Q UANTITATIVE F INANCE V OLUME 1 (2001) 237–245 RESEARCH PAPER I NSTITUTE OF P HYSICS P UBLISHING quant.iop.org What good is a volatility model? Robert F Engle and Andrew J Patton Department of Finance, NYU Stern School of Business and Department of Economics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0508, USA Received 15 October 2000 Abstract A volatility model must be able to forecast volatility; this is the central requirement in almost all financial applications. In this paper we outline some stylized facts about volatility that should be incorporated in a model: pronounced persistence and mean-reversion, asymmetry such that the sign of an innovation also affects volatility and the possibility of exogenous or pre-determined variables influencing volatility. We use data on the Dow Jones Industrial Index to illustrate these stylized facts, and the ability of GARCH-type models to capture these features. We conclude with some challenges for future research in this area. 1. Introduction asset returns and consequently will pay very little attention to expected returns. A volatility model should be able to forecast volatility. Virtually all the financial uses of volatility models entail 1.1. Notation forecasting aspects of future returns. Typically a volatility First we will establish notation. Let Pt be the asset price at time model is used to forecast the absolute magnitude of returns, t and rt = ln(Pt )−ln(Pt −1) be the continuously compounded but it may also be used to predict quantiles or, in fact, the entire return on the asset over the period t − 1tot. We define the density. Such forecasts are used in risk management, derivative conditional mean and conditional variance as: pricing and hedging, market making, market timing, portfolio selection and many other financial activities. In each, it is the mt = Et−1[rt ] (1) predictability of volatility that is required. A risk manager h = E (r − m )2 must know today the likelihood that his portfolio will decline t t−1[ t t ] (2) in the future. An option trader will want to know the volatility where Et−1[u] is the expectation of some variable u given the that can be expected over the future life of the contract. To information set at time t −1 which is often denoted E[u|t−1]. hedge this contract he will also want to know how volatile Without loss of generality this implies that Rt is generated is this forecast volatility. A portfolio manager may want to according to the following process: sell a stock or a portfolio before it becomes too volatile. A R = m h ε , E ε = V ε = . market maker may want to set the bid–ask spread wider when t t + t t where t−1[ t ] 0 and t−1[ t ] 1 the future is believed to be more volatile. In this paper we are often concerned with the conditional There is now an enormous body of research on volatility variance of the process and the distribution of returns. Clearly models. This has been surveyed in several articles and the distribution of ε is central in this definition. Sometimes a continues to be a fruitful line of research for both practitioners model will assume: and academics. As new approaches are proposed and tested, it is helpful to formulate the properties that these models should {εt }∼i.i.d.F ( ) (3) satisfy. At the same time, it is useful to discuss properties that where F is the cdf of ε. standard volatility models do not appear to satisfy. We can also define the unconditional moments of the We will concern ourselves in this paper only with the process. The mean and variance are naturally defined as volatility of univariate series. Many of the same issues will 2 2 arise in multivariate models. We will focus on the volatility of µ = E[rt ],σ= E[(rt − µ) ] (4) 1469-7688/01/020237+09$30.00 © 2001 IOP Publishing Ltd PII: S1469-7688(01)21478-X 237 R F Engle andAJPatton Q UANTITATIVE F INANCE and the unconditional distribution is defined as 1.2. Types of volatility models There are two general classes of volatility models in (rt − µ)/σ ∼ G (5) widespread use. The first type formulates the conditional where G is the cdf of the normalized returns. variance directly as a function of observables. The simplest A model specified as in equations (1), (2) and (2) will examples here are the ARCH and GARCH models which will imply properties of (4) and (5) although often with considerable be discussed in some detail in section 3. computation. A complete specification of (4) and (5) however, The second general class formulates models of volatility does not imply conditional distributions since the degree of that are not functions purely of observables. These dependence is not formulated. Consequently, this does not might be called latent volatility or (misleadingly) stochastic deliver forecasting relations. Various models for returns and volatility models. For example, a simple stochastic volatility volatilities have been proposed and employed. Some, such specification is: as the GARCH type of models, are formulated in terms √ of the conditional moments. Others, such as stochastic rt = mt + νt εt volatility models, are formulated in terms of latent variables β νt = ωνt− exp(κηt ) which make it easy to evaluate unconditional moments and 1 ε ,η ∼ n( , ). distributions but relatively difficult to evaluate conditional t t i 0 1 moments. Still others, such as multifractals or stochastic ν structural break models, are formulated in terms of the Notice that is not simply a function of elements of the unconditional distributions. These models often require information set of past returns, and therefore it cannot be reformulation to give forecasting relations. the conditional variance or the one step variance forecast. Higher moments of the process often figure prominently Intuitively, this happens because there are two shocks and only ν in volatility models. The unconditional skewness and kurtosis one observable so that current and past are never observed are defined as usual by precisely. The conditional variance in this model is well defined but difficult to compute since it depends on a nonlinear 3 4 E[(rt − µ) ] E[(rt − µ) ] filtering problem defined as (2). ξ = ,ζ= . (6) σ 3 σ 4 Latent volatility models can be arbitrarily elaborate with structural breaks at random times and with random amplitudes, The conditional skewness and kurtosis are similarly defined multiple factors, jumps and fat-tailed shocks, fractals and 3 4 multifractals, and general types of nonlinearities. Such Et−1[(rt − mt ) ] Et−1[(rt − mt ) ] st = ,kt = . (7) models can typically be simulated but are difficult to estimate h3/2 h2 t−1 t−1 and forecast. A general first-order representation could be expressed in terms of a latent vector ν and a vector of shocks Furthermore, we can define the proportional change in η . conditional variance as √ h − h rt = mt + ν ,t εt = t t−1 . 1 variance return h (8) ν = f(ν , η ) t−1 t t−1 t ε ∼ G. Some of the variance return is predictable and some is an η innovation. The volatility of the variance (VoV) is therefore the standard deviation of this innovation. This definition is This system can be simulated if all the functions and directly analogous to price volatility: distributions are known. Yet the forecasts and conditional variances must still be computed. Many of the stylized facts = V( ). VoV variance return (9) about volatility are properties of the volatility forecasts so a model like this is only a starting point in checking consistency A model will also generate a term structure of volatility. h ≡ E r2 with the data. Defining t+k|t t [ t+k], the term structure of volatility is the forecast standard deviation of returns of various maturities, all starting at date t. Thus for an asset with maturity at time t + k, this is defined as 2. Stylized facts about asset price volatility k k ν ≡ V r =∼ E r2 . A number of stylized facts about the volatility of financial asset t+k|t t t+j t t+j (10) j=1 j=1 prices have emerged over the years, and been confirmed in numerous studies. A good volatility model, then, must be able The term structure of volatility summarizes all the forecasting to capture and reflect these stylized facts. In this section we properties of second moments. From such forecasts, several document some of the common features of asset price volatility specific features of volatility processes are easily defined. processes. 238 Q UANTITATIVE F INANCE What good is a volatility model? 2.1. Volatility exhibits persistence generally interpreted as meaning that there is a normal level The clustering of large moves and small moves (of either sign) of volatility to which volatility will eventually return. Very in the price process was one of the first documented features long run forecasts of volatility should all converge to this same of the volatility process of asset prices. Mandelbrot (1963) normal level of volatility, no matter when they are made. While and Fama (1965) both reported evidence that large changes in most practitioners believe this is a characteristic of volatility, the price of an asset are often followed by other large changes, they might differ on the normal level of volatility and whether and small changes are often followed by small changes. This it is constant over all time and institutional changes. behaviour has been reported by numerous other studies, such More precisely, mean reversion in volatility implies that as Baillie et al (1996), Chou (1988) and Schwert (1989).