A Workout in Computational Finance, with

Quantitative skills are a prerequisite for anyone looking to work in the finance industry today, and within the industry, any risk professional who wants to collaborate with, or work in most front office departments need a thorough grounding in numerical methods, and the ability to assess their quality, their advantages and their limitations. A Workout in Computational Finance provides a comprehensive introduction to the different numerical methods used in computational finance today. As well as giving a thorough grounding to each method, the book will reveals the numerical 'traps', practitioners can fall into using each method, revealing their strengths and limitations. Each method will be referenced with practical, real-world examples in the areas of valuation, risk analysis and calibration of specific financial instruments and models with a strong emphasis on robust schemes for the numerical treatment of problems within computational finance. The numerical methods treated in this book cover: - solution of PDE/PIDE using finite differences or finite elements including stabilization techniques for convection dominated equations - fast and stable solvers for the solution of the resulting sparse grid systems - stabilization and regularization techniques for the inverse problems resulting from the calibration of financial models to market data - Monte Carlo and Quasi Monte Carlo techniques for the simulation of high dimensional systems - local and global optimization tools to solve the minimization problem The book will be accompanied by a website, featuring a wide range of interactive examples available for download in Mathematica notebooks, in which the distinguishing features of the various methods are demonstrated. These examples, based on the authors' experience in numerical software development, will enable readers to engage one on one with the numerical methods in question, and understand and manipulate the impact of varying parameter settings, gaining valuable insight on how different results can be accomplished through different numerical and computational methods. A Workout in Computational Finance will guide risk and quantitative finance practitioners through the landscape of different numerical methods, and issue the danger signals around the traps and problems occurring if he or she does not pay enough attention to the numerical schemes involved in calibration and pricing processes.

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.

1 Introduction and Reading Guide 9

2 Binomial Trees 19

2.1 Equities and Some Basic Options 19

2.2 The One Period Model 20

2.3 The Multiperiod Binomial Model 22

2.4 Black-Scholes and Trees 23

2.5 Strengths and Weaknesses of Binomial Trees 24

2.6 Conclusion 28

3 Finite Di_erences and the Black Scholes PDE 31

3.1 A Continuous Time Model for Equity Prices 31

3.2 Black Scholes Model: From the SDE to the PDE 34

3.3 Finite Di_erences 38

3.4 Time Discretisation 43

3.5 Stability Considerations 45

3.6 Finite Di_erences and the Heat Equation 46

3.7 Appendix: Error Analysis 50

4 Mean Reversion and Trinomial Trees 57

4.1 Some Fixed Income Terms 57

4.2 Black76 for Caps and Swaptions 62

4.3 One Factor Short Rate Models 64

4.4 The Hull-White Model in More Detail 66

4.5 Trinomial Trees 67

5 Upwinding Techniques 75

5.1 Derivation of a PDE for Short Rate Models 75

5.2 Upwind Schemes 77

5.3 Hull-White and Upwinding 84

6 Boundary, Terminal, Interface Conditions 93

6.1 Terminal Conditions for Equity Options 93

6.2 Terminal Conditions for Fixed Income Instruments 95

6.3 Callability and Bermudan Options 96

6.4 Dividends 97

6.5 Snowballs and TARNs 98

6.6 Boundary Conditions 100

7 Finite Element Methods 105

7.1 Introduction 105

7.2 Grid Generation 107

7.3 Elements 110

7.5 2D Hull-White and Streamline Di_usion 132

7.6 Appendix: Higher Order Elements 136

8 Solving Systems of Linear Equations 145

8.1 Direct Methods 146

8.2 Iterative Solvers 152

9 Monte Carlo Simulation 163

9.1 The Principles of Monte Carlo Integration 163

9.2 Pricing Derivatives with Monte Carlo Methods 165

9.3 An Introduction to the Libor Market Model 170

9.4 Random Number Generation 178

10 Advanced Monte Carlo Techniques 199

10.1 Variance Reduction Techniques 199

10.2 Quasi Monte Carlo Method 210

10.3 Brownian Bridge Technique 219

11 Least Squares Monte Carlo 223

11.1 American Options 224

11.2 Least Squares Monte Carlo 226

11.3 Examples 233

12 Characteristic Function Methods 241

12.1 Equity Models 242

13 PIDE 259

13.1 A PIDE for Jump Models 259

13.2 Numerical Solution of the PIDE 260

13.3 Appendix: Numerical Integration via Newton-Cotes Formulae 265

14 Correlation and Copulas 269

14.1 Correlation 270

14.2 Copulas 274

15 Parameter Calibration 293

15.1 Implied Black Scholes Volatilities 294

15.2 Calibration Problems for Yield Curves 294

15.3 Reversion Speed and Volatility 299

15.4 Local Volatility 300

15.5 Identifying Parameters in Volatility Models 303

16 Optimisation Techniques 309

16.1 Model Calibration and Optimisation 312

16.2 Heuristically Inspired Algorithms 316

16.3 Heston Model Calibration 318

16.4 Portfolio Optimisation 323

17 Risk Management 329

17.1 Value at Risk and Expected Shortfall 329

17.2 Principal component analysis 337

17.3 Extreme Value Theory 339

18 Parallel Computing in Finance 347

18.1 A Short Introduction to Parallel Computing 347

18.2 Di_erent Levels of Parallelisation 351

18.3 GPU Programming 351

18.4 Parallel (Q)MC 353

18.5 Parallel Calibration Algorithms 354

19 Large Software Systems 363

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A Workout in Computational Finance, with Website

Summary

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