Volatility Analysis: Measuring, Modeling, and Trading Uncertainty

Types of Volatility

• Historical volatility — realized standard deviation of past returns
• Implied volatility — market's expectation of future vol, embedded in option prices
• Forward volatility — expected vol between two future dates (extracted from options)
• Local volatility — vol that varies by price level and time (Dupire model)

The VIX Index (CBOE Volatility Index) measures 30-day implied volatility of S&P 500 options — often called the "fear gauge." The gap between implied and realized volatility is the market's compensation for uncertainty itself, and that gap is where most volatility strategies earn their edge.

Volatility Modeling

# GARCH(1,1) — the workhorse volatility model
from arch import arch_model

returns = prices.pct_change().dropna() * 100
model = arch_model(returns, vol='Garch', p=1, q=1)
result = model.fit(disp='off')

# Conditional volatility forecast
forecasts = result.forecast(horizon=5)
predicted_vol = forecasts.variance.iloc[-1].values

# EWMA — faster-reacting alternative
ewma_vol = returns.ewm(span=21).std() * np.sqrt(252)

GARCH captures volatility clustering — the empirical fact that high-vol days follow high-vol days. EWMA reacts faster but forgets faster; GARCH balances persistence with responsiveness. For risk applications, the choice matters less than the error model: always validate forecasts against realized vol, and re-estimate parameters as regimes shift rather than trusting a single fitted window.

Volatility Risk Premium

Historically, implied volatility exceeds realized volatility by 2-4 percentage points on average. This volatility risk premium is compensation for bearing vol risk.

Strategies that harvest this premium:

• Sell covered calls — earn premium from time decay
• Short strangles — collect premium from both sides (high risk)
• Put-write strategies — systematically sell puts on indices

Caution: Vol selling strategies have negative skew — small steady gains punctuated by large losses during vol spikes. The premium is real, but it is precisely the compensation for the tail risk you are taking. Size vol positions by what a 2-3 sigma spike would cost, and treat VIX-term-structure signals (contango vs backwardation) as regime filters rather than entry triggers.