A Workout in Computational Finance

Name: A Workout in Computational Finance
Brand: Wiley
Price: 50.99 EUR
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Andreas Binder Michael Aichinger(Author)

Wiley (Publisher)

Published on 6. August 2013

336 pages

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978-1-119-97348-5 (ISBN)

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A comprehensive introduction to various numerical methods usedin computational finance today Quantitative skills are a prerequisite for anyone working infinance or beginning a career in the field, as well as riskmanagers. A thorough grounding in numerical methods is necessary,as is the ability to assess their quality, advantages, andlimitations. This book offers a thorough introduction to eachmethod, revealing the numerical traps that practitioners frequentlyfall into. Each method is referenced with practical, real-worldexamples in the areas of valuation, risk analysis, and calibrationof specific financial instruments and models. It features a strongemphasis on robust schemes for the numerical treatment of problemswithin computational finance. Methods covered include PDE/PIDEusing finite differences or finite elements, fast and stablesolvers for sparse grid systems, stabilization and regularizationtechniques for inverse problems resulting from the calibration offinancial models to market data, Monte Carlo and Quasi Monte Carlotechniques for simulating high dimensional systems, and local andglobal optimization tools to solve the minimization problem.

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"Mathematical Finance needs both: a well-founded theorybased on stochastic calculus as well as numerical valuation schemesthat work. In A Workout in Computational Finance the authorsput emphasis on the numerical aspects and present an impressiverange of numerical methods. All these techniques have beenimplemented by their group and can be used as a starting point forbuilding a professional software system." --Walter Schachermayer, Full Professor forMathematical Finance, University of Vienna "With their strong background in numerical simulation ofindustrial problems, the authors succeed to develop the concepts ofdifferent numerical schemes which are useful for computationalfinance and essential for valuation, risk analysis and the riskmanagement of financial instruments. Especially in times ofdifficult market environments, the mathematical and algorithmicfoundation of software used in banking must be a solid one whichavoids additional traps of poor implementation. A Workout inComputational Finance gives clear recommendations for thepreferred numerical methods for various models and instruments. Thebook will be utmost useful for practitioners but it also will be ofgreat interest for researchers in the field." --Gerhard Larcher, Institute of MathematicalFinance, Kepler Universität "The authors cover a broad range of numerical techniquesfor differential equations in computational finance, such as finiteelements, trees, Monte Carlo, Fourier techniques and parametercalibration. Using sound, yet compact mathematical reasoning, theycapture the substance of models for interest rate and equityderivatives, and provide hands-on guidance to numerics, coveringall sorts of practical challenges. A vast number of numericalresults illustrate potential implementation pitfalls and themitigation techniques presented. With its strong focus on tangibleusability this book is a highly valuable manual for students aswell as professionals." --Robert Maringer, Head Valuation ControlSwitzerland, Credit Suisse "For shaping your body you should go to a gym, while forbuilding up your numerical toolkit you need a workout incomputational finance. This modern treatment of numerical methodsin quantitative finance addresses problems that professionalsworking in the field face on a daily basis. The very clearpresentation of the material also makes it a perfect fit forstudents having a background in the theory of mathematical financewho want to gain insight on how practical problems are tackled inthe industry." --Philipp Mayer, Financial Modeling, ING FinancialMarkets, Brussels

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Content

Intro
A Workout in Computational Finance
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
2.5.1 Ease of Implementation
2.5.2 Oscillations
2.5.3 Non-recombining Trees
2.5.4 Exotic Options and Trees
2.5.5 Greeks and Binomial Trees
2.5.6 Grid Adaptivity and 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
3.6.1 Numerical Results
3.7 Appendix: Error Analysis
4 Mean Reversion and Trinomial Trees
4.1 Some Fixed Income Terms
4.1.1 Interest Rates and Compounding
4.1.2 Libor Rates and Vanilla Interest Rate Swaps
4.2 Black76 for Caps and Swaptions
4.3 One-Factor Short Rate Models
4.3.1 Prominent 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
5.2.1 Model Equation
5.3 A Puttable Fixed Rate Bond under the Hull-White One Factor Model
5.3.1 Bond Details
5.3.2 Model Details
5.3.3 Numerical Method
5.3.4 An Algorithm in Pseudocode
5.3.5 Results
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
6.6.1 Double Barrier Options and Dirichlet Boundary Conditions
6.6.2 Artificial Boundary Conditions and the Neumann Case
7 Finite Element Methods
7.1 Introduction
7.1.1 Weighted Residual Methods
7.1.2 Basic Steps
7.2 Grid Generation
7.3 Elements
7.3.1 1D Elements
7.3.2 2D Elements
7.4 The Assembling Process
7.4.1 Element Matrices
7.4.2 Time Discretization
7.4.3 Global Matrices
7.4.4 Boundary Conditions
7.4.5 Application of the Finite Element Method to Convection-Diffusion-Reaction Problems
7.5 A Zero Coupon Bond Under the Two Factor Hull-White Model
7.6 Appendix: Higher Order Elements
7.6.1 3D Elements
7.6.2 Local and Natural Coordinates
8 Solving Systems of Linear Equations
8.1 Direct Methods
8.1.1 Gaussian Elimination
8.1.2 Thomas Algorithm
8.1.3 LU Decomposition
8.1.4 Cholesky Decomposition
8.2 Iterative Solvers
8.2.1 Matrix Decomposition
8.2.2 Krylov Methods
8.2.3 Multigrid Solvers
8.2.4 Preconditioning
9 Monte Carlo Simulation
9.1 The Principles of Monte Carlo Integration
9.2 Pricing Derivatives with Monte Carlo Methods
9.2.1 Discretizing the Stochastic Differential Equation
9.2.2 Pricing Formalism
9.2.3 Valuation of a Steepener under a Two Factor Hull-White Model
9.3 An Introduction to the Libor Market Model
9.4 Random Number Generation
9.4.1 Properties of a Random Number Generator
9.4.2 Uniform Variates
9.4.3 Random Vectors
9.4.4 Recent Developments in Random Number Generation
9.4.5 Transforming Variables
9.4.6 Random Number Generation for Commonly Used Distributions
10 Advanced Monte Carlo Techniques
10.1 Variance Reduction Techniques
10.1.1 Antithetic Variates
10.1.2 Control Variates
10.1.3 Conditioning
10.1.4 Additional Techniques for Variance Reduction
10.2 Quasi Monte Carlo Method
10.2.1 Low-Discrepancy Sequences
10.2.2 Randomizing QMC
10.3 Brownian Bridge Technique
10.3.1 A Steepener under a Libor Market Model
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
11.2.1 Algorithm: Extended LSMC Method for Bermudan Options
11.2.2 Notes on Basis Functions and Regression
11.3 Examples
11.3.1 A Bermudan Callable Floater under Different Short-rate Models
11.3.2 A Bermudan Callable Steepener Swap under a Two Factor Hull-White Model
11.3.3 A Bermudan Callable Steepener Cross Currency Swap in a 3D IR/FX Model Framework
12 Characteristic Function Methods for Option Pricing
12.1 Equity Models
12.1.1 Heston Model
12.1.2 Jump Diffusion Models
12.1.3 Infinite Activity Models
12.1.4 Bates Model
12.2 Fourier Techniques
12.2.1 Fast Fourier Transform Methods
12.2.2 Fourier-Cosine Expansion Methods
13 Numerical Methods for the Solution of PIDEs
13.1 A PIDE for Jump Models
13.2 Numerical Solution of the PIDE
13.2.1 Discretization of the Spatial Domain
13.2.2 Discretization of the Time Domain
13.2.3 A European Option under the Kou Jump Diffusion Model
13.3 Appendix: Numerical Integration via Newton-Cotes Formulae
14 Copulas and the Pitfalls of Correlation
14.1 Correlation
14.1.1 Pearson's ?
14.1.2 Spearman's ?
14.1.3 Kendall's ?
14.1.4 Other Measures
14.2 Copulas
14.2.1 Basic Concepts
14.2.2 Important Copula Functions
14.2.3 Parameter estimation and sampling
14.2.4 Default Probabilities for Credit Derivatives
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
15.4.1 Dupire's Inversion Formula
15.4.2 Identifying Local Volatility
15.4.3 Results
15.5 Identifying Parameters in Volatility Models
15.5.1 Model Calibration for the FTSE-100
16 Optimization Techniques
16.1 Model Calibration and Optimization
16.1.1 Gradient-Based Algorithms for Nonlinear Least Squares Problems
16.2 Heuristically Inspired Algorithms
16.2.1 Simulated Annealing
16.2.2 Differential Evolution
16.3 A Hybrid Algorithm for Heston Model Calibration
16.4 Portfolio Optimization
17 Risk Management
17.1 Value at Risk and Expected Shortfall
17.1.1 Parametric VaR
17.1.2 Historical VaR
17.1.3 Monte Carlo VaR
17.1.4 Individual and Contribution VaR
17.2 Principal Component Analysis
17.2.1 Principal Component Analysis for Non-scalar Risk Factors
17.2.2 Principal Components for Fast Valuation
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
18.3.1 CUDA and OpenCL
18.3.2 Memory
18.4 Parallelization of Single Instrument Valuations using (Q)MC
18.5 Parallelization of Hybrid Calibration Algorithms
18.5.1 Implementation Details
18.5.2 Results
19 Building Large Software Systems for the Financial Industry
Bibliography
Index

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A Workout in Computational Finance

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