# Stochastic Simulation and Applications in Finance with MATLAB Programs

# Stochastic Simulation and Applications in Finance with MATLAB Programs

ISBN: 978-1-118-46737-4 May 2012 354 Pages

## Description

*Stochastic Simulation and Applications in Finance with MATLAB Programs*explains the fundamentals of Monte Carlo simulation techniques, their use in the numerical resolution of stochastic differential equations and their current applications in finance. Building on an integrated approach, it provides a pedagogical treatment of the need-to-know materials in risk management and financial engineering.

The book takes readers through the basic concepts, covering the most recent research and problems in the area, including: the quadratic re-sampling technique, the Least Squared Method, the dynamic programming and Stratified State Aggregation technique to price American options, the extreme value simulation technique to price exotic options and the retrieval of volatility method to estimate Greeks. The authors also present modern term structure of interest rate models and pricing swaptions with the BGM market model, and give a full explanation of corporate securities valuation and credit risk based on the structural approach of Merton. Case studies on financial guarantees illustrate how to implement the simulation techniques in pricing and hedging.

NOTE TO READER: The CD has been converted to URL. Go to the following website www.wiley.com/go/huyhnstochastic which provides MATLAB programs for the practical examples and case studies, which will give the reader confidence in using and adapting specific ways to solve problems involving stochastic processes in finance.

**Preface**

**1 Introduction to Probability**

1.1 Intuitive Explanation

1.2 Axiomatic Definition

**2 Introduction to Random Variables**

2.1 Random Variables

2.2 Random Vectors

2.3 Transformation of Random Variables

2.4 Transformation of Random Vectors

2.5 Approximation of the Standard Normal Cumulative Distribution Function

**3 Random Sequences**

3.1 Sum of Independent Random Variables

3.2 Law of Large Numbers

3.3 Central Limit Theorem

3.4 Convergence of Sequences of Random Variables

**4 Introduction to Computer Simulation of Random Variables**

4.1 Uniform Random Variable Generator

4.2 Generating Discrete Random Variables

4.3 Simulation of Continuous Random Variables

4.4 Simulation of Random Vectors

4.5 Acceptance-Rejection Method

4.6 Markov Chain Monte Carlo Method (MCMC)

**5 Foundations of Monte Carlo Simulations**

5.1 Basic Idea

5.2 Introduction to the Concept of Precision

5.3 Quality of Monte Carlo Simulations Results

5.4 Improvement of the Quality of Monte Carlo Simulations or Variance Reduction Techniques

5.5 Application Cases of Random Variables Simulations

**6 Fundamentals of Quasi Monte Carlo (QMC) Simulations**

6.1 Van Der Corput Sequence (Basic Sequence)

6.2 Halton Sequence

6.3 Faure Sequence

6.4 Sobol Sequence

6.5 Latin Hypercube Sampling

6.6 Comparison of the Different Sequences

**7 Introduction to Random Processes**

7.1 Characterization

7.2 Notion of Continuity, Differentiability and Integrability

7.3 Examples of Random Processes

**8 Solution of Stochastic Differential Equations**

8.1 Introduction to Stochastic Calculus

8.2 Introduction to Stochastic Differential Equations

8.3 Introduction to Stochastic Processes with Jump

8.4 Numerical Solutions of some Stochastic Differential Equations (SDE)

8.5 Application case: Generation of a Stochastic Differential Equation using the Euler and Milstein Schemes

8.6 Application Case: Simulation of a Stochastic Differential Equation with Control and Antithetic Variables

8.7 Application Case: Generation of a Stochastic Differential Equation with Jumps

**9 General Approach to the Valuation of Contingent Claims**

9.1 The Cox, Ross and Rubinstein (1979) Binomial Model of Option Pricing

9.2 Black and Scholes (1973) and Merton (1973) Option Pricing Model

9.3 Derivation of the Black-Scholes Formula using the Risk-Neutral Valuation Principle

**10 Pricing Options using Monte Carlo Simulations**

10.1 Plain Vanilla Options: European put and Call

10.2 American options

10.3 Asian options

10.4 Barrier options

10.5 Estimation Methods for the Sensitivity Coefficients or Greeks

**11 Term Structure of Interest Rates and Interest Rate Derivatives**

11.1 General Approach and the Vasicek (1977) Model

11.2 The General Equilibrium Approach: The Cox, Ingersoll and Ross (CIR, 1985) model

11.3 The Affine Model of the Term Structure

11.4 Market Models

**12 Credit Risk and the Valuation of Corporate Securities**

12.1 Valuation of Corporate Risky Debts: The Merton (1974) Model

12.2 Insuring Debt Against Default Risk

12.3 Valuation of a Risky Debt: The Reduced-Form Approach

**13 Valuation of Portfolios of Financial Guarantees**

13.1 Valuation of a Portfolio of Loan Guarantees

13.2 Valuation of Credit Insurance Portfolios using Monte Carlo Simulations

**14 Risk Management and Value at Risk (VaR)**

14.1 Types of Financial Risks

14.2 Definition of the Value at Risk (VaR)

14.3 The Regulatory Environment of Basle

14.4 Approaches to compute VaR

14.5 Computing VaR by Monte Carlo Simulations

**15 VaR and Principal Components Analysis (PCA)**

15.1 Introduction to the Principal Components Analysis

15.2 Computing the VaR of a Bond Portfolio

**Appendix A: Review of Mathematics**

A.1 Matrices

A.1.1 Elementary Operations on Matrices

A.1.2 Vectors

A.1.3 Properties

A.1.4 Determinants of Matrices

A.2 Solution of a System of Linear Equations

A.3 Matrix Decomposition

A.4 Polynomial and Linear Approximation

A.5 Eigenvectors and Eigenvalues of a Matrix

**Appendix B: MATLAB ^{®}**

**Functions**

**References and Bibliography**

**Index**

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