If one does not know whether the hedging volatility is accurate, it is better to use a higher value than a lower value

The standard error of the value of the portfolio (option plus hedge) is zero if ah = rj0. If the hedging volatility is correct the option can be perfectly hedged. But rj0 is unknown, and if the implied (hedging) volatility is not equal to the process volatility, the standard error of the portfolio value increases with the volatility error, as shown in Figure 5.7. It is not symmetric: if the process volatility is less than the hedging volatility then there is less uncertainty in the value of the hedged portfolio; if the process volatility is greater than the hedging volatility then there is more uncertainty in the value of the hedged portfolio. This means that if one does not know whether the hedging volatility is accurate, it is better to use a higher value than a lower value. That is, there will be less uncertainty in the value of the hedged portfolio if one over-hedges the position.

Part II

Modelling the Market Risk of Portfolios

The first two chapters in this part of the book introduce principal component analysis (PCA) and covariance matrices as key tools for the statistical analysis of market risk. Investment appraisal and the risk management of financial assets will, typically, be modelled by systems that have hundreds of different risk factors. The purpose of PCA is to reduce dimensions, so that only the most important sources of information are used, and to orthogonalize the variables, so that covariance matrices are diagonal. Chapter 6 begins with the traditional application of PCA to modelling term structures of interest rates and futures. Then some of the latest PCA applications, to models for implied volatilities and for overcoming data problems, are described.

Chapter 7 covers the use of covariance matrices in risk and investment analysis. An algebraic approach is taken to introduce the variance of a linear portfolio as a quadratic form of a positive semi-definite covariance matrix. Section 7.2 shows how covariance matrices are used to diversify investments and obtain minimum risk portfolios, and to generate the efficient frontier for optimal asset allocation. Optimal portfolios are determined by the risk attitude of investors, and so this section also contains a brief introduction to utility theory. The chapter concludes by describing methods for generating large covariance matrices, including a new orthogonal method that employs PCA. The linear theme is continued in Chapter 8, which examines fundamental factor models and their implications for diversifiable and non-diversifiable risk.

The first three chapters of this part provide the background for the comprehensive and detailed description of value-at-risk (VaR) models in Chapter 9. Following a summary of the Basel regulations for market risk capital, the chapter gives an overview of various alternative measures of risk. The core of this chapter is in the description of different VaR models and in the supporting examples from the text and on the CD. The chapter concludes with a brief account of VaR model validation and the application of stress testing and scenario analysis to daily risk management.

The final chapter in this part concerns the testing, modelling and application of non-normal distributions in financial markets. It begins with a description of

Part II: Modelling the Market Risk of Portfolios

standard statistical tests for skewness and kurtosis and a brief description of the extreme value and hyperbolic distributions. The main focus of this chapter is on normal mixture densities, their estimation and their application to risk measurement, option pricing and hedging. It makes concrete the ideas on modelling uncertainty in volatility that were introduced at the end of Part I.