0.06

Volatility smile surfaces can change so quickly from day to day that it may be misguided to expect stability in parameter estimates of a GARCH model that is based on snapshots of option prices from the market

0 -0.04

0.02

Figure 4.12 GARCH smile of the FTSE 100 index on 2 March 1998.

Table 4.10: N-GARCH parameter estimates from two different days

Interestingly, both these observations could have been made from the time series models of the FTSE 100 GARCH volatilities in Figures 4.7e and 4.10b. At this rather general level the GARCH models estimated on snapshots of option market data are corroborated by the GARCH estimates based on time series of the underlying. However, volatility smile surfaces can change so quickly from day to day that it may be misguided to expect stability in parameter estimates of a GARCH model that is based on snapshots of option prices from the market.

4.5 Multivariate GARCH

This section begins by discussing the main use of time-varying correlations, which is to construct short-term hedge ratios that accurately reflect current market conditions. Then it reviews some of the most common multivariate GARCH models, and discusses the computational problems that are inevitable if one attempts direct estimation of full multivariate GARCH models for large-dimensional systems. The last part of this section describes how to deal with

the complications that arise when using GARCH to build the large covariance matrices that are necessary for netting the risks from all positions in a large trading book.

4.5.1 Time-Varying Correlation

Computational problems are inevitable if one attempts direct estimation of full multivariate GARCH models for large-dimensional systems

In §1.4 the distinction was drawn between unconditional and conditional correlation. Conditional correlation is a time-varying parameter: there is not one true value for the parameter, as so many economic models would assume. Of course, all correlation estimates will change over time, whether the model is based on constant correlation or time-varying correlation. In a constant correlation model this variation in the estimates with time is only due to sampling error, but in a time-varying parameter model this variation in the estimates is also ascribed to changes in the true value of the parameter.

Conditional correlations can be estimated by any -dimensional GARCH model (n > 1), but estimates of the same conditional correlation parameter can be quite different, depending on the model chosen. In this section only bivariate GARCH models are discussed. Like univariate GARCH models, bivariate GARCH models normally converge rapidly, in fact the only problems that might arise are lack of proper specification by the user, or the use of inappropriate data (e.g. there may be problems if the data generation processes are not jointly stationary).

The simplest parameterization of a bivariate GARCH(1,1) model is the diagonal vech parameterization.26 In addition to the two conditional mean equations, one for each return, it has the conditional variance equations:

P>i.,-i

co2 + a2E2,, i >2 2,/-1 0"i2,, = +a36u 1E2i( 1 +P3a12,, i.

(4.21)

where £] and e2 are the unexpected returns from the two conditional mean equations. This model is fairly restrictive on the dynamics of conditional correlation. For example, yesterdays variances o2, ] and o\t { do not enter the equation for todays covariance, so the model does not capture the increase in correlation that often accompanies increased volatilities. This is a severe limitation, and additional constraints on the coefficients in (4.21) are necessary to ensure positive definiteness of the covariance matrices.

The bivariate diagonal vech model (4.21) is just one of a large number of different parameterizations of the same bivariate normal GARCH(1,1) model. These parameterizations will be discussed in §4.5.2 and §4.5.3. For the moment

6The vech operator stacks all columns of a matrix into a column vector. Thus if the columns of X are X], . . ., \n then vech(X) is the column vector that stacks x, above X2 above... above x„.

the only point to make is that when a different parameterization is used for the three GARCH(1,1) equations, the estimates of the same time-varying correlation will be different. Figure 4.13 compares diagonal vech correlation estimates with orthogonal GARCH and BEKK correlation estimates (these are described in §4.5.2 and §7.4). Figure 4.13a does not include the BEKK correlation because there were difficulties in convergence. In fact the convergence problems for univariate GARCH models that were outlined in §4.3.3 are even more substantial in multivariate GARCH.

If the model does converge then the mean-reverting term structure forecasts that have been described for univariate GARCH models in §4.4 can be extended to correlation. Simply iterate the GARCH conditional variance and covariance forecasts27 and then take the sum over the next h days. These / -day variance and covariance forecasts are then converted to correlation term structures using

Pt.h - a12,/,/i/°"l,(,/i°"2,(,/i-

Correlation-dependent market parameters, such as market betas or statistical hedge ratios, may also be regarded as time-varying. In §8.1 the market sensitivity p in a CAPM model is defined as the ratio of the covariance between the market return X and the asset return to the variance of the market. It can also be written 3 = pv, where v denotes the relative volatility and p is the correlation (cf. (1.5)). The definition of a conditional market beta may be derived from the definitions of conditional covariances and variances in the natural way. A time-varying beta is defined as

P, = °A>,,/oi,« = P,V,

where, at time t, oXyl is the conditional covariance, o\, is the conditional variance of the market, p, is the conditional correlation and v, is the ratio of conditional volatilities.

In §8.2.1 we note that, for risk management purposes, it is better to use estimates of market betas that are based on recent daily data, so that they respond to current market conditions, rather than the equally weighted estimates that are calculated when a CAPM is estimated by ordinary least squares using monthly data over several years (§A.1.4). In the covariance VaR models described in §9.3 the net betas for each risk factor should be estimated from models that assume that the true beta is not constant over time. As a rule the time-varying beta estimates of any stock will change rapidly and frequently. It is normal for a time-varying beta to oscillate between values much less than 1 and much more than 1 in the space of a few days. This casts some doubt on the wisdom of categorizing stocks into high and low risk according to their ordinary least squares market betas.

27Note that cov,(-Ri ,j,,R2.,./,) = S,/=i covr(ri,i+/ ru+j)- Ignoring non-contemporaneous covariances gives

CT12.!,/i = S/=i 12. -

Correlation-dependent market parameters, such as market betas or statistical hedge ratios, may also be regarded as time-varying

This casts some doubt on the wisdom of categorizing stocks into high and low risk according to their ordinary least squares market betas