Introduction: Modelling Stock Market Volatility—Bridging the Gap to Continuous Time

Tim Bollerslev; Peter E. Rossi

Until recent years, there has been a curious dichotomy between the sorts of time-series models favored for empirical work with economic and financial time series and the models typically used in theoretical work on asset pricing. Since its introduction in 1982 by Engle, empirical work with financial time series has been dominated by variants of the autoregressive conditionally heteroskedastic (ARCH) model. On the other hand, much of the theoretical development of contingent claims pricing models has been based on continuous-time models of the sort which can be represented by stochastic differential equations. The reason for these divergent paths is one of mathematical and statistical convenience. Discrete-time ARCH models have been favored over continuous-time models for statistical inference because of the relative ease of estimation. On the theoretical side, application of various “no arbitrage” conditions is most easily accomplished via the Itô differential calculus requires a continuous-time formulation of the problem. The purpose of this volume is to bring together work from a new literature which seeks to forge a closer link between discrete-time and continuous-time models along with new work on practical estimation methods for continuous-time models.

A great deal of the seminal work in this area is due to Dan Nelson of the University of Chicago. During an academic career cut short by cancer at 10 years, Dan completed a remarkably comprehensive and coherent research agenda which has provided a complete answer to many of the questions surrounding the relationship between discrete-time ARCH models and continuous-time stochastic volatility models. Thus, this volume serves, in part, to honor Dan and his remarkable achievements (see Bollerslev and Rossi, 1995, for a comprehensive bibliography of Dan’s work).

Discrete-time modelling of stock return and other financial time series has been dominated by models formulated to capture various qualitative features of this data. The ARCH model was introduced to capture the predictability of volatility or variance and the so-called volatility clustering as well as the thick-tailed marginal distribution of the time-series process. The single most widely used ARCH formulation is the GARCH(1, 1) model introduced by Bollerslev in 1986:

t=μt+σtzt{zt}i.i.d.,E[Zt]=0,Var(zt)=1;σt2=ω+α(σtzt)2+βσt−12.

This model allows for a smoothly evolving σt process as a function of past squared innovations. As a scale mixture model, the GARCH (1, 1) gives rise to a thick-tailed marginal distribution of the data. Application of GARCH models to financial data has produced two common empirical findings:

(1) The variance equation parameter estimates are often close to a unit root (for GARCH(1, 1), α + β = 1), especially when the models are fit to high frequency data.

(2) In spite of the scale mixing achieved by the ARCH model, nonnormal innovations are required to fit the most stock return series.

The first finding of unit roots caused a good deal of confusion in the early ARCH literature. From a forecasting perspective, the corresponding IGARCH model with α + β = 1.0 behaves like a random walk; i.e., t(σt+s2)=σt+12+(s-1)ω→∞ almost surely. However, as Nelson shows in “Stationarity and Persistence in the GARCH(1, 1) Model,” the persistence of a shock to the conditional variance should be very carefully interpreted. The behavior of a martingale can differ markedly from the behavior of a random walk. Strict stationarity and ergodicity of the GARCH(1, 1) model require geometric convergence of β+αzt2}, or [ln(β+αzt2)]<0, which is a much less stringent condition than arithmetic convergence, or [β+αzt2]<1, required for the model to be covariance stationary. Thus, IGARCH models can be strictly stationary models in which no second moments exist. This important insight also underlies subsequent work on the consistency and asymptotic normality of maximum-likelihood-based estimators for GARCH-type models.

Another problematic feature of the GARCH formulation is that parameter inequality restrictions are required to keep the variance function positive. This, coupled with the tendency of GARCH model outliers to be negative, led to the development of the EGARCH model introduced in Nelson’s two chapters, “Modelling Stock Market Volatility Changes” and “Conditional Heteroskedasticity in Asset Returns: A New Approach.” In the EGARCH model, (σt2) is parameterized as an ARMA model in the absolute size and the sign of the lagged innovations. For example, the AR(1)-EGARCH model is given by

(σt2)=ω+βln(σt-12)+θzt-1+γ[|zt-1|-E|zt-1|].

The log formulation enforces variance positivity and the use of both the absolute size and sign of lagged innovations gives the model some robust variance estimation characteristics. As in the GARCH(1, 1) model, γ, β = 0, large price changes are still followed by large price changes, but with θ < 0 this effect is accentuated for negative price changes, a stylized feature of equity returns often referred to as the “leverage effect.” Whether due to changes in leverage or not, a stock market crash is typically followed by a period of much higher volatility than a corresponding upward run in the prices, and the EGARCH model was an instant hit since it parsimoniously captured this phenomenon. In the short time since its introduction more than 100 empirical studies have employed the model, and EGARCH estimation is now also available as a standard procedure in a number of commercial statistical software packages.

While the univariate EGARCH model has enjoyed considerable empirical success, many interesting questions in financial economics necessarily call for a multivariate modelling approach. However, the formulation of multivariate ARCH models poses a number of practical problems, including parameter parsimony and positive definiteness of the conditional covariance matrix estimators. The bivariate version of the EGARCH model in Braun, Nelson, and Sunier ( Good News, Bad News, Volatility, and Betas"), designed explicitly to capture any “leverage effects” in the conditional β’s of equity returns, represents a particularly elegant solution to both of these problems.

The empirical finding that the persistence in volatility generally increases as the frequency at which the data is sampled increases suggests that it would be interesting to conduct a limiting experiment with ARCH models (obviously, the fact that stock prices are quoted in eighths means that this limiting experiment will ultimately break down at some intraday sampling frequency). In “ARCH Models as Diffusion Approximations,” Nelson establishes weak convergence results for sequences of stochastic difference equations (e.g., ARCH models) to stochastic differential equations as the length of the sampling interval between the observations diminishes. For instance, consider the sequence of GARCH(1, 1) models observed at finer and finer time intervals h with conditional variance parameters ωh = ωh, αh = α(h/2)1/2, and βh = 1 – α(h/2)1/2 – θh, and conditional mean h=hcσt2. Under suitable regularity conditions, the diffusion limit of this process equals

yt=cσt2dt+σtdW1,t,dσt2≃(ω-θσt2)dt+ασt2dW2,t,

where W1,t and W2,t denote independent Brownian motions. Similarly, the sequence of AR(1)-EGARCH models, with βh = 1 – βh and the other parameters as defined in this chapter, converges weakly to the

(ln(yt))=θσt2dt+σtdW1,t,d(ln(σt2))=-β(ln(σt2)-α)dt+dW2,t,

diffusion processes commonly employed in the theoretical options pricing literature.

The continuous-record asymptotics introduced by Nelson can be applied to a number of other issues that arise in the use of ARCH models. For instance, one could regard the ARCH model as merely a device which can be used to perform filtering or smoothing estimation of unobserved volatilities. Nelson’s insights are that, even when misspecified, appropriately defined sequences of ARCH models may still serve as consistent estimators for the volatility of the true underlying diffusion, in the sense that the difference between the true instantaneous volatility and the ARCH filter estimates converges to zero in probability as the length of the sampling frequency diminishes. Although the formal proofs for this important result as developed in “Filtering and Forecasting with Misspecified ARCH Models I” are somewhat complex, as with most important insights, the intuition is fairly straightforward. In...