Essential Mathematics for Market Risk Management E-Book

Simon Hubbert

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Series
• Sprache: Englisch

Seitenzahl: 386

Veröffentlichungsjahr: 2011

Beschreibung

Everything you need to know in order to manage risk effectively within your organization You cannot afford to ignore the explosion in mathematical finance in your quest to remain competitive. This exciting branch of mathematics has very direct practical implications: when a new model is tested and implemented it can have an immediate impact on the financial environment. With risk management top of the agenda for many organizations, this book is essential reading for getting to grips with the mathematical story behind the subject of financial risk management. It will take you on a journey--from the early ideas of risk quantification up to today's sophisticated models and approaches to business risk management. To help you investigate the most up-to-date, pioneering developments in modern risk management, the book presents statistical theories and shows you how to put statistical tools into action to investigate areas such as the design of mathematical models for financial volatility or calculating the value at risk for an investment portfolio. * Respected academic author Simon Hubbert is the youngest director of a financial engineering program in the U.K. He brings his industry experience to his practical approach to risk analysis * Captures the essential mathematical tools needed to explore many common risk management problems * Website with model simulations and source code enables you to put models of risk management into practice * Plunges into the world of high-risk finance and examines the crucial relationship between the risk and the potential reward of holding a portfolio of risky financial assets This book is your one-stop-shop for effective risk management.

Leseprobe

Table of Contents

Series Page

Title Page

Copyright

Dedication

Preface

Chapter 1: Introduction

1.1 Basic Challenges in Risk Management

1.2 Value at Risk

1.3 Further Challenges in Risk Management

Chapter 2: Applied Linear Algebra for Risk Managers

2.1 Vectors and Matrices

2.2 Matrix Algebra in Practice

2.3 Eigenvectors and Eigenvalues

2.4 Positive Definite Matrices

Chapter 3: Probability Theory for Risk Managers

3.1 Univariate Theory

3.2 Multivariate Theory

3.3 The Normal Distribution

Chapter 4: Optimization Tools

4.1 Background Calculus

4.2 Optimizing Functions

4.3 Over-determined Linear Systems

4.4 Linear Regression

Chapter 5: Portfolio Theory I

5.1 Measuring Returns

5.2 Setting Up the Optimal Portfolio Problem

5.3 Solving the Optimal Portfolio Problem

Chapter 6: Portfolio Theory II

6.1 The Two-Fund Investment Service

6.2 A Mathematical Investigation of the Optimal Frontier

6.3 A Geometrical Investigation of the Optimal Frontier

6.4 A Further Investigation of Covariance

6.5 The Optimal Portfolio Problem Revisited

Chapter 7: The Capital Asset Pricing Model (CAPM)

7.1 Connecting the Portfolio Frontiers

7.2 The Tangent Portfolio

7.3 The CAPM

7.4 Applications of CAPM

Chapter 8: Risk Factor Modelling

8.1 General Factor Modelling

8.2 Theoretical Properties of the Factor Model

8.3 Models Based on Principal Component Analysis (PCA)

Chapter 9: The Value at Risk Concept

9.1 A Framework for Value at Risk

9.2 Investigating Value at Risk

9.3 Tail Value at Risk

9.4 Spectral Risk Measures

Chapter 10: Value at Risk under a Normal Distribution

10.1 Calculation of Value at Risk

10.2 Calculation of Marginal Value at Risk

10.3 Calculation of Tail Value at Risk

10.4 Sub-Additivity of Normal Value at Risk

Chapter 11: Advanced Probability Theory for Risk Managers

11.1 Moments of a Random Variable

11.2 The Characteristic Function

11.3 The Central Limit Theorem

11.4 The Moment-Generating Function

11.5 The Log-normal Distribution

Chapter 12: A Survey of Useful Distribution Functions

12.1 The Gamma Distribution

12.2 The Chi-Squared Distribution

12.3 The Non-central Chi-Squared Distribution

12.4 The F-Distribution

12.5 The t-Distribution

Chapter 13: A Crash Course on Financial Derivatives

13.1 The Black–Scholes Pricing Formula

13.2 Risk-Neutral Pricing

13.3 A Sensitivity Analysis

Chapter 14: Non-linear Value at Risk

14.1 Linear Value at Risk Revisited

14.2 Approximations for Non-linear Portfolios

14.3 Value at Risk for Derivative Portfolios

Chapter 15: Time Series Analysis

15.1 Stationary Processes

15.2 Moving Average Processes

15.3 Auto-regressive Processes

15.4 Auto-regressive Moving Average Processes

Chapter 16: Maximum Likelihood Estimation

16.1 Sample Mean and Variance

16.2 On the Accuracy of Statistical Estimators

16.3 The Appeal of the Maximum Likelihood Method

Chapter 17: The Delta Method for Statistical Estimates

17.1 Theoretical Framework

17.2 Sample Variance

17.3 Sample Skewness and Kurtosis

Chapter 18: Hypothesis Testing

18.1 The Testing Framework

18.2 Testing Simple Hypotheses

18.3 The Test Statistic

18.4 Testing Compound Hypotheses

Chapter 19: Statistical Properties of Financial Losses

19.1 Analysis of Sample Statistics

19.2 The Empirical Density and Q–Q Plots

19.3 The Auto-correlation Function

19.4 The Volatility Plot

19.5 The Stylized Facts

Chapter 20: Modelling Volatility

20.1 The RiskMetrics Model

20.2 ARCH Models

20.3 GARCH Models

20.4 Exponential GARCH

Chapter 21: Extreme Value Theory

21.1 The Mathematics of Extreme Events

21.2 Domains of Attraction

21.3 Extreme Value at Risk

21.4 Practical Issues

Chapter 22: Simulation Models

22.1 Estimating the Quantile of a Distribution

22.2 Historical Simulation

22.3 Monte Carlo Simulation

Chapter 23: Alternative Approaches to VaR

23.1 The t-Distributed Assumption

23.2 Corrections to the Normal Assumption

Chapter 24: Backtesting

24.1 Quantifying the Performance of VaR

24.2 Testing the Proportion of VaR Exceptions

24.3 Testing the Independence of VaR Exceptions

References

Index

For other titles in the Wiley Finance series please see www.wiley.com/finance

This edition first published 2012

© 2012 John Wiley & Sons, Ltd

Registered office

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. 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 or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.

Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

Library of Congress Cataloging-in-Publication Data

Hubbert, Simon.

Essential mathematics for market risk management / Simon Hubbert.—2nd ed.

p. cm.—(The Wiley finance series)

Includes bibliographical references and index.

ISBN 978-1-119-97952-4 (hardback)

1. Risk management—Mathematical models. 2. Capital market—Mathematical models. I. Title.

HD61.H763 2012

658.15′50151—dc23

2011039267

A catalogue record for this book is available from the British Library.

ISBN 978-1-119-97952-4 (hardback) ISBN 978-1-119-95301-2 (ebk)

ISBN 978-1-119-95302-9 (ebk) ISBN 978-1-119-95303-6 (ebk)

For my parents, Michelle and Nancy.

And dedicated to the memory of my brother, Craig.

Preface

The aim of this book is to provide the reader with a clear exposition of some of the fundamental mathematical tools and techniques that are frequently used in financial risk management. The book has been written with a wide audience in mind. For instance, it should appeal to numerate graduates who seek an accessible and self-contained account of the science behind the evolving story of financial risk management. In addition, it should also be of interest to the market practitioner who is interested in gaining a deeper understanding of the mathematical theory which underpins some of the most commonly used quantitative (black-box) techniques.

Most of the existing books devoted to financial risk management tend to fall into two categories, those that tackle a large number of topics with only brief overviews of the mathematical ideas (e.g., Hull (2007), Dowd (2002) and Jorion (2006)) and, on the other hand, rigorous mathematical expositions that are too advanced for an introductory level (e.g., McNeil, Frey and Embrechts (2005) and Moix (2001)). In view of this I have designed this book to occupy the middle ground, namely one that delivers an accessible yet thorough mathematical account of a broad sweep of carefully selected topics that an experienced risk manager is likely to encounter on a regular basis. In order to maintain focus I have devoted the book entirely to the mathematics of market risk management; there are already a whole host of excellent texts that cover the science of credit risk management, Bielecki, and Rutkowski (2010) and Schönbucher (2003) being excellent examples. The book, as its title suggests, is focused firmly on the essential mathematics of the subject and so, by design, it should equip the reader with the required scientific background to either embark on a rewarding career in risk management or to study the subject at a more advanced level. In particular, it is hoped that this text will serve as a useful companion to Alexander (2008a), Alexander (2008b) and Christoffersen (2003); three excellent books which place the emphasis firmly on practical examples and implementation.

The book itself has evolved from two courses on risk management that I teach regularly at Birkbeck, University of London. Both courses form part of a wider qualification in financial engineering, one at graduate diploma level and the other at masters level. The graduate diploma courses at Birkbeck are aimed at students who are familiar with basic calculus, linear algebra and probability theory, and they are designed to serve as a stepping stone to the more technically demanding masters level courses. Students who take this route invariably perform extremely well and, in view of this, the book represents a blend of introductory material (from the graduate diploma) and advanced topics (from the masters course). The field of market risk management is so vast that one could devote an entire textbook to several of its sub-branches (e.g., volatility modelling, simulation methods, extreme value theory) and thus I do not claim that this text represents an exhaustive account of state-of-the-art topics in this field. However, it is hoped that the book will inspire the reader to go on and investigate these topics in more depth.

It is a pleasure to thank the people who have helped make this book possible. I would like to acknowledge my colleagues Brad Baxter and Raymond Brummelhuis at Birkbeck for their support and encouragement. I also gratefully appreciate many of my past students for their valuable feedback on the structure and content of the book; special thanks go to Mafalda Alabort Jordan who provided many of the figures that appear in Chapter 19.

Chapter 2

Applied Linear Algebra for Risk Managers

Many of the problems in risk management are said to be high dimensional because they involve a large number of underlying variables. For instance, problems involving financial portfolios are high dimensional because a portfolio is made up of many financial assets and its value is determined by the monetary amounts that are invested in these assets. Applied linear algebra is the branch of mathematics that provides the framework needed to set up and solve these problems and, in this chapter, we present the most crucial results.

2.1 Vectors and Matrices