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by Michael Aichinger, Andreas Binder
A Workout in Computational Finance
Cover Page
Title Page
Copyright
Dedication
Contents
Acknowledgements
About the Authors
1 Introduction and Reading Guide
2 Binomial Trees
2.1 Equities and Basic Options
2.2 The One Period Model
2.3 The Multiperiod Binomial Model
2.4 Black-Scholes and Trees
2.5 Strengths and Weaknesses of Binomial Trees
2.6 Conclusion
3 Finite Differences and the Black-Scholes PDE
3.1 A Continuous Time Model for Equity Prices
3.2 Black-Scholes Model: From the SDE to the PDE
3.3 Finite Differences
3.4 Time Discretization
3.5 Stability Considerations
3.6 Finite Differences and the Heat Equation
3.7 Appendix: Error Analysis
4 Mean Reversion and Trinomial Trees
4.1 Some Fixed Income Terms
4.2 Black76 for Caps and Swaptions
4.3 One-Factor Short Rate Models
4.4 The Hull-White Model in More Detail
4.5 Trinomial Trees
5 Upwinding Techniques for Short Rate Models
5.1 Derivation of a PDE for Short Rate Models
5.2 Upwind Schemes
5.3 A Puttable Fixed Rate Bond under the Hull-White One Factor Model
6 Boundary, Terminal and Interface Conditions and their Influence
6.1 Terminal Conditions for Equity Options
6.2 Terminal Conditions for Fixed Income Instruments
6.3 Callability and Bermudan Options
6.4 Dividends
6.5 Snowballs and TARNs
6.6 Boundary Conditions
7 Finite Element Methods
7.1 Introduction
7.2 Grid Generation
7.3 Elements
7.4 The Assembling Process
7.5 A Zero Coupon Bond Under the Two Factor Hull-White Model
7.6 Appendix: Higher Order Elements
8 Solving Systems of Linear Equations
8.1 Direct Methods
8.2 Iterative Solvers
9 Monte Carlo Simulation
9.1 The Principles of Monte Carlo Integration
9.2 Pricing Derivatives with Monte Carlo Methods
9.3 An Introduction to the Libor Market Model
9.4 Random Number Generation
10 Advanced Monte Carlo Techniques
10.1 Variance Reduction Techniques
10.2 Quasi Monte Carlo Method
10.3 Brownian Bridge Technique
11 Valuation of Financial Instruments with Embedded American/Bermudan Options within Monte Carlo Frameworks
11.1 Pricing American options using the Longstaff and Schwartz algorithm
11.2 A Modified Least Squares Monte Carlo Algorithm for Bermudan Callable Interest Rate Instruments
11.3 Examples
12 Characteristic Function Methods for Option Pricing
12.1 Equity Models
12.2 Fourier Techniques
13 Numerical Methods for the Solution of PIDEs
13.1 A PIDE for Jump Models
13.2 Numerical Solution of the PIDE
13.3 Appendix: Numerical Integration via Newton-Cotes Formulae
14 Copulas and the Pitfalls of Correlation
14.1 Correlation
14.2 Copulas
15 Parameter Calibration and Inverse Problems
15.1 Implied Black-Scholes Volatilities
15.2 Calibration Problems for Yield Curves
15.3 Reversion Speed and Volatility
15.4 Local Volatility
15.5 Identifying Parameters in Volatility Models
16 Optimization Techniques
16.1 Model Calibration and Optimization
16.2 Heuristically Inspired Algorithms
16.3 A Hybrid Algorithm for Heston Model Calibration
16.4 Portfolio Optimization
17 Risk Management
17.1 Value at Risk and Expected Shortfall
17.2 Principal Component Analysis
17.3 Extreme Value Theory
18 Quantitative Finance on Parallel Architectures
18.1 A Short Introduction to Parallel Computing
18.2 Different Levels of Parallelization
18.3 GPU Programming
18.4 Parallelization of Single Instrument Valuations using (Q)MC
18.5 Parallelization of Hybrid Calibration Algorithms
19 Building Large Software Systems for the Financial Industry
Bibliography
Index
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Copyright
A Workout in Computational Finance
Michael Aichinger
Andreas Binder
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