Simulation Techniques in Financial Risk Management E-Book

Ngai Hang Chan

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Statistics in Practice
• Sprache: Englisch

Seitenzahl: 267

Veröffentlichungsjahr: 2015

Beschreibung

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.

Leseprobe

Table of Contents

Cover

Wiley Series in Statistics in Practice

Title Page

Copyright

Dedication

List of Figures

List of Tables

Preface

Preface to the Second Edition

Preface to the First Edition

Chapter 1: Preliminaries of VBA

1.1 Introduction

1.2 Basis Excel VBA

1.3 VBA Programming Fundamentals

Chapter 2: Basic Properties of Futures and Options

2.1 Introduction

2.2 Options

2.3 Exercises

Chapter 3: Introduction to Simulation

3.1 Questions

3.2 Simulation

3.3 Examples

3.4 Stochastic Simulations

3.5 Exercises

Chapter 4: Brownian Motions and Itô's Rule

4.1 Introduction

4.2 Wiener and Itô's Processes

4.3 Stock Price

4.4 Itô's Formula

4.5 Exercises

Chapter 5: Black–Scholes Model and Option Pricing

5.1 Introduction

5.2 One Period Binomial Model

5.3 The Black–Scholes–Merton Equation

5.4 Black–Scholes Formula

5.5 Exercises

Chapter 6: Generating Random Variables

6.1 Introduction

6.2 Random Numbers

6.3 Discrete Random Variables

6.4 Acceptance-Rejection Method

6.5 Continuous Random Variables

6.6 Exercises

Chapter 7: Standard Simulations in Risk Management

7.1 Introduction

7.2 Scenario Analysis

7.3 Standard Monte Carlo

7.4 Exercises

7.5 Appendix

Chapter 8: Variance Reduction Techniques

8.1 Introduction

8.2 Antithetic Variables

8.3 Stratified Sampling

8.4 Control Variates

8.5 Importance Sampling

8.6 Exercises

Chapter 9: Path Dependent Options

9.1 Introduction

9.2 Barrier Option

9.3 Lookback Option

9.4 Asian Option

9.5 American Option

9.6 Greek Letters

9.7 Exercises

Chapter 10: Multiasset Options

10.1 Introduction

10.2 Simulating European Multiasset Options

10.3 Case Study: On Estimating Basket Options

10.4 Dimension Reduction

10.5 Exercises

Chapter 11: Interest Rate Models

11.1 Introduction

11.2 Discount Factor and Bond Prices

11.3 Stochastic Interest Rate Models and Their Simulations

11.4 Hull–White Model

11.5 Fixed Income Derivatives Pricing

11.6 Exercises

Chapter 12: Markov Chain Monte Carlo Methods

12.1 Introduction

12.2 Bayesian Inference

12.3 Simulating Posteriors

12.4 Markov Chain Monte Carlo

12.5 Metropolis–Hastings Algorithm

12.6 Exercises

References

Index

Wiley Series in Statistics in Practice

End User License Agreement

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Guide

Cover

Table of Contents

Preface

Begin Reading

List of Illustrations

Chapter 1: Preliminaries of VBA

Figure 1.1 Excel [Options].

Figure 1.2 Developer mode selection.

Figure 1.3 Excel in developer mode.

Figure 1.4 Macro security.

Figure 1.5 Visual basic editor.

Figure 1.6 The macro recorder.

Figure 1.7 The recorded codes.

Figure 1.8 Creating command button.

Figure 1.9 Assigning a macro to a command button.

Chapter 2: Basic Properties of Futures and Options

Figure 2.1 Payoffs of option positions.

Figure 2.2 The prices of Hang Seng Index options on September 12, 2014.

Chapter 3: Introduction to Simulation

Figure 3.1 Densities of a lognormal distribution with mean and variance , that is, and and a standard normal distribution.

Chapter 4: Brownian Motions and Itô's Rule

Figure 4.1 Sample paths of the process for different

n

and the same sequence of .

Figure 4.2 Sample paths of Brownian motions on [0,1].

Figure 4.3 Geometric Brownian motion.

Chapter 5: Black–Scholes Model and Option Pricing

Figure 5.1 One period binomial tree.

Chapter 6: Generating Random Variables

Figure 6.1 Sample paths of the portfolio.

Figure 6.2 Sample paths of the assets and the portfolio.

Chapter 7: Standard Simulations in Risk Management

Figure 7.1 The shape of GED density function.

Figure 7.2 Left tail of GED.

Figure 7.3 QQ plot of normal quantiles against daily Dow Jones returns.

Figure 7.4 Determine the maximum of graphically.

Figure 7.5 QQ plot GED(1.21) quantiles against Dow Jones return quantiles.

Figure 7.6 Simulations of the call price against the size.

Figure 7.7 The log likelihood against .

Chapter 8: Variance Reduction Techniques

Figure 8.1 Illustration of payoffs for antithetic comparisons.

Figure 8.2 Payoff on a straddle as a function of input normal

Z

based on the parameters , , , and .

Figure 8.3 Simulations of 500 standard normal random numbers by standard Monte Carlo.

Figure 8.4 Simulations of 500 standard normal random numbers by stratified sampling.

Chapter 9: Path Dependent Options

Figure 9.1 The exercising region of the American put option.

Figure 9.2 Exercise regions of the American style Asian option.

Figure 9.3 The strike against the delta of a down-and-out call option.

Chapter 10: Multiasset Options

Figure 10.1 The distribution of simulated price.

Figure 10.2 The historical price of shocks.

Figure 10.3 Simulating terminal asset prices.

Chapter 11: Interest Rate Models

Figure 11.1 Cash flow of a 3-year coupon bond.

Figure 11.2 Yield to maturity .

Figure 11.3 Discount curve .

Figure 11.4 Simulated sample path of .

Figure 11.5 .

Figure 11.6 .

Chapter 12: Markov Chain Monte Carlo Methods

Figure 12.1 A sample path of the jump-diffusion model.

List of Tables

Chapter 1: Preliminaries of VBA

Table 1.1 Numeric Data Type

Table 1.2 VBA Logical Operators

Table 1.3 Common Built-In Math Functions in VBA

Chapter 2: Basic Properties of Futures and Options

Table 2.1 Strategy A for Longing Futures Contracts

Table 2.2 Data on Stock and Futures Prices

Table 2.3 Payoffs of Different Options with Strike Price

K

Table 2.4 Payoffs of Portfolios A and B

Table 2.5 Payoffs of Portfolios C and D

Table 2.6 Payoffs of Portfolios E and F

Table 2.7 Properties of Stock Options

Chapter 7: Standard Simulations in Risk Management

Table 7.1 Sales Record

Table 7.2 Probability Mass Function

Table 7.3 Policy Simulation and Evaluation

Table 7.4 Simulated Prices of the First and the Last 10 Weeks

Table 7.5 The Discounted Call Prices for the First 20 Paths

Chapter 8: Variance Reduction Techniques

Table 8.1 Effects of Stratification for Simulated Option Prices with Different Bin Sizes

Table 8.2 Effects of Stratification for Simulated Option Prices with Restricted Normal

Chapter 9: Path Dependent Options

Table 9.1 Sample Paths

Table 9.2 Regression at

Table 9.3 Optimal Decision at

Table 9.4 Regression at

Table 9.5 Optimal Decision at

Chapter 12: Markov Chain Monte Carlo Methods

Table 12.1 Conjugate Priors

Table 12.2 Performance of the Gibbs Sampling

Table 12.3 Jump-Diffusion Estimation for Dow Jones

WILEY SERIES IN STATISTICS IN PRACTICE

Advisory Editor, MARIAN SCOTT, University of Glasgow, Scotland, UK

Founding Editor, VIC BARNETT, Nottingham Trent University, UK

Statistics in Practice is an important international series of texts which provide detailed coverage of statistical concepts, methods, and worked case studies in specific fields of investigation and study.

With sound motivation and many worked practical examples, the books show in down-to-earth terms how to select and use an appropriate range of statistical techniques in a particular practical field within each titl's special topic area.

The books provide statistical support for professionals and research workers across a range of employment fields and research environments. Subject areas covered include medicine and pharmaceutics; industry, finance, and commerce; public services; the earth and environmental sciences, and so on.

The books also provide support to students studying statistical courses applied to the above areas. The demand for graduates to be equipped for the work environment has led to such courses becoming increasingly prevalent at universities and colleges.

It is our aim to present judiciously chosen and well-written workbooks to meet everyday practical needs. Feedback of views from readers will be most valuable to monitor the success of this aim.

A complete list of titles in this series appears at the end of the volume.

SIMULATION TECHNIQUES IN FINANCIAL RISK MANAGEMENT

Second Edition

NGAI HANG CHAN AND HOI YING WONG

The Chinese University of Hong Kong

Copyright © 2015 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, [phone], fax [phone], or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, [phone], fax [phone], or online at http://www.wiley.com/go/permissions.

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Library of Congress Cataloging-in-Publication Data:

Chan, Ngai Hang.

Simulation techniques in financial risk management / Ngai Hang Chan and Hoi Ying Wong. – Second edition.

pages cm. – (Statistics in practice)

Includes bibliographical references and index.

ISBN 978-1-118-73581-7 (hardback)

1. Finance–Simulation methods. 2. Risk management–Simulation methods. I. Wong, Hoi Ying, 1974- II. Title.

HG173.C47 2015

338.5–dc23

2015001921

Cover image courtesy of iStockphoto © pawel.gaul

To our families

N.H. Chan and H.Y. Wong

Preface

Preface to the Second Edition

This book has now been in print for almost 10 years and has seen several printings. During this period, the field of quantitative finance has experienced abrupt changes, some for better and some for worse. But it has been very gratifying to us to have heard from many readers that this book has been helpful to them in dealing with the ever-changing financial landscape. It appears that to some extent at least the original objectives set out in the first edition have been realized. This book can be used either as an introductory text to simulations at the senior undergraduate or as a Master's level course. It can also be used as a complimentary source to the more specialized treatise by Chan and Wong (2013) entitled Handbook of Financial Risk Management: Simulations and Case Studies.

This second edition has been thoroughly revised and enhanced. Many of these changes were results of teaching different courses in simulation for financial risk managers over the years. In addition to cleaning up as many errors and misprints as possible, the following specific changes have been incorporated in this revision.

Many readers suggested more exercises with worked solutions. As a result, we enlarge the problems and answers section in light of these requests.

Because the use of VBA in Excel has been common in the financial industry, the current edition incorporates this suggestion. We have now replaced all S-Plus codes with VBA codes.

Due to the advent in IT technology, a new website has been set up for readers to download the VBA computer codes.

http://www.sta.cuhk.edu.hk/Book/SRMS/

As long as the website is available, we no longer print computer codes, so that more space can be used for expanded topics.

Likewise, suggested solutions to exercises at the end of each chapter are now available via online supplementary materials.

To make the book self-contained, two new chapters,

Chapters 1

and

2

, have been added.

Chapter 1

introduces basic concepts of Excel VBA, and

Chapter 2

introduces basic concepts of derivatives.

Corresponding to

Chapter 9

in the first edition,

Chapter 11

of this edition is expanded to discuss in detail a one-factor interest rate model and the calibration to yield curves.

More examples have been added to illustrate the concept of MCMC, in particular the Metropolis–Hastings algorithm.

Finally, we would like to thank colleagues and students alike, who have been giving us suggestions and ideas throughout the years. In particular, we would like to thank the editorial assistance of Dr. Warwick Yuen and Mr. Tom Ng of CUHK and Ms. Sari Friedman and Mr. Jon Gurstelle of Wiley. We also want to express our gratitude to the Research Grants Council of HKSAR for support at various stages of our work on this revision.

Ngai Hang Chan and Hoi Ying WongShatin, Hong Kong

Preface to the First Edition

Risk management is an important subject in finance. Despite its popularity, risk management has a broad and diverse definition that varies from individual to individual. One fact remains, however. Every modern risk management method comprises a significant amount of computations. To assess the success of a risk management procedure, one has to rely heavily on simulation methods. A typical example is the pricing and hedging of exotic options in the derivative market. These over-the-counter options experience very thin trading volume, and yet their nonlinear features forbid the use of analytical techniques. As a result, one has to rely on simulations in order to examine their properties. It is therefore not surprising that simulation has become an indispensable tool in the financial and risk management industry today.

Although simulation as a subject has a long history by itself, the same cannot be said about risk management. To fully appreciate the power and usefulness of risk management, one has to acquire a considerable amount of background knowledge across several disciplines: finance, statistics, mathematics, and computer science. It is the synergy of various concepts across these different fields that marks the success of modern risk management. Although many excellent books have been written on the subject of simulation, none has been written from a risk management perspective. It is therefore timely and important to have a text that readily introduces the modern techniques of simulation and risk management to the financial world.

This text aims at introducing simulation techniques for practitioners in the financial and risk management industry at an intermediate level. The only prerequisite is a standard undergraduate course in probability at the level of Hogg and Tanis (2006), say, and some rudimentary exposure to finance. The present volume stems from a set of lecture notes used at the Chinese University of Hong Kong. It aims at striking a balance between theory and applications of risk management and simulations, particularly along the financial sector. The book comprises three parts.

Part one consists of the first three chapters. After introducing the motivations of simulation in

Chapter 1

, basic ideas of Wiener processes and Itô's calculus are introduced in

Chapters 2

and

3

. The reason for this inclusion is that many students have experienced difficulties in this area because they lack the understanding of the theoretical underpinnings of these topics. We try to introduce these topics at an operational level so that readers can immediately appreciate the complexity and importance of stochastic calculus and its relationship with simulations. This will pave the way for a smooth transition to option pricing and Greeks in later chapters. For readers familiar with these topics, this part can be used as a review.

Chapters 4

–

6

comprise the second part of the book. This part constitutes the main core of an introductory course in risk management. It covers standard topics in a traditional course in simulation, but at a much higher and succinct level. Technical details are left in the references, but important ideas are explained in a conceptual manner. Examples are also given throughout to illustrate the use of these techniques in risk management. By introducing simulations this way, both students with strong theoretical background and students with strong practical motivations get excited about the subject early on.

The remaining

Chapters 7

–

10

constitute part 3 of the book. In this part, more advanced and exotic topics of simulations in financial engineering and risk management are introduced. One distinctive feature in these chapters is the inclusion of case studies. Many of these cases have strong practical bearings such as pricing of exotic options, simulations of Greeks in hedging, and the use of Bayesian ideas to assess the impact of jumps. By means of these examples, it is hoped that readers can acquire a first-hand knowledge about the importance of simulations and apply them to their work.

Throughout the book, examples from finance and risk management have been incorporated as much as possible. This is done throughout the text, starting at the early chapter that discusses VaR of Dow to pricing of basket options in a multiasset setting. Almost all of the examples and cases are illustrated with Splus and some with Visual Basics. Readers would be able to reproduce the analysis and learn about either Splus or Visual Basics by replicating some of the empirical work.

Many recent developments in both simulations and risk management, such as Gibbs sampling, the use of heavy-tailed distributions in VaR calculation, and principal components in multiasset settings are discussed and illustrated in detail. Although many of these developments have found applications in the academic literature, they are less understood among practitioners. Inclusion of these topics narrows the gap between academic developments and practical applications.

In summary, this text fills a vacuum in the market of simulations and risk management. By giving both conceptual and practical illustrations, this text not only provides an efficient vehicle for practitioners to apply simulation techniques, but also demonstrates a synergy of these techniques. The examples and discussions in later chapters make recent developments in simulations and risk management more accessible to a larger audience.

Several versions of these lecture notes have been used in a simulation course given at the Chinese University of Hong Kong. We are grateful for many suggestions, comments, and questions from both students and colleagues. In particular, the first author is indebted to Professor John Lehoczky at Carnegie Mellon University, from whom he learned the essence of simulations in computational finance. Part 2 of this book reflects many of the ideas of John and is a reminiscence of his lecture notes at Carnegie Mellon. We would also like to thank Yu-Fung Lam and Ka-Yung Lau for their help in carrying out some of the computational tasks in the examples and for producing the figures in LaTeX, and to Mr. Steve Quigley and Ms. Susanne Steitz, both from Wiley, for their patience and professional assistance in guiding the preparation and production of this book. Financial support from the Research Grant Council of Hong Kong throughout this project is gratefully acknowledged. Last, but not least, we would like to thank our families for their understanding and encouragement in writing this book. Any remaining errors are, of course, our sole responsibility.

Ngai Hang Chan and Hoi Ying WongShatin, Hong Kong

1Preliminaries of VBA

1.1 Introduction

This chapter introduces the elementary programming skills in Visual Basic for Applications (VBA) that we use for numerical computation of the examples in the book. Experienced readers can read this chapter as a quick review.

1.2 Basis Excel VBA

Microsoft Excel is widely used in the financial industry for performing financial calculations. VBA is a common programming language linked to Excel and other Microsoft Office software that was developed to automatically control and perform repetitive actions. In this section, we guide readers on how to start a VBA in Microsoft Excel and give some popular algorithms for performing repetitions. In most cases, simple algorithms will be sufficient to perform the computations in the examples and exercises. We provide the illustrations in Excel 2010, although other versions can be set up in a similar way. For a comprehensive reference, readers are referred to other books.

1.2.1 Developer Mode and Security Level

For first-time users of VBA in Excel, it is more convenient to switch on the developer mode, where many of the VBA functions can be easily accessed. To open the developer mode, follow the following steps:

Click [File] → [Options] (

Fig. 1.1

) → [Customize Ribbon] (

Fig. 1.2

) → [Developer].

Figure 1.1 Excel [Options].

Figure 1.2 Developer mode selection.

Figure 1.3 shows the ribbons at the top of Excel after switching on the developer mode. Macros refer to the codes executed in the VBA language. To execute the macros promptly, users are recommended to turn down the security level as follows:

Click [Macro Security] (

Fig. 1.3

) → Macro Settings [Enable all macros] (

Fig. 1.4

).

Figure 1.3 Excel in developer mode.

Figure 1.4 Macro security.

1.2.2 Visual Basic Editor

To edit the VBA codes, Microsoft provides a Visual Basic editor (VBE) in Excel for editing the macros. Macros are created, edited, and debugged in the VBE before being executed. A macro is usually created as a Sub or Function procedure that can perform automatic tasks, while a module consists of one or more macros. Similarly, a project has one or more modules. and are reserved keywords in VBA. Users need to avoid using keywords when defining new variables. The codes in the VBE are saved together with the Excel worksheet. In Excel 2010, these worksheets can be saved as .xlsm as an Excel Macro-Enabled Workbook file. To open and edit macros in VBE, follow the following procedure: