Stochastic Simulation and Applications in Finance with MATLAB Programs

by Huynh, Huu Tue; Lai, Van Son; Soumare, Issouf

• ISBN13: 9781118467374
• ISBN10: 111846737X
• Edition: 1st
• Format: eBook
• Copyright: 2012-05-25
• Publisher: Wiley

Summary

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.

The book also includes an accompanying CD-ROM 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.

"This book provides a very useful set of tools for those who are interested in the simulation method of asset pricing and its implementation with MatLab. It is pitched at just the right level for anyone who seeks to learn about this fascinating area of finance. The collection of specific topics thoughtfully selected by the authors, such as credit risk, loan guarantee and value-at-risk, is an additional nice feature, making it a great source of reference for researchers and practitioners. The book is a valuable contribution to the fast growing area of quantitative finance."

-Tan Wang, Sauder School of Business, UBC

“This book is a good companion to text books on theory, so if you want to get straight to the meat of implementing the classical quantitative finance models here's the answer.”

—Paul Wilmott, wilmott.com

“This powerful book is a comprehensive guide for Monte Carlo methods in finance. Every quant knows that one of the biggest issues in finance is to well understand the mathematical framework in order to translate it in programming code. Look at the chapter on Quasi Monte Carlo or the paragraph on variance reduction techniques and you will see that Huu Tue Huynh, Van Son Lai and Issouf Soumaré have done a very good job in order to provide a bridge between the complex mathematics used in finance and the programming implementation. Because it adopts both theoretical and practical point of views with a lot of applications, because it treats about some sophisticated financial problems (like Brownian bridges, jump processes, exotic options pricing or Longstaff-Schwartz methods) and because it is easy to understand, this handbook is valuable for academics, students and financial engineers who want to learn the computational aspects of simulations in finance.”

—Thierry Roncalli, Head of Investment Products and Strategies, SGAM Alternative Investments & Professor of Finance, University of Evry

Author Biography

HUU TUE HUYNH obtained his D.Sc. in communication theory from Laval University, Canada. From 1969 to 2004 he was a faculty member of Laval University. He left Laval University to become Chairman of the Department of data processing at the College of Technology of The Vietnam National University, Hanoi. Since 2007 he has been Rector of the Bac Ha International University, Vietnam. His main recent research interest covers Fast Monte Carlo methods and applications.

VAN SON LAI is Professor of Finance at the Business School of Laval University, Canada. He obtained his Ph.D. in Finance from the University of Georgia, USA and a master degree in water resources engineering from the University of British Columbia, Canada. He is also a CFA charterholder from the CFA Institute and a registered P.Eng. in the Province of British Columbia. An established teacher and researcher in banking, financial engineering, and risk management, he has extensively published in mainstream banking, economics, and finance journals.

ISSOUF SOUMARÉ is currently associate professor of finance and managing director of the Laboratory for Financial Engineering at Laval University. His research and teaching interests included risk management, financial engineering and numerical methods in finance. He has published his theoretical and applied finance works in economics and finance journals. Dr Soumaré holds a PhD in Finance from the University of British Columbia, Canada, MSc in Financial Engineering from Laval University, Canada, MSc in Statistics and Quantitative Economics and MSc and BSc in Applied Mathematics from Ivory Coast. He is also a certified Professional Risk Manager (PRM) of the Professional Risk Managers’ International Association (PRMIA).

Table of Contents

Contents

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