3.3 Constant Volatility and the Square Root of Time Rule

Term structure volatility forecasts are forecasts of the volatility of / -day returns for every maturity h. Denote by rth the return over the next h days at time ?, so that approximately

r,h = In Pt+h - In P„

where P, denotes the price at time t (see footnote 2 in §1.1). Converting the forecasted variance V{rlh) for every h to a volatility gives the volatility term structure.

The underlying model for both equally weighted moving averages and EWMAs is a constant volatility model, so term structure volatility forecasts that are consistent with moving average models will be constant. Note that

= r,i + r,

M-1,1

/+A-l,b

so if the return process is independent and identically distributed (i.i.d.) with constant variance a2, taking variances of the above gives V(rth) - ho2. If there are A returns a year, then the number of / -day returns per year is , and so annualizing V(rlh) into a volatility gives

/ -day volatility = 100 ( / 2) = 100 ( 2) = 1-day volatility.

In general, if 1-period returns are i.i.d. then / -period standard deviations are just *Jh times the 1-period standard deviation: this is the square root of time rule. It is widely used, although it may not be supported by empirical observations.

Term structure volatility forecasts that are consistent with moving average models will be constant

The constant volatility assumption underlies both moving average statistical models and Black-Scholes type pricing models. In that sense the pricing model and the estimation model are coherent. However, constant volatility term structures are not generally observed in market implied volatilities (§2.2.2). Volatility term structures normally mean-revert, with short-term volatility lying either above or below the long-term mean, depending on whether current conditions are high or low volatility. It is unrealistic to assume that current levels of volatility will remain unchanged for ever. One of the main advantages of GARCH models is that their term structures reflect the mean-reversion of volatility that is expected in financial markets (§4.4.1).

If 1-period returns are i.i.d. then h-period standard deviations are just y/h times the 1-period standard deviation: this is the square root of time rule

GARCH Models

The moving average models of volatility and correlation that were discussed in Chapter 3 assume that asset returns are independent and identically distributed. There is no time-varying volatility assumption in any weighted moving average method. They only provide an estimate of the unconditional volatility, assumed to be a constant, and the current estimate is taken as the forecast. The volatility estimates do change over time, but this can only be ascribed to noise or sampling errors in a moving average model. There is nothing else in the model that allows for variation in volatility.

However, the returns in many financial markets are not well modelled by an independent and identically distributed process. At very high frequencies returns may show signs of autocorrelation, so they are not independent. Lower-frequency returns may not be autocorrelated, but there is often a strong autocorrelation in squared returns, and so again returns are not independent (§13.1.3).

Positive autocorrelation in squared returns indicates that financial market volatility comes in clusters where tranquil periods of small returns are interspersed with volatile periods of large returns. The technical term given to this is autoregressive conditional heteroscedasticity. This phenomenon has been documented for a very long time. As long ago as 1963, Benoit Mandlebrot observed that financial returns time series exhibit periods of volatility interspersed with tranquillity, where Large returns follow large returns, of either sign. However, it is only relatively recently that useful models of volatility clustering have been developed.

This chapter describes a framework for modelling time-varying volatility that has engendered a truly remarkable number of academic papers. Autoregressive conditional heteroscedasticity (ARCH) models of volatility and correlation were first introduced by Rob Engle (1982). Consequently a bibliography of research papers on GARCH during the last twenty years would run to hundreds of pages, so it is not an easy task to cut through all this literature to find the work that is most relevant to practitioners. Sometimes academic research seems to exist only to perpetuate more research in the area, but. of course, one never knows whether research that presently seems to lack real-world application will have practical relevance in the future. There is currently

Financial market volatility comes in clusters where tranquil periods of small returns are interspersed with volatile periods of large returns. The technical term given to this is autoregressive conditional heteroscedasticitx

At the present time the majority of practitioners do not use GARCH models at all, and if they do it is in a relatively basic form

so much academic interest in GARCH models that it is quite possible that most financial institutions will be using sophisticated GARCH models some time in the future. Nevertheless, at the present time the majority of practitioners do not use GARCH models at all, and if they do it is in a relatively basic form. Part of the problem is the divide between academic research and practical applications. This chapter aims to present some of the most relevant academic research on GARCH models in a form that is accessible to practitioners. Many empirical examples are provided in the text and the reader should use the MBRM-GARCH spreadsheet and the PcGive demo on the CD, to generate their own models for the data used in this chapter. The aim is to highlight some of the problems that are most important for practical applications and to help direct academic research towards the areas that have most practical relevance.

Section 4.1 describes the nature of GARCH models from both financial and statistical perspectives. Volatility clustering and the leverage effect are both accommodated in a GARCH framework by simply extending the linear regression model with another equation called the conditional variance equation. The section ends by emphasizing the tremendous scope of this framework for financial modelling. Section 4.2 outlines the mathematical specification of several univariate GARCH models - not all of them by any means, as this would take up a whole book on its own. There is a vast literature on different specifications of the conditional variance equation, so this section just gives a brief overview of some of the most common GARCH models. Their estimation by maximum likelihood methods is detailed in §4.3, which emphasizes the choice of data period, and the way it affects long-term volatility and the stability of GARCH parameters.

Section 4.4 looks at some of the most important practical applications of GARCH models. Although it has received relatively little attention in the academic literature, one of the most useful applications of GARCH models is the generation of volatility term structure forecasts that converge to a long-term average level of volatility as the maturity increases. In §4.4.1 we describe how to construct these forecasts, and how to relate their properties to the GARCH parameters. Univariate GARCH models are also used to price and hedge options. Some of the fundamental concepts for GARCH option pricing are discussed in §4.4.2. The section ends with an explanation of how to use GARCH models for smile fitting and forecasting.

The last part of this chapter examines some of the extensive research on multivariate GARCH models. The bivariate GARCH models described in §4.5.1 have found some very useful applications to the computation of time-varying hedge ratios and the pricing of options that are based on two correlated assets. Bivarate GARCH models are relatively easy to estimate, but GARCH models on more than two variables have been less successful. Section 4.5.2 reviews some different ways of parameterizing multivariate GARCH models: the computational aspects of multivariate GARCH become more and more problematic as the dimension increases. At the moment there is no