A workout in computational finance / Michael Aichinger, Andreas Binder.

Format:

Book

Author/Creator:

Aichinger, Michael, 1979-

Contributor:

Binder, Andreas, 1964-

Series:

Wiley finance series.

Wiley finance series

Language:

English

Subjects (All):

Finance--Mathematical models.

Finance.

Physical Description:

1 online resource (354 p.)

Edition:

1st edition

Place of Publication:

Hoboken, N.J. : John Wiley & Sons, Inc., 2013.

System Details:

text file

Summary:

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.

Contents:

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 ρ

14.1.2 Spearman's ρ

14.1.3 Kendall's ρ

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.

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

9781119973515

1119973511

9781119973485

1119973481

9781119973492

111997349X

OCLC:

842323326