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10 · Volatility models

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Time Series, lesson 10

Volatility models

Exponential weighting and the idea behind GARCH.

You know volatility clustering, rolling volatility, and the AR(1) pull back to an average.

Part 1. Weigh recent days more

A rolling window counts each day fully and then drops it. Instead, let every past day count, with a weight that shrinks by a fixed factor λ ("lambda") for each day back. Set λ and score it against the true volatility of the practice stock.

Fading weightsLive

A small λ makes yesterday count for a lot, so the estimate jumps. A λ near 1 spreads the weight over months, so it's smooth and slow. In between, it tracked the truth more closely than any plain window could, and no day ever falls off a cliff.

EWMA volatility

An exponentially weighted moving average of squared returns: σ²ₜ = λσ²ₜ₋₁ + (1 − λ)r²ₜ₋₁, read "today's variance is lambda times yesterday's variance, plus one minus lambda times yesterday's squared return". The day k back gets weight (1 − λ)λᵏ⁻¹.

Example 1.1

One update, with λ = 0.94

  1. σ yesterday = 1%·r yesterday = −3%

    Yesterday's estimate, and a big move yesterday.

  2. 0.94 × 1²+0.06 × 3²=0.94 + 0.54=1.48

    In %²: mostly the old estimate, plus 6% of the new squared move.

  3. √1.48=1.22%

    One −3% day lifts the estimate from 1% to 1.22%.

Your turn

With λ = 0.94, how much weight does yesterday's squared return get, and the one from 10 days ago?

Yesterday gets 1 − λ = 6%. Ten days back gets 0.06 × 0.94⁹ = 3.4%. Half of all the weight sits in the last 11 days, since 0.94¹¹ ≈ 0.5.

Part 2. Pulled back to normal

Keep going with Pro

You've done the first part. Pro opens the rest of this lesson, and every other one.

Still to come
  1. Pulled back to normal
  2. A worked problem
  3. The same thing, for real
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Next lessonForecasting honestlyOut-of-sample tests, walk-forward and the baseline to beat. Start lesson 11

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