What drives the shape of the volatility term structure, and how does mean reversion flatten it?

The volatility term structure describes how implied (or expected) volatility changes across different option maturities. Unlike the yield curve for interest rates, which is usually upward-sloping, the volatility term structure can take several shapes depending on market conditions.

Common Shapes

1. Flat — Volatility roughly constant across maturities. Typical in calm, low-news environments.
2. Upward-sloping (normal) — Short-term vol < long-term vol. Common when current vol is unusually low and expected to rise.
3. Downward-sloping (inverted) — Short-term vol > long-term vol. Common after a market shock when near-term uncertainty is high but expected to dissipate.
4. Humped — Vol peaks at some intermediate maturity.

Mean Reversion and the Term Structure

Mean reversion is the key force that shapes the long end. If volatility follows a mean-reverting process:

sigma_t = sigma_bar + (sigma_0 - sigma_bar) x e^{-kappa t}

Where kappa is the speed of mean reversion and sigma_bar is the long-run average volatility.

When current vol is HIGH (sigma_0 > sigma_bar):

The model predicts volatility will decline toward sigma_bar. This creates a downward-sloping term structure because:

• Short-dated options price in today's high vol
• Long-dated options price in the expectation that vol will revert lower

When current vol is LOW (sigma_0 < sigma_bar):

The model predicts volatility will rise. This creates an upward-sloping term structure.

Numerical Example

Lakefield Securities models equity vol with:

• Current vol: sigma_0 = 35% (elevated after earnings shock)
• Long-run vol: sigma_bar = 20%
• Mean reversion speed: kappa = 2.0 (fast reversion)

Expected vol at various horizons:

• 3 months: 20% + 15% x e^{-2 x 0.25} = 20% + 15% x 0.6065 = 29.1%
• 6 months: 20% + 15% x e^{-2 x 0.5} = 20% + 15% x 0.3679 = 25.5%
• 1 year: 20% + 15% x e^{-2 x 1.0} = 20% + 15% x 0.1353 = 22.0%
• 2 years: 20% + 15% x e^{-2 x 2.0} = 20% + 15% x 0.0183 = 20.3%

The term structure is steeply inverted at the short end and flattens toward the long-run level. Faster mean reversion (higher kappa) makes this convergence happen sooner.

FRM relevance: This directly affects how you calibrate GARCH and stochastic vol models, and how you interpret VIX term structure data.

For more on volatility modeling, check our FRM Part I course.