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
 
2.6 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

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.