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Report this Document. Flag for inappropriate content. Download now. Save Save Alexander, Market Models. Original Title: Alexander, Market Models. Related titles. Carousel Previous Carousel Next. The World Is Flat 3. Jump to Page. Search inside document. 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 under the terms of the Copyright, Designs and Patents Act or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher.
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.
The password for the CD is available from http:;:www. Vili Contents 3. It aims to provide a rigorous explanation of the theoretical ideas, but in practical and very clear terms. As concepts are introduced, real-world examples are provided in the text and, interactively, on the accompanying CD. This book aims to provide academi- cally acceptable answers to the questions that are really important for practitioners, It is written for a wide audience of practitioners, academics and students interested in the data analysis of financial asset prices.
It aims to help practitioners cut through the vast literature on financial market models, to focus on the most important and useful theoretical concepts. For academics the book highlights interesting research problems that are relevant to the day-to-day work of risk managers and investment analysts. For students, the comprehensive and self-contained nature of the text should appeal.
The book is divided into three parts: Part I: Volatility and Correlation Analysis covers the estimation and forecasting of volatility and correlation for the pricing and hedging of options portfolios. A detailed summary of the content is provided in the introduction to each part. At the end of the book a low-level technical appendix is included; this covers the basic statistical theory that 1s necessary for the book to be self-contained. However, it is not always easy to straddle the divide between academic research and the practice of risk management and investment analysis.
A common language, a common terminology and, above all, a common approach are necessary. It is hoped that this book will help to enhance the communication between these two schools.
In more than ten years of software design and development for financial institutions I have been the architect but not usually the implementer of the model. It would not have been possible to build many of the models presented in this book without the expertise provided by many mathematicians and financial engineers and the guidance of senior colleagues.
These include: Dr. Their contributions are acknowledged in various parts of this text, but here I would like to say a special thanks, to all of them, for their support. Steffen and Sujit have written excellent VBA code with graphical user interfaces for the spreadsheets that accompany each chapter. More details about Steffen and Sujit may be found on the CD.
The optimization based principal component analysis spreadsheets were kindly provided by my esteemed colleague at the ISMA Centre, Ubbo Wiersema. Many thanks also to Dr. Mamdouh Barakat of MB Risk Management, for tailoring Universal Excel Add-ins to some of the data used in the book and for allowing readers a limited free license for this software. Only two people apart from myself have read the entire manuscript: Richard Leigh and Jacques Pezier.
He made numerous contributions through constructive criticism and insightful feedback on most parts of the book. Any remaining oversights are my responsibility, but [ shall probably blame him nevertheless, for not catching them! With this aim in mind, a CD has been provided that contains examples of many of the models that are described in the text. Most of the chapters have an associated spreadsheet that illustrates how important parameters, such as volatility, and important quantities, such as option prices or value-at-risk, may be obtained using the models described.
These spreadsheets contain individual help files that explain their use, with references to the text that covers the technical background of the model. The reader may wish to use the programs as a basis for their own working models, but it should be stressed that the CD is provided free and for educational purposes only.
It is not guaranteed to work and no additional software or hardware support will be given to the user. Neither are the spreadsheets guaranteed to be free of errors. It is hoped that this forum will provide a means for users to exchange ideas on the aspects of model development that are covered in the text.
There are approximately figures in this book, and more than a few have had their glorious Technicolor suppressed by the confines of monochrome print. An appealing feature of the CD is that it contains the original colour versions of these figures and, in most cases, the supporting data. The CD includes free demonstration versions of commercial software that are particularly relevant to the subjects covered in the book, in addition to the Market Models spreadsheets mentioned above.
Doornik and David F. Wiersema reproduced with permission All material contained within this CD product is protected by copyright, whether or not a copyright notice appears on the particular screen where the material is displayed. No part of the material may be reproduced or transmitted in any form or by any means, or stored in a computer for retrieval purposes or otherwise, without written permission from Wiley, unless this is expressly permitted in a copyright notice or usage statement accompanying the materials.
The authors and Publisher expressly disclaim all implied warranties, including merchantability or fitness for any particular purpose. There will be no duty on the authors or Publisher to correct any errors or defects in the software. Part I Volatility and Correlation Analysis Part I provides insights into the pricing and hedging of options through the understanding of volatility and correlation, and the uncertainty which surrounds these key determinants of portfolio risk.
The first chapter introduces volatility and correlation as parameters of the stochastic processes that are used to model variations in financial asset prices. They are not observable in the market and can only be measured in the context of a model. Option pricing, which models asset prices in continuous time, is covered in Chapter 2.
This chapter focuses on the consequences of using the Black- Scholes model to price options. Although there can only be one true volatility for the underlying price process, different volatilities are implied by the market prices of options on the same underlying asset.
The relationship between underlying price changes and changes in the implied volatility of an option is analysed to support the use of different volatility assumptions for pricing and hedging. Statistical forecasts of volatility and correlation employ discrete time series models on historical return data. Chapter 3 explains how to obtain moving average estimates of volatility and correlation and outlines their advantages and limitations. A weighted average is a method for estimation. The current estimate of volatility or correlation is sometimes used as a forecast, but this requires returns to be independent and identically distributed, an assumption which is not always supported by empirical evidence.
Chapter 4 introduces generalized autoregressive conditional heteroscedasticity GARCH models, which are based on more realistic assumptions about asset price dynamics. This chapter aims to cut through a vast academic literature on the subject to present the concepts and models that are most relevant to practitioners.
A step-by-step guide to the implementation of the GARCH models that are commonly used by risk managers and investment analysts is followed by a description of the application of GARCH models to option pricing and hedging. However, a forecast is an expectation, taken under some probability measure, and the expectation of a square root is not equal to the square root of an expectation, The last chapter in this part of the book examines this and other key issues surrounding the use of volatility and correlation forecasts.
Quite different results can be obtained, depending on the model used and on the market conditions so, since volatility can only be measured in the context of a model, how does one assess the accuracy of a volatility forecast? Rather than employ point forecasts of volatility, this part of the book ends by advocating the use of standard errors, or other measures of uncertainty in volatility forecasts, to improve the valuation of options.
Part I introduces some challenging concepts that will be returned to later as further models are introduced. This is a vast subject that has been approached from two different technical perspectives.
On the one hand, the option pricing school models the variation in asset prices in continuous time; this perspective will be taken in Chapter 2.
On the other hand, the statistical forecasting school models volatility and correlation from the perspective of a discrete time series analyst; this is the approach used in Chapters 3 and 4. The basic concepts are introduced within a unified framework that, I hope, will be accessible to both schools. Some of these concepts are quite complex and their exposition has necessitated many footnotes and numerous pointers to other parts of the book.
First, volatility and correlation are described as parameters of stochastic processes that are used to model variations in financial asset prices. Then the differing needs of various market participants to assess volatility and correlation are examined. The needs of the analyst will Figure 1. Implied volatility and statistical volatility normally refer to the same process volatility, but volatility estimates often turn out to be quite different and because volatility can only be measured in the context of a model it is very difficult to assess the accuracy of estimates and forecasts.
The chapter concludes with remarks on the decisions about the data and the models that will need to be made when volatility and correlation forecasts are implemented. Financial asset prices are random variables, not deterministic variables.! Variations of financial asset prices over a short holding period are often assumed to be lognormal random variables. Therefore returns to financial assets, the relative price changes, are usually measured by the difference in log prices, which will be normally distributed.
The two density functions shown in Figure 1. The most common measure of dispersion is the standard deviation o of a random variable, that is, the square root of its variance. Each outcome is determined by a chance event, and so has a probability measure. If there is a strong negative correlation then upward movements in one series are associated with downward movements in the other.
A simple statistical measure of co-movements between two random variables is covariance, the first product moment about the mean of the joint density function. Covariance is determined not only by the degree of co-movement but also by the size of the returns.
Since covariance is not independent of the units of measurement, it is a difficult measure to use for comparisons. It is better to use the correlation, which is a standardized form of covariance that is independent of the units of measurement. Normalizing the covariance as we have in 1. High negative correlation indicates that the returns are still highly co-dependent, but they tend to move in opposite directions.
If two random variables are statistically independent then a good estimate of their correlation should be insignificantly different from zero. We use the term orthogonal to describe such variables. However, the converse is not true. That is, orthogonality zero correlation does not imply independence, because two variables could have zero covariance and still be related the higher moments of their joint density function could be different from zero.
In financial markets, where there is often a non-linear dependence between returns, correlation may not be an appropriate measure of co-dependency. Comparison of the ordinary least squares OLS formula with 1. Thus correlation is only a linear measure of association. If a regression line were fitted to the data in Figure 1. We tend to use parameter notation because it is concise. Fitting a line to the data in Figure 1. Correlation is a limited measure of dependency. Very often correlation estimates in financial markets lack robustness?
A copula is a function of several variables: in fact it is a multivariate uniform distribution function. If u,, Copulas are used to combine marginal distributions into multivariate distributions.
They are unique: for any given multivariate distribution with continuous marginal distributions there is a unique copula that represents it. They are also invariant under strictly increasing transformations of the marginal distributions. Copulas have long been recognized as a powerful tool for modelling dependence between random variables. A useful general reference text on copulas is Nelsen Here are some simple examples of copulas: i CQ,.
Uy 11 C u,.. Copula ii corresponds to counter-monotonic dependency, which is similar to negative correlation. Copula iii corresponds to co-monotonic dependency, which 1s similar to positive correlation.!! There it is argued that return data have all the memory taken out of them before the analysis even begins. So return data can only be used to pick up short-term associations between returns series.
To investigate the possibility of any long-run associations it is necessary to use a long-memory model, such as a cointegration analysis on the price series. Understanding Volatility and Correlation 9 4 a e.
Highly dependent returns and a high correlation; b low correlation. In the last few years copulas have been used as a powerful tool in financial risk management Embrechts et al. They have important applications to the aggregation of individual loss distributions into an overall loss distribution, particularly when correlation is difficult to assess, as it is, for example, in operational risk measurement. Thus the concerns of a portfolio manager focus not on the total volatility of a portfolio, but on the volatility that is collinear with the market.
Of course, if the relative volatility of the portfolio is very high it can have a large irreducible risk even when the correlation with the market is low as long as it is positive. Similarly, what matters for pricing an option is only the volatility of underlying price movements and not the trend in prices.
Whatever the trend in an asset price, an option position can be hedged by the proper position on the underlying asset. Thus the estimation and forecasting of volatility and correlation is at the heart of financial risk modelling: x» Traders writing options need to forecast the volatility of the price process over the lifetime of the option.
Implied volatility is the volatility forecast over the life of an option that equates an observed market price with the model price of an option. This implied volatility is the volatility of the geometric Brownian motion process that is assumed to govern price variations from now until the option matures, that will equate the model price with the market price Chapter 2.
In that sense it is more accurate to refer to the Black— Scholes implied volatility, or for short, the Black-Scholes volatility. Search this site. It introduces the econometric techniquesthat are commonly applied to finance with a critical and selectiveexposition, emphasising the areas of econometrics, such as GARCH,cointegration and copulas that are required for resolving problemsin market risk analysis.
The book covers material for aone-semester graduate course in applied financial econometrics in avery pedagogical fashion as each time a concept is introduced anempirical example is given, and whenever possible this isillustrated with an Excel spreadsheet. All together, the Market Risk Analysis four volume setillustrates virtually every concept or formula with a practical,numerical example or a longer, empirical case study.
Across allfour volumes there are approximately numerical and empiricalexamples, graphs and figures and 30 case studies many of whichare contained in interactive Excel spreadsheets available from thethe accompanying CD-ROM.
Building on the three previous volumes this book provides by far the most comprehensive, rigorous and detailed treatment of market VaR models.
Risk Measurement in. Better then never, though i am quite late in start reading this one. Book Description. File sharing network. File upload progressor. It rests on the basic knowledge of financial mathematics and statistics gained from Volume I, of factor.
Chan H. Contents Preface About the. Spot and Futures. It rests on the basic knowledge of financial mathematics and statistics gained from Volume I, of factor models, principal component analysis, statistical models of volatility and correlation and copulas from Volume II and, from Volume III, knowledge of pricing and hedging financial instruments and of mapping portfolios of similar instruments to risk factors.
A unifying characteristic of the series is the pedagogical approach to practical examples that are relevant to market risk analysis in practice. All together, the Market Risk Analysis four volume set illustrates virtually every concept or formula with a practical, numerical example or a longer, empirical case study.
Across all four volumes there are approximately numerical and empirical examples, graphs and figures and 30 case studies many of which are contained in interactive Excel spreadsheets available from the the accompanying CD-ROM. Empirical examples and case studies specific to this volume include: Parametric linear value at risk VaR models: normal, Student t and normal mixture and their expected tail loss ETL ; New formulae for VaR based on autocorrelated returns; Historical simulation VaR models: how to scale historical VaR and volatility adjusted historical VaR; Monte Carlo simulation VaR models based on multivariate normal and Student t distributions, and based on copulas; Examples and case studies of numerous applications to interest rate sensitive, equity, commodity and international portfolios; Decomposition of systematic VaR of large portfolios into standard alone and marginal VaR components; Backtesting and the assessment of risk model risk; Hypothetical factor push and historical stress tests, and stress testing based on VaR and ETL.
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