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.
Autorentext
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.
Klappentext
Quantitative skills are a prerequisite for anyone looking to work in the finance industry today. Within the industry, any risk professional who wants to collaborate with, or work in most front office departments needs a thorough grounding in numerical methods, and the ability to assess their quality, their advantages and their limitations.
A Workout in Computational Finance delivers a profound and hands-on account of numerical methods used in modern quantitative finance, covering valuation and risk analysis of financial instruments from vanilla bonds to complex structures. The presented algorithms include, amongst others, tree methods, finite differences and finite elements, efficient Monte Carlo methods and Fourier techniques. Local and global optimisation techniques as well as stabilising regularisation methods for model calibration are thoroughly analysed.
The authors originate from the fields of theoretical physics and industrial mathematics, respectively, and have spent their professional careers creating efficient software solutions for producing industries and for financial industries. This book develops algorithms from the ground up, thus giving the reader a sound overview of their relative strengths and weaknesses. It is aimed at practitioners in the financial industry, for whom this is key knowledge in order to achieve optimal results with available data. It also enables junior quants with an IT background to implement numerical algorithms that work right away.
A Workout in Computational Finance is accompanied by a range of worked-out examples available from www.unrisk.com/Workout.
Zusammenfassung
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.
Inhalt
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 7…