Chapters 3 and 4). Applying the model to historical data will generate statistical estimates of volatility for the past, where historical data are available. It will also generate forecasts of volatility from now until some future point in time, called the risk horizon. It is convenient to present the statistical estimates (or forecasts) of volatility and correlation between all asset (or risk factor) returns in a portfolio in the form of a covariance matrix (Chapter 7).

Unlike prices, volatility and correlation are not directly observable in the market. They can only be estimated in the context of a model. It is important to understand that implied and statistical volatility models normally provide estimates or forecasts of the same thing - that is, the volatility parameter in some assumed underlying price process.12 The volatility of the stochastic process that governs price movements (or equivalently returns) is called the process volatility.

In practice there can be substantial differences between implied and statistical forecasts of the process volatility; there are many reasons for this, some of which are discussed in §2.1.3.

Our knowledge of the process volatility will depend on the model of the price process. The Black-Scholes model (§2.1.1) assumes that the underlying price process is a geometric Brownian motion, which has a constant volatility. More advanced price process models may assume other diffusion or jump models where the process volatility is stochastic. When volatility is stochastic it may have its own diffusion or jump model; so the price process model may have two random factors, one to represent the price process for a given volatility and another to represent the volatility process. Often we seek to represent the volatility process and, in particular, its dependence on the underlying price through time, by a volatility surface (§2.2.3). In multi-factor models, this dependence - and, in particular, the correlation between price changes and volatility changes - will be a key determinant of option prices; empirical modelling of the relationship between prices and volatility will be discussed in §2.3 and §6.3.

Unlike prices, volatility and correlation are not directly observable in the market. They can only be estimated in the context of a model

Realized volatility is a realization of the process volatility. It can be measured using historical price data. For example, if the price process is one of constant volatility then the realized volatility is the sample standard deviation of the observed returns.13 If the price process has a time-varying volatility that is governed by a GARCH model, then the realized volatility is the GARCH volatility that is estimated over the historical data period. Realized volatility, the ex-post estimate of the process volatility, is a very difficult thing to forecast

- is normally assumed that underlying asset returns are generated by a stationary stochastic process. In a stochastic process the return that is observed at time / is an observation on a density/(r,; 9,) where 6, are the parameters of the density function at time t and the functional form ) is the same throughout the process. The term stationary is defined in §11.1.2.

1? That is. s - v/[(r, - )"/{n - 1)], where r is the average return over the sample of size n. Often we assume = 0 so the standard delation has the unbiased estimate (see §3.1.1).

ex ante; it will be greatly affected by an extreme market movement occurring at any time up to the risk horizon of the model.

Remember, volatility can only be observed in the context of a model. Most implied volatilities are based on the Black-Scholes model, and equally weighted moving average models are still the most common method of estimating statistical volatilities and correlations. These models are based on similar assumptions, but they are not very good models. The assumptions upon which these models are based just do not hold in practice. Therefore, much of the discussion in Part I of the book will focus on the implications of using the wrong model for volatility.

1.3 Constant and Time-Varying Volatility Models

Constant volatility models only refer to the unconditional volatility of a returns process. This is a finite constant a, the same throughout the whole data generation process. It can be defined in terms of the variance parameter of the unconditional distribution of a stationary returns process. In fact, unconditional volatility is only defined if one assumes that the asset return series is generated by a stationary stochastic process (§11.1.2), but this assumption seems far more reasonable than many other assumptions that are commonly made in financial models.

One of the properties of a stationary series is that it has a finite unconditional variance a2. To understand what this means, suppose one were to take together all the returns that were observed over some historic period, forgetting about any dynamic ordering. Consider a single density function that could have generated them (some idea of this density function may be obtained by plotting the histogram of the observed returns). This density is called the unconditional density, and its associated distribution is the unconditional distribution of the return process. The variance of this distribution is the unconditional variance and its square root is the unconditional volatility.14

Time-varying volatility models describe a process for the conditional volatility. A conditional distribution, in this context, is a distribution that governs a return at a particular instant in time15 and the conditional volatility at time t is the square root of the variance of the conditional distribution at time /. The conditional mean at time t is denoted E,(r,) or \i, and the conditional variance at time t is denoted V,(r,) or a2. An estimation procedure for the time-varying parameters of the conditional distributions is based on a model where anything that has happened in the past is not considered to be an observation on the

14 The unconditional mean and variance operators are written £() and K(). The unconditional mean is denoted E(rt) or and the unconditional variance is denoted V(rt) or a2.

15 In more general terms, a conditional distribution is any distribution that is conditioned on a set of known values for some of the variables, that is, on an information set. In time series models the information set at time t, I,, is often taken as all the past values that were realized in the process.

Suppose one were to take together all the returns that were observed over some historic period. Consider a single density function that could have generated them. The variance of this distribution is the unconditional variance

0.25 0.2 0.15 0.1 0.05

0 \ -0.05 -0.1 -0.15 (a) -0.2

Conditional variance is the same throughout the process

0.251 Conditional variance is

lower during tranquil 02 { i times than during . . , volatile times

U.I 0.1 0.05 0

-0.05 -0.1 -0.15 (b) -0.2 J

Figure 1.4 The assumption of (a) constant and (b) time-varying volatility.

current random variable. Its value is known, and so past observations become part of the information set. That is, the actual rather than the expected values of anything that happened in the past will be used to estimate the current value of a time-varying volatility parameter. Put another way, the current (and future) conditional distributions of the random variable will be conditioned on the current information set.16

Figure 1.4 illustrates the distinction between constant and time-varying volatility models. The majority of time-varying volatility models assume that returns are normally distributed, in which case each conditional distribution is completely determined by its conditional mean and its conditional variance. Both the conditional mean and the conditional variance could change at every

The actual rather than the expected values of anything that happened in the past will be used to estimate the current value of a time-varying volatility parameter

\ simple example to illustrate the difference between the conditional and the unconditional mean and variance is given b\ the AR(1) model y, = aytt -f e,, where the e, are independent and identically distributed with mean 0 and variance rr. denoted e, ~ i.i.d (0, n2). The conditional mean at time tis E,(ay: ) + E,(e,) = ay, , and the conditional variance is Vt(ay, {)+ H,(e,) = a2 because y, t is known at time (. In §11.1.2 it is shown that 1 = 0 and V(y,) = rr2/(l - a2).