Simulation Techniques in Financial Risk Management

Praise for the First Edition

“…a nice, self-contained introduction to simulation and computational techniques in finance…”

 – Mathematical Reviews

Simulation Techniques in Financial Risk Management, Second Edition takes a unique approach to the field of simulations by focusing on techniques necessary in the fields of finance and risk management. Thoroughly updated, the new edition expands on several key topics in these areas and presents many of the recent innovations in simulations and risk management, such as advanced option pricing models beyond the Black–Scholes paradigm, interest rate models, MCMC methods including stochastic volatility models simulations, model assets and model-free properties, jump diffusion, and state space modeling. The Second Edition also features:

• Updates to primary software used throughout the book, Microsoft Office® Excel® VBA
• New topical coverage on multiple assets, model-free properties, and related models
• More than 300 exercises at the end of each chapter, with select answers in the appendix, to help readers apply new concepts and test their understanding
• Extensive use of examples to illustrate how to use simulation techniques in risk management
• Practical case studies, such as the pricing of exotic options; simulations of Greeks in hedging; and the use of Bayesian ideas to assess the impact of jumps, so readers can reproduce the results of the studies
• A related website with additional solutions to problems within the book as well as Excel VBA and S-Plus computer code for many of the examples within the book

Simulation Techniques in Financial Risk Management, Second Edition is an invaluable resource for risk managers in the financial and actuarial industries as well as a useful reference for readers interested in learning how to better gauge risk and make more informed decisions. The book is also ideal for upper-undergraduate and graduate-level courses in simulation and risk management.

List of Figures xi

List of Tables xiii

Preface xv

1 Preliminaries of VBA 1

1.1 Introduction 1

1.2 Basis Excel VBA 1

1.2.1 Developer Mode and Security Level 2

1.2.2 Visual Basic Editor 2

1.2.3 The Macro Recorder 5

1.2.4 Setting Up a Command Button 6

1.3 VBA Programming Fundamentals 8

1.3.1 Declaration of Variables 8

1.3.2 Types of Variables 8

1.3.3 Declaration of Multivariable 9

1.3.4 Declaration of Constants 9

1.3.5 Operators 9

1.3.6 User-Defined Data Types 10

1.3.7 Arrays and Matrices 11

1.3.8 Data Input and Output 12

1.3.9 Conditional Statements 12

1.3.10 Loops 13

1.3.11 Sub Procedures and Function Procedures 15

1.3.12 VBA’s Built-In Functions 18

2 Basic Properties of Futures and Options 19

2.1 Introduction 19

2.1.1 Arbitrage and Hedging 19

2.1.2 Forward Contracts 20

2.1.3 Futures Contracts 23

2.2 Options 26

2.3 Exercises 31

3 Introduction to Simulation 35

3.1 Questions 35

3.2 Simulation 35

3.3 Examples 36

3.3.1 Quadrature 36

3.3.2 Monte Carlo 37

3.4 Stochastic Simulations 38

3.5 Exercises 40

4 Brownian Motions and Itô’s Rule 41

4.1 Introduction 41

4.2 Wiener and Itô’s Processes 41

4.3 Stock Price 46

4.4 Itô’s Formula 47

4.5 Exercises 54

5 Black–Scholes Model and Option Pricing 57

5.1 Introduction 57

5.2 One Period Binomial Model 58

5.3 The Black–Scholes–Merton Equation 61

5.4 Black–Scholes Formula 67

5.5 Exercises 72

6 Generating Random Variables 75

6.1 Introduction 75

6.2 Random Numbers 75

6.3 Discrete Random Variables 76

6.4 Acceptance-Rejection Method 78

6.5 Continuous Random Variables 79

6.5.1 Inverse Transform 80

6.5.2 The Rejection Method 81

6.5.3 Multivariate Normal 83

6.6 Exercises 87

7 Standard Simulations in Risk Management 89

7.1 Introduction 89

7.2 Scenario Analysis 89

7.2.1 Value at Risk 91

7.2.2 Heavy-Tailed Distribution 92

7.2.3 Case Study: VaR of Dow Jones 94

7.3 Standard Monte Carlo 96

7.3.1 Mean, Variance, and Interval Estimation 97

7.3.2 Simulating Option Prices 99

7.3.3 Simulating Option Delta 102

7.4 Exercises 104

7.5 Appendix 105

8 Variance Reduction Techniques 107

8.1 Introduction 107

8.2 Antithetic Variables 107

8.3 Stratified Sampling 112

8.4 Control Variates 120

8.5 Importance Sampling 125

8.6 Exercises 131

9 Path Dependent Options 133

9.1 Introduction 133

9.2 Barrier Option 133

9.3 Lookback Option 135

9.4 Asian Option 136

9.5 American Option 138

9.5.1 Simulation: Least Squares Approach 138

9.5.2 Analyzing the Least Squares Approach 141

9.5.3 American Style Path Dependent Options 144

9.6 Greek Letters 145

9.7 Exercises 148

10 Multiasset Options 151

10.1 Introduction 151

10.2 Simulating European Multiasset Options 152

10.3 Case Study: On Estimating Basket Options 153

10.4 Dimension Reduction 155

10.5 Exercises 158

11 Interest Rate Models 161

11.1 Introduction 161

11.2 Discount Factor and Bond Prices 161

11.3 Stochastic Interest Rate Models and Their Simulations 165

11.4 Hull–White Model 167

11.5 Fixed Income Derivatives Pricing 171

11.6 Exercises 174

12 Markov Chain Monte Carlo Methods 177

12.1 Introduction 177

12.2 Bayesian Inference 177

12.3 Simulating Posteriors 179

12.4 Markov Chain Monte Carlo 180

12.4.1 Gibbs Sampling 180

12.4.2 Case Study: The Effect of Jumps on Dow Jones 183

12.5 Metropolis–Hastings Algorithm 188

12.6 Exercises 196

References 199

Index 203

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