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A Workout in Computational Finance

Book Description

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.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright
  4. Dedication
  5. Contents
  6. Acknowledgements
  7. About the Authors
  8. 1 Introduction and Reading Guide
  9. 2 Binomial Trees
    1. 2.1 Equities and Basic Options
    2. 2.2 The One Period Model
    3. 2.3 The Multiperiod Binomial Model
    4. 2.4 Black-Scholes and Trees
    5. 2.5 Strengths and Weaknesses of Binomial Trees
    6. 2.6 Conclusion
  10. 3 Finite Differences and the Black-Scholes PDE
    1. 3.1 A Continuous Time Model for Equity Prices
    2. 3.2 Black-Scholes Model: From the SDE to the PDE
    3. 3.3 Finite Differences
    4. 3.4 Time Discretization
    5. 3.5 Stability Considerations
    6. 3.6 Finite Differences and the Heat Equation
    7. 3.7 Appendix: Error Analysis
  11. 4 Mean Reversion and Trinomial Trees
    1. 4.1 Some Fixed Income Terms
    2. 4.2 Black76 for Caps and Swaptions
    3. 4.3 One-Factor Short Rate Models
    4. 4.4 The Hull-White Model in More Detail
    5. 4.5 Trinomial Trees
  12. 5 Upwinding Techniques for Short Rate Models
    1. 5.1 Derivation of a PDE for Short Rate Models
    2. 5.2 Upwind Schemes
    3. 5.3 A Puttable Fixed Rate Bond under the Hull-White One Factor Model
  13. 6 Boundary, Terminal and Interface Conditions and their Influence
    1. 6.1 Terminal Conditions for Equity Options
    2. 6.2 Terminal Conditions for Fixed Income Instruments
    3. 6.3 Callability and Bermudan Options
    4. 6.4 Dividends
    5. 6.5 Snowballs and TARNs
    6. 6.6 Boundary Conditions
  14. 7 Finite Element Methods
    1. 7.1 Introduction
    2. 7.2 Grid Generation
    3. 7.3 Elements
    4. 7.4 The Assembling Process
    5. 7.5 A Zero Coupon Bond Under the Two Factor Hull-White Model
    6. 7.6 Appendix: Higher Order Elements
  15. 8 Solving Systems of Linear Equations
    1. 8.1 Direct Methods
    2. 8.2 Iterative Solvers
  16. 9 Monte Carlo Simulation
    1. 9.1 The Principles of Monte Carlo Integration
    2. 9.2 Pricing Derivatives with Monte Carlo Methods
    3. 9.3 An Introduction to the Libor Market Model
    4. 9.4 Random Number Generation
  17. 10 Advanced Monte Carlo Techniques
    1. 10.1 Variance Reduction Techniques
    2. 10.2 Quasi Monte Carlo Method
    3. 10.3 Brownian Bridge Technique
  18. 11 Valuation of Financial Instruments with Embedded American/Bermudan Options within Monte Carlo Frameworks
    1. 11.1 Pricing American options using the Longstaff and Schwartz algorithm
    2. 11.2 A Modified Least Squares Monte Carlo Algorithm for Bermudan Callable Interest Rate Instruments
    3. 11.3 Examples
  19. 12 Characteristic Function Methods for Option Pricing
    1. 12.1 Equity Models
    2. 12.2 Fourier Techniques
  20. 13 Numerical Methods for the Solution of PIDEs
    1. 13.1 A PIDE for Jump Models
    2. 13.2 Numerical Solution of the PIDE
    3. 13.3 Appendix: Numerical Integration via Newton-Cotes Formulae
  21. 14 Copulas and the Pitfalls of Correlation
    1. 14.1 Correlation
    2. 14.2 Copulas
  22. 15 Parameter Calibration and Inverse Problems
    1. 15.1 Implied Black-Scholes Volatilities
    2. 15.2 Calibration Problems for Yield Curves
    3. 15.3 Reversion Speed and Volatility
    4. 15.4 Local Volatility
    5. 15.5 Identifying Parameters in Volatility Models
  23. 16 Optimization Techniques
    1. 16.1 Model Calibration and Optimization
    2. 16.2 Heuristically Inspired Algorithms
    3. 16.3 A Hybrid Algorithm for Heston Model Calibration
    4. 16.4 Portfolio Optimization
  24. 17 Risk Management
    1. 17.1 Value at Risk and Expected Shortfall
    2. 17.2 Principal Component Analysis
    3. 17.3 Extreme Value Theory
  25. 18 Quantitative Finance on Parallel Architectures
    1. 18.1 A Short Introduction to Parallel Computing
    2. 18.2 Different Levels of Parallelization
    3. 18.3 GPU Programming
    4. 18.4 Parallelization of Single Instrument Valuations using (Q)MC
    5. 18.5 Parallelization of Hybrid Calibration Algorithms
  26. 19 Building Large Software Systems for the Financial Industry
  27. Bibliography
  28. Index