Historical Volatility (HV)
The realized standard deviation of the asset's returns over a historical lookback window, typically annualized.
Historical (realized) volatility is computed as σ = std(log returns) × √252 for daily data. Common lookback: 20 or 30 trading days. Comparing HV to IV is a key trading signal: if IV > HV, options may be expensive; if HV > IV, options may be cheap. Note that HV is backward-looking while IV is forward-looking — they can differ substantially during regime changes.
Implied Volatility (IV)
The volatility that, when input into the BSM model, makes the model price equal to the market price of an option.
Volatility Smile
The pattern where implied volatility is higher for deep in-the-money and out-of-the-money options than for ATM options.
Black-Scholes-Merton Model (BSM)
The foundational option pricing formula that gives the fair value of a European call or put as a function of spot, strike, rate, volatility, and time.
Delta (Δ)
The sensitivity of an option's price to a $1 change in the underlying spot price.
Gamma (Γ)
The rate of change of delta with respect to the spot price — the curvature of the option's value.
Theta (Θ)
The rate at which an option loses value as time passes — time decay per calendar day.