List of Algorithms
Acknowledgements
1.Introduction
1.1 About this book
1.2 Principles
1.3 on software
1.4 on approximations andaccuracy
1.5 Summary: the theme of the book
Part One Fundamentals
2. Numerical analysisin a nutshell
2.1 Computer arithmetic
Representation of real numbers
Machine precision
Example of limitations of floating point arithmetic
2.2 Measuringerrors
2.3 Approximating derivatives with finite differences
Approximating first-order derivatives
Approximating second-order derivatives
Partial derivatives
How to choose h
Truncation error for forward difference
2.4 Numerical instability and ill-conditioning
Example of a numerically unstable algorithm
Example of an ill-conditioned problem
2.5Condition number of a matrix
Comments and examples
2.6 A primer on algorithmic and computational complexity
2.6.1 Criteria for comparison
Order of complexity and classification
2.A Operation count for basiclinear algebra operations
3. Linear equations and Least Squares problems
Choice of method
3.1 Direct methods
3.1.1 Triangular systems
3.1.2 LU factorization
3.1.3 Cholesky factorization
3.1.4 QRdecomposition
3.1.5 Singular value decomposition
3.2 Iterative methods
3.2.1 Jacobi, Gauss-Seidel, and SOR
Successive overrelaxation
3.2.2 Convergence of niterative methods
3.2.3 General structure of algorithms for iterative methods
3.2.4 Block iterative methods
3.3 Sparse linear systems
3.3.1 Tridiagonal systems
3.3.2 Irregular sparse matrices
3.3.3 Structural properties of sparse matrices
3.4 The Least Squares problem
3.4.1 Method of normal equations
3.4.2 Least Squares via QR factorization
3.4.3 Least Squares via SVD decomposition
3.4.4 Final remarks
The backslash operator in Matlab
4. Finite difference methods
4.1 An example of a numerical solution
A first numerical approximation
A second numerical approximation
4.2 Classification of differential equations
4.3 The Black-Scholes equation
4.3.1 Explicit, implicit, and θ-methods
4.3.2 Initial and boundary conditions and definition of the
grid
4.3.3 Implementation of the θ-method with Matlab
4.3.4 Stability
4.3.5 Coordinate transformation of space variables
4.4 American options
4.A A note on Matlab''s function spdiags
5.Binomialtrees
5.1 Motivation
Matching moments
5.2 Growing the tree
5.2.1 Implementing a tree
5.2.2 Vectorization
5.2.3 Binomial expansion
5.3 Early exerase
5.4 Dividends
5.5 The Greeks
Greeks from the tree
Part Two Simulation
6. Generatmg random numbers
6.1 Monte Carlo methods and sampling
6.1.1 How it allbegan
6.1.2 Financialapplications
6.2 Uniform random number generators
6.2.1 Congruential generators
6.2.2 Mersenne Twister
6.3 Nonuniform distributions
6.3.1 The inversion method
6.3.2 Acceptance-rejection method
6.4 Specialized methods for selected distributions
6.4.1 Normal distribution
6.4.2 Higher order moments and the Cornish-Fisher expansion
6.4.3 Further distributions
6.5 Sampling from a discrete set
6.5.1 Discrete uniform selection
6.5.2 Roulette wheel selection
6.5.3 Random permutations and shuffling
6.6 Sampling errors-and how to reduce them
6.6.1 The basic problem
6.6.2 Quasi-Monte Carlo
6.6.3 Stratified sampling
6.6.4 Variance reduction
6.7Drawing from empirical distributions
6.7.1 Data randomization
6.7.2 Bootstrap
6.8 Controlled experiments and experimental design
6.8.1 Replicability and ceteris paribus analysis
6.8.2 Available random number generators in Matlab
6.8.3 Uniform random numbers from Matlab''s rand function
6.8.4 Gaussian random numbers from Matlab''s randn function
6.8.5 Remedies
7.Modelingdependenaes
7.1 Transformation methods
7.1.1 Linear correlation
7.1.2 Rank correlation
7.2 Markov chains
7.2.1 Concepts
7.2.2 The Metropolis algorithm
7.3 Copula models
7.3.1 Concepts
7.3.2 Simulation using copulas
8. A gentle introduction to financial simulation
8.1 Setting the stage
8.2 Single-period simulations
8.2.1 Terminal asset prices
8.2.2 l-over-N portfolios
8.2.3 European options
8.2.4 VaR of a covered put portfolio
8.3 Simple price processes
8.4 Processes with memoryin thelevels of returns
8.4.1 Efficient versus adaptive markets
8.4.2 Moving averages
8.4.3 Autoregressive models
8.4.4 Autoregressive moving average （ARMA） models
8.4.5 Simulating ARMA models
8.4.6 Models withlong-term memory
8.5 Time-varying volatility
8.5.1 Theconcepts
8.5.2 Autocorrelated time-varying volatility
8.5.3 Simulating GARCH processes
8.5.4 Selected further autoregressive volatility models
8.6 Adaptive expectations and patternsin price processes
8.6.1 Price-earningsmodels
8.6.2 Models with learning
8.7Historical simulation
8.7.1 Backtesting
8.7.2 Bootstrap
8.8 Agent-based models and complexity
9. Financial simulation at work: some case studies
9.1Constant proportion portfolio insurance （CPPI）
9.1.1 Basicconcepts
9.1.2 Bootstrap
9.2 VaR estimation with Extreme Value Theory
9.2.1 Basicconcepts
9.2.2 Scaling the data
9.2.3 Using Extreme Value Theory
9.3 Option pricing
9.3.1 Modeling prices
9.3.2 Pricingmodels
9.3.3 Greeks
9.3.4 Quasi-Monte Carlo
Part Three Optimization
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