Most statistical models for forecasting volatility are actually models for forecasting variance: the volatility forecast is taken as the square root of the variance forecast. 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. For example, the principal component models of implied volatility in §6.3, the orthogonal method for generating covariance matrices in §7.4 and the normal mixture density models of §10.3 will all continue the exposition of ideas that are introduced in Part I.

Understanding Volatility and Correlation

This chapter introduces some of the concepts that are fundamental to the analysis of volatility and correlation of financial assets. 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 1.1 Volatility and scale.

determine whether an option pricing (implied volatility) approach or a statistical modelling (covariance matrix) approach is required (or both). 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.

1.1 The Statistical Nature of Volatility and Correlation

Financial asset prices are observed in the present, and will have been observed in the past, but it is not possible to determine exactly what they will be in the future. Financial asset prices are random variables, not deterministic variables.1 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.2

Volatility is a measure of the dispersion in a probability density. The two density functions shown in Figure 1.1 have the same mean but the density function indicated by the dotted line has greater dispersion than the density indicated by the continuous line.3 The most common measure of dispersion is the standard deviation a of a random variable, that is, the square root of its variance.

1 A random variable, also called a stochastic variable or variate, is a real-valued function that is defined over a sample space with probability measure. A value x of a random variable X may be thought of as a number that is associated with a chance outcome. Each outcome is determined by a chance event, and so has a probability measure. This probability measure is represented by the probability density function of the random variable. For any probability density function g(x), the corresponding distribution function is defined as G(x) = Prob(X<x) - g(x) dx. It is not necessary to specify both density and distribution: given the density one can calculate the distribution, and conversely since g(x) = G\x).

" The normal density function (. ) is defined by two parameters, the mean and the variance a2: (. ) = ((2 2 /2) (\( - \i)2/a2) for - oo < x < oo. This gives the familiar symmetric bell-shaped curve, which is centred on the mean u and has a dispersion that is determined by the variance a2.

A random variable is said to be lognormally distributed when its logarithm is normally distributed. A lognormal density function is not symmetrical; it is bounded by zero on the low side but can, in theory, reach infinitely high values. For this reason it is commonly assumed that financial assets (bonds and shares) and possibly commodity prices are better represented by lognormal than by normal variates. Conversely, investors compare financial assets on the basis of their returns; it is therefore returns that are comparable whatever the price of the underlying asset, and it is simplest to assume that returns arc normally distributed. It follows that the price is lognormally distributed; indeed if r, = (P, - , ,)/ , , is normally distributed then = 1 + r, and ,/ , ,) « r,

(note that when is small, ln(l + .v) x). Therefore ln(P,/P0) is normally distributed and P,/Po is lognormally distributed. Note that this argument is based on investment assets and would not apply to interest rates. The argument has also shown that the return over small time intervals is approximated by the first difference in the log prices.

3 If a random variable X has density function fix) then its mean is u = E(X) = J xf\x)dx. The mean is like the centre of gravity of a density. It is a fundamental parameter of any density, the parameter that describes the location of the density. It is also called the first moment of the density function. The variance is a2 = V(X) = \{x - u)"/(.y)(/.v = E(X2) - [E(X)]1. This parameter measures the dispersion of the density function about the mean. It is also called the second moment about the mean of the density function.