金融工程中的蒙特卡罗方法:[英文本].影印版

1 Foundations .
1.1 Principles of Monte Carlo
1.1.1 Introduction
1.1.2 First Examples
1.1.3 Efficiency of Simulation Estimators
1.2 Principles of Derivatives Pricing
1.2.1 Pricing and Replication
1.2.2 Arbitrage and Risk-Neutral Pricing
1.2.3 Change of Numeraire
1.2.4 The Market Price of Risk

2 Generating Random Numbers and Random Variables
2.1 Random Number Generation
2.1.1 General Considerations
2.1.2 Linear Congruential Generators
2.1.3 Implementation of Linear Congruential Generators
2.1.4 Lattice Structure
2.1.5 Combined Generators and Other Methods
2.2 General Sampling Methods
2.2.1 Inverse Transform Method
2.2.2 Acceptance-Rejection Method
2.3 Normal Random Variables and Vectors
2.3.1 Basic Properties
2.3.2 Generating Univariate Normals
2.3.3 Generating Multivariate Normals

3 Generating Sample Paths
3.1 Brownian Motion
3.1.1 One Dimension
3.1.2 Multiple Dimensions
3.2 Geometric Brownian Motion
3.2.1 Basic Properties
3.2.2 Path-Dependent Options
3.2.3 Multiple Dimensions
3.3 Gaussian Short Rate Models
3.3.1 Basic Models and Simulation
3.3.2 Bond Prices
.3.3 Multifactor Models
3.4 Square-Root Diffusions
3.4.1 Transition Density
3.4.2 Sampling Gamma and Poisson
3.4.3 Bond Prices
3.4.4 Extensions
3.5 Processes with Jumps
3.5.1 A Jump-Diffusion Model
3.5.2 Pure-Jump Processes
3.6 Forward Rate Models: Continuous Rates
3.6.1 The HJM Framework
3.6.2 The Discrete Drift
3.6.3 Implementation
3.7 Forward Rate Models: Simple Rates
3.7.1 LIBOR Market Model Dynamics
3.7.2 Pricing Derivatives
3.7.3 Simulation
3.7.4 Volatility Structure and Calibration

4 Variance Reduction Techniques
4.1 Control Variates
4.1.1 Method and Examples
4.1.2 Multiple Controls
4.1.3 Small-Sample Issues
4.1.4 Nonlinear Controls
4.2 Antithetic Variates
4.3 Stratified Sampling
4.3.1 Method and Examples
4.3.2 Applications
4.3.3 Poststratification
4.4 Latin Hypercube Sampling
4.5 Matching Underlying Assets
4.5.1 Moment Matching Through Path Adjustments
4.5.2 Weighted Monte Carlo
4.6 Importance Sampling
4.6.1 Principles and First Examples
4.6.2 Path-Dependent Options
4.7 Concluding Remarks

5 Quasi-Monte Carlo
5.1 General Principles
5.1.1 Discrepancy
5.1.2 Van der Corput Sequences
5.1.3 The Koksma-Hlawka Bound
5,1.4 Nets and Sequences
5.2 Low-Discrepancy Sequences
5.2.1 Halton and Hammersley
5.2.2 Faure
5.2.3 Sobol'
5.2.4 Further Constructions ..
5.3 Lattice Rules
5.4 Randomized QMC
5.5 The Finance Setting
5.5.1 Numerical Examples
5.5.2 Strategic Implementation
5.6 Concluding Remarks

6 Discretization Methods
6.1 Introduction
6.1.1 The Euler Scheme and a First Refinement
6.1.2 Convergence Order
6.2 Second-Order Methods
6.2.1 The Scalar Case
6.2.2 The Vector Case
6.2.3 Incorporating Path-Dependence
6.2.4 Extrapolation
6.3 Extensions
6.3.1 General Expansions
6.3.2 Jump-Diffusion Processes
6.3.3 Convergence of Mean Square Error
6.4 Extremes and Barrier Crossings: Brownian Interpolation
6.5 Changing Variables
6.6 Concluding Remarks

7 Estimating Sensitivities
7.1 Finite-Difference Approximations
7.1.1 Bias and Variance
7.1.2 Optimal Mean Square Error
7.2 Pathwise Derivative Estimates
7.2.1 Method and Examples
7.2.2 Conditions for Unbiasedness
7.2.3 Approximations and Related Methods
7.3 The Likelihood Ratio Method
7.3.1 Method and Examples
7.3.2 Bias and Variance Properties
7.3.3 Gamma
7.3.4 Approximations and Related Methods
7.4 Concluding Remarks

8 Pricing American Options
8.1 Problem Formulation
8.2 Parametric Approximations
8.3 Random Tree Methods
8.3.1 High Estimator
8.3.2 Low Estimator
8.3.3 Implementation
8.4 State-Space Partitioning
8.5 Stochastic Mesh Methods
8.5.1 General Framework
8.5.2 Likelihood Ratio Weights
8.6 Regression-Based Methods and Weights
8.6.1 Approximate Continuation Values
8.6.2 Regression and Mesh Weights
8.7 Duality
8.8 Concluding Remarks

9 Applications in Risk Management
9.1 Loss Probabilities and Value-at-Risk
9.1.1 Background
9.1.2 Calculating VAR
9.2 Variance Reduction Using the Delta-Gamma Approximation
9.2.1 Control Variate
9.2.2 Importance Sampling
9.2.3 Stratified Sampling
9.3 A Heavy-Tailed Setting
9.3.1 Modeling Heavy Tails
9.3.2 Delta-Gamma Approximation
9.3.3 Variance Reduction
9.4 Credit Risk
9.4.1 Default Times and Valuation
9.4.2 Dependent Defaults
9.4.3 Portfolio Credit Risk
9.5 Concluding Remarks

A Appendix: Convergence and Confidence Intervals
A.1 Convergence Concepts
A.2 Central Limit Theorem and Confidence Intervals
B Appendix: Results from Stochastic Calculus
B.1 It6's Formula
B.2 Stochastic Differential Equations
B.3 Martingales
B.4 Change of Measure
C Appendix: The Term Structure of Interest Rates
C.1 Term Structure Terminology
C.2 Interest Rate Derivatives
References
Index