathematical finance and 25 journal articles in the fields of industrial mathematics and of mathematical 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 2.6 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 3.7
Appendix: Error Analysis 36 4 Mean Reversion and Trinomial Trees 39 4.1
Some Fixed Income Terms 39 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 5.3 A Puttable Fixed Rate Bond under the
Hull-White One Factor Model 63 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 7 Finite Element Methods 81 7.1
Introduction 81 7.2 Grid Generation 83 7.3 Elements 85 7.4 The Assembling
Process 90 7.5 A Zero Coupon Bond Under the Two Factor Hull-White Model 105
7.6 Appendix: Higher Order Elements 107 8 Solving Systems of Linear
Equations 117 8.1 Direct Methods 118 8.2 Iterative Solvers 122 9 Monte
Carlo Simulation 133 9.1 The Principles of Monte Carlo Integration 133 9.2
Pricing Derivatives with Monte Carlo Methods 134 9.3 An Introduction to the
Libor Market Model 139 9.4 Random Number Generation 146 10 Advanced Monte
Carlo Techniques 161 10.1 Variance Reduction Techniques 161 10.2 Quasi
Monte Carlo Method 169 10.3 Brownian Bridge Technique 175 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 11.3 Examples 186 12
Characteristic Function Methods for Option Pricing 193 12.1 Equity Models
194 12.2 Fourier Techniques 201 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 13.3 Appendix: Numerical Integration via Newton-Cotes Formulae 214
14 Copulas and the Pitfalls of Correlation 217 14.1 Correlation 218 14.2
Copulas 221 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 15.5
Identifying Parameters in Volatility Models 248 16 Optimization Techniques
253 16.1 Model Calibration and Optimization 255 16.2 Heuristically Inspired
Algorithms 258 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 17.2 Principal Component Analysis 276 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 18.4 Parallelization
of Single Instrument Valuations using (Q)MC 290 18.5 Parallelization of
Hybrid Calibration Algorithms 291 19 Building Large Software Systems for
the Financial Industry 297 Bibliography 301 Index 307
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 2.6 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 3.7
Appendix: Error Analysis 36 4 Mean Reversion and Trinomial Trees 39 4.1
Some Fixed Income Terms 39 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 5.3 A Puttable Fixed Rate Bond under the
Hull-White One Factor Model 63 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 7 Finite Element Methods 81 7.1
Introduction 81 7.2 Grid Generation 83 7.3 Elements 85 7.4 The Assembling
Process 90 7.5 A Zero Coupon Bond Under the Two Factor Hull-White Model 105
7.6 Appendix: Higher Order Elements 107 8 Solving Systems of Linear
Equations 117 8.1 Direct Methods 118 8.2 Iterative Solvers 122 9 Monte
Carlo Simulation 133 9.1 The Principles of Monte Carlo Integration 133 9.2
Pricing Derivatives with Monte Carlo Methods 134 9.3 An Introduction to the
Libor Market Model 139 9.4 Random Number Generation 146 10 Advanced Monte
Carlo Techniques 161 10.1 Variance Reduction Techniques 161 10.2 Quasi
Monte Carlo Method 169 10.3 Brownian Bridge Technique 175 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 11.3 Examples 186 12
Characteristic Function Methods for Option Pricing 193 12.1 Equity Models
194 12.2 Fourier Techniques 201 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 13.3 Appendix: Numerical Integration via Newton-Cotes Formulae 214
14 Copulas and the Pitfalls of Correlation 217 14.1 Correlation 218 14.2
Copulas 221 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 15.5
Identifying Parameters in Volatility Models 248 16 Optimization Techniques
253 16.1 Model Calibration and Optimization 255 16.2 Heuristically Inspired
Algorithms 258 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 17.2 Principal Component Analysis 276 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 18.4 Parallelization
of Single Instrument Valuations using (Q)MC 290 18.5 Parallelization of
Hybrid Calibration Algorithms 291 19 Building Large Software Systems for
the Financial Industry 297 Bibliography 301 Index 307
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 2.6 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 3.7
Appendix: Error Analysis 36 4 Mean Reversion and Trinomial Trees 39 4.1
Some Fixed Income Terms 39 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 5.3 A Puttable Fixed Rate Bond under the
Hull-White One Factor Model 63 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 7 Finite Element Methods 81 7.1
Introduction 81 7.2 Grid Generation 83 7.3 Elements 85 7.4 The Assembling
Process 90 7.5 A Zero Coupon Bond Under the Two Factor Hull-White Model 105
7.6 Appendix: Higher Order Elements 107 8 Solving Systems of Linear
Equations 117 8.1 Direct Methods 118 8.2 Iterative Solvers 122 9 Monte
Carlo Simulation 133 9.1 The Principles of Monte Carlo Integration 133 9.2
Pricing Derivatives with Monte Carlo Methods 134 9.3 An Introduction to the
Libor Market Model 139 9.4 Random Number Generation 146 10 Advanced Monte
Carlo Techniques 161 10.1 Variance Reduction Techniques 161 10.2 Quasi
Monte Carlo Method 169 10.3 Brownian Bridge Technique 175 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 11.3 Examples 186 12
Characteristic Function Methods for Option Pricing 193 12.1 Equity Models
194 12.2 Fourier Techniques 201 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 13.3 Appendix: Numerical Integration via Newton-Cotes Formulae 214
14 Copulas and the Pitfalls of Correlation 217 14.1 Correlation 218 14.2
Copulas 221 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 15.5
Identifying Parameters in Volatility Models 248 16 Optimization Techniques
253 16.1 Model Calibration and Optimization 255 16.2 Heuristically Inspired
Algorithms 258 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 17.2 Principal Component Analysis 276 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 18.4 Parallelization
of Single Instrument Valuations using (Q)MC 290 18.5 Parallelization of
Hybrid Calibration Algorithms 291 19 Building Large Software Systems for
the Financial Industry 297 Bibliography 301 Index 307
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 2.6 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 3.7
Appendix: Error Analysis 36 4 Mean Reversion and Trinomial Trees 39 4.1
Some Fixed Income Terms 39 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 5.3 A Puttable Fixed Rate Bond under the
Hull-White One Factor Model 63 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 7 Finite Element Methods 81 7.1
Introduction 81 7.2 Grid Generation 83 7.3 Elements 85 7.4 The Assembling
Process 90 7.5 A Zero Coupon Bond Under the Two Factor Hull-White Model 105
7.6 Appendix: Higher Order Elements 107 8 Solving Systems of Linear
Equations 117 8.1 Direct Methods 118 8.2 Iterative Solvers 122 9 Monte
Carlo Simulation 133 9.1 The Principles of Monte Carlo Integration 133 9.2
Pricing Derivatives with Monte Carlo Methods 134 9.3 An Introduction to the
Libor Market Model 139 9.4 Random Number Generation 146 10 Advanced Monte
Carlo Techniques 161 10.1 Variance Reduction Techniques 161 10.2 Quasi
Monte Carlo Method 169 10.3 Brownian Bridge Technique 175 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 11.3 Examples 186 12
Characteristic Function Methods for Option Pricing 193 12.1 Equity Models
194 12.2 Fourier Techniques 201 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 13.3 Appendix: Numerical Integration via Newton-Cotes Formulae 214
14 Copulas and the Pitfalls of Correlation 217 14.1 Correlation 218 14.2
Copulas 221 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 15.5
Identifying Parameters in Volatility Models 248 16 Optimization Techniques
253 16.1 Model Calibration and Optimization 255 16.2 Heuristically Inspired
Algorithms 258 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 17.2 Principal Component Analysis 276 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 18.4 Parallelization
of Single Instrument Valuations using (Q)MC 290 18.5 Parallelization of
Hybrid Calibration Algorithms 291 19 Building Large Software Systems for
the Financial Industry 297 Bibliography 301 Index 307