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Compare the different methods for estimating conditional volatility, noting how they are alike and how they differ.

FRM® Part I · Valuation and Risk Models · Measuring and Monitoring Volatility · FRM1.VR.3.d

The idea

Conditional volatility can be estimated several ways, and all of them rest on the same idea: tomorrow's variance is a weighted average of the squares of past returns, so the methods differ mainly in the weights. An equally weighted rolling window gives every observation in the window the same weight and zero outside, so estimates jump when a large return enters or leaves, and the window length trades noise against responsiveness. EWMA weights decline geometrically by a decay factor λ, so recent data count most and shocks fade smoothly, but there is no long-run level to revert to. GARCH(1,1) also has declining weights and adds a constant, so variance reverts toward a long-run value, at the cost of estimating three parameters by maximum likelihood. EWMA is the special case of GARCH in which ω is zero and α + β is 1. Implied volatility is the forward-looking alternative.

A practice question

From the question bank, with the reason each wrong option is wrong.

In a rolling 250-day equally weighted estimate of volatility, what happens on the day a single very large return leaves the window?

  1. The estimate rises, because the window now contains fewer observations.Why not: Wrongly assumes dropping an observation raises volatility; the removed large squared return lowers the average.
  2. The estimate declines smoothly because older returns carry exponentially smaller weights.Why not: Describes EWMA weighting, not equal weighting, in which every window observation carries the same weight until it drops out.
  3. The estimate drops abruptly, even though nothing notable happened in the market that day. - correct
  4. The estimate is unchanged, because the sample mean absorbs the shock.Why not: Confuses the effect on the mean with the effect on variance; the squared deviation is removed from the sum.

An equally weighted window gives each of the last N returns weight 1/N and the rest zero, so a large shock inflates the estimate for exactly N days and then disappears at once. EWMA and GARCH fade shocks gradually.

A harder one

Which GARCH(1,1) parameter set is identical to an EWMA model with lambda = 0.94, where GARCH variance = omega + alpha x u² + beta x previous variance?

  1. omega = 0, alpha = 0.94, beta = 0.06Why not: Swaps alpha and beta; lambda multiplies the previous variance, so it is beta.
  2. omega = 0.06, alpha = 0.94, beta = 0Why not: Puts the weights in the wrong places and adds a constant that EWMA does not have.
  3. omega = 0.00001, alpha = 0.06, beta = 0.94Why not: Keeps a positive constant, but EWMA requires omega = 0, and with persistence 1 a positive omega makes variance explode.
  4. omega = 0, alpha = 0.06, beta = 0.94 - correct

EWMA: variance = 0.94 x previous variance + 0.06 x u². Matching terms gives beta = lambda = 0.94, alpha = 1 - lambda = 0.06, and omega = 0, so persistence alpha + beta equals 1.

The hardest tier

Over a long history every daily return has an absolute size of 1% except one, a loss of 5% that occurred 20 trading days ago (the 20th most recent return). Zero means are assumed. Estimate the volatility with a 250-day equally weighted window and with EWMA using lambda = 0.94, where the return k days back carries weight (1 - lambda) x lambda^(k-1). Which pair is closest?

  1. EWMA 1.20% and equally weighted 1.05% - correct
  2. EWMA 1.05% and equally weighted 1.20%Why not: Swaps the two; the shock is 20 days old and carries 0.4% weight in the window but about 1.9% weight in the EWMA.
  3. EWMA 1.56% and equally weighted 1.05%Why not: Treats the shock as yesterday's return and gives it the full (1 - lambda) weight of 0.06.
  4. EWMA 1.20% and equally weighted 1.00%Why not: Dismisses the shock in the window as negligible; it still adds 24/250 to the variance.

Equally weighted variance = (249 x 1 + 25) / 250 = 1.096, volatility 1.05%. The EWMA weight on the shock is 0.06 x 0.94^19 = 0.0185, so variance = 1 + 0.0185 x 24 = 1.444, volatility 1.20%. EWMA reacts more because it weights recent data more heavily.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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