Apr 16

GARCH(1,1)

by David Harper, CFA, FRM, CIPM

FRM | |

IStock_000000351970Small

In a previous post we looked at different approaches to estimating current volatility. Aside from implicit (market implied) volatility, we divided historically informed volatilty into two types:

  • Unconditional volatility does not care about the time series. For example, “the volatility will have a mean of 10% and a standard deviation of 5%, regardless of yesterday's volatility.”
  • Conditional volatility cares about the time series. In other words, it is time-varying. For example, “tomorrow’s volatility will be very near today’s volatility, or at least it will be 94% weighted accordingly”

But unconditional volatility is not realistic. In fact, Linda Allen argues that conditional volatility is empirically observed and probably is the culprit behind fat-tailed asset returns.

We further parsed conditional volatility into an either an unweighted scheme (i.e., a simple average of recent variances) or a weighted scheme:

GarchOverview

Weighted schemes assign greater weights to more recent data points (variances). This is intuitive, but there is a judgement call. The trade-off is between sample size and recency. Larger sample sizes are better but they require more distant variances, which are less relevant. On the other hand, if we only use recent data, our sample size is smaller.

EWMA is a special case of GARCH(1,1)

The most popular approach to estimating current volatility is GARCH(1,1). GARCH stands for General Autoregressive Conditional Heteroskedasticity and GARCH(1,1) is the simplest flavor of GARCH(m,n). As Linda Allen says, GARCH is a "statistical time series model that enables [us] to model volatility as time varying and predictable." As shown in purple above, EWMA is a special case of the GARCH model; you can read more about exponentially weighted moving average (EWMA) here. 

The idea behind GARCH is that volatility is a function of lagged squared returns and lagged variances. In the illustration below, assume that (i-1) is the most recent period, like yesterday.

Then ui-1 is yesterday's return and sigmai-1 is yesterday's volatility (standard deviation). If we go back (lag) three periods, for example, then we have an historical series of squared periodic returns and variances (i.e., squared volatilities):

GarchOverview2 

If we add a constant, then we have a GARCH(m,n) model where (m) indicates the number of lagged squared returns and (n) indicates the number of lagged variances.

A GARCH(1,1) model lags on only one squared return and only one variance. That's the meaning of (1,1). But we need to weight the terms, and we need to give weight to the constant too. The constant is the long-run average variance; it exerts a gravitational pull on the time series. The more weight we assign to the long-run variance, the less "persistent" the time series and the more the time series is pulled toward the mean or exhibits a tendency to "regress to the mean."

GARCH(1,1) has three terms. Each is a weighting factor multiplied by, respectively, the long-run variance, a single lagged return squared, and a single lagged variance:

Garch_400w

Once more, the GARCH(1,1) model incorporates long-run variance (VL), the most recent squared return(u2), and the the most recent variance (sigma2). It weighs each of these and the weights must sum to one (gamma + alpha + beta = 1).

EWMA is a special case of GARCH(1,1)

With only two steps, GARCH(1,1) becomes EMWA:

  • We set gamma (i.e., the weight assigned to the long run variance) to zero, and the first term drops out. Note: this means the time series does not exhibit mean reversion.
  • We insist that alpha and beta sum to one ("unity").

The current volatility becomes:

Emwa

Note that

  • lambda replaces alpha, and
  • (1-lambda) replaces beta (since alpha+beta=1)

Comments

  1. Be the first to leave a comment!

Leave a Comment