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A Workout in Computational Finance

ISBN: 978-1-119-97349-2
336 pages
August 2013
A Workout in Computational Finance (111997349X) cover image


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.

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Table of Contents

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

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Author Information

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.

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“Mathematical Finance needs both: a well-founded theory based on stochastic calculus as well as numerical valuation schemes that work. In A Workout in Computational Finance the authors put emphasis on the numerical aspects and present an impressive range of numerical methods. All these techniques have been implemented by their group and can be used as a starting point for building a professional software system.”
Walter Schachermayer, Full Professor for Mathematical Finance, University of Vienna

“With their strong background in numerical simulation of industrial problems, the authors succeed to develop the concepts of different numerical schemes which are useful for computational finance and essential for valuation, risk analysis and the risk management of financial instruments. Especially in times of difficult market environments, the mathematical and algorithmic foundation of software used in banking must be a solid one which avoids additional traps of poor implementation. A Workout in Computational Finance gives clear recommendations for the preferred numerical methods for various models and instruments. The book will be utmost useful for practitioners but it also will be of great interest for researchers in the field.”
Gerhard Larcher, Institute of Mathematical Finance, Kepler Universität

“The authors cover a broad range of numerical techniques for differential equations in computational finance, such as finite elements, trees, Monte Carlo, Fourier techniques and parameter calibration. Using sound, yet compact mathematical reasoning, they capture the substance of models for interest rate and equity derivatives, and provide hands-on guidance to numerics, covering all sorts of practical challenges. A vast number of numerical results illustrate potential implementation pitfalls and the mitigation techniques presented. With its strong focus on tangible usability this book is a highly valuable manual for students as well as professionals.”
Robert Maringer, Head Valuation Control Switzerland, Credit Suisse

“For shaping your body you should go to a gym, while for building up your numerical toolkit you need a workout in computational finance. This modern treatment of numerical methods in quantitative finance addresses problems that professionals working in the field face on a daily basis. The very clear presentation of the material also makes it a perfect fit for students having a background in the theory of mathematical finance who want to gain insight on how practical problems are tackled in the industry.”
Philipp Mayer, Financial Modeling, ING Financial Markets, Brussels

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