Volatility — Realized, Implied, & VIX

Medium3–4 hrsVolatilityImplied VolVIXVol SurfaceGARCH

Overview

Volatility is the most important parameter in options pricing — and a tradeable asset in its own right. The key distinction: realized vol is backward-looking (what happened), implied vol is forward-looking (what the market expects). The 'vol risk premium' — implied vol persistently exceeds realized vol — is one of the most robust phenomena in finance. This is why selling options generates positive expected returns, but with large tail risk.

How to Recognize

→'What does it mean when VIX spikes?'
→Questions about vol surface, skew, and term structure
→Comparing implied vol to realized vol
→How do traders make money on volatility?

Step-by-Step Approach

1.Realized vol: historical std of log returns × √(annualization factor)
2.Implied vol: reverse-engineer σ from observed option price via Black-Scholes
3.VIX: forward-looking 30-day implied vol of S&P 500 (fear gauge)
4.Vol skew: OTM puts have higher IV than calls (crash protection demand)

Key Formulas

\sigma_{realized} = \sqrt{252} \times \text{std}(\log(S_t/S_{t-1}))

02Rule: daily move

\approx \sigma_{annual}/16

(since

\sqrt{252}\approx 16

)

03Vol of vol (VVIX): measures uncertainty about future implied vol

04Vol risk premium:

E[\sigma_{implied}] > E[\sigma_{realized}]

historically

Worked Examples

0Quick Example

VIX=20 means market prices in ±1.25% S&P daily move (20%/16). VIX=40 means ±2.5% daily moves are 'priced in'.

Problem

Daily returns over 5 days: +1%, −0.5%, +2%, −1%, +0.5%. Compute the 5-day realized vol and annualize it.

Solution

Log returns ≈ simple returns for small moves: {0.01, −0.005, 0.02, −0.01, 0.005}.

Mean = (0.01−0.005+0.02−0.01+0.005)/5 = 0.02/5 = 0.004.

Variance = Σ(rᵢ − mean)² / (n−1) = [(0.006)²+(−0.009)²+(0.016)²+(−0.014)²+(0.001)²]/4.

= [0.000036+0.000081+0.000256+0.000196+0.000001]/4 = 0.000570/4 = 0.0001425.

Daily std = √0.0001425 = 1.19%. Annualized = 1.19% × √252 ≈ 18.9%.

Answer

Realized vol ≈ 18.9% annualized. Quick path: daily std ≈ 1.2%, × 16 = 19.2% (close enough).

Common Mistakes

!
Annualizing vol by × 252 instead of × √252. Variance scales linearly, vol scales with √time.
!
Confusing implied vol (from options) with realized vol (from historical returns). They measure different things.
!
Thinking VIX=20 means '20% annual return volatility expected' — it means 20% annualized implied vol, which translates to ~±1.25% per day.

Practice Problems

Click "Show Answer" to reveal

VIX = 30. What daily move is 'priced in' by the options market?

Hint: Daily vol = annual vol / √252 ≈ annual vol / 16.

A stock has IV=25% for ATM options. You believe the stock will realize only 15% vol over the next month. What strategy would you implement?

Hint: Implied vol > realized vol → sell options (collect the premium).

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