If BSM were perfectly correct, implied volatility would be the same for all strikes and maturities. It isn't. The implied volatility surface is one of the most information-rich objects in financial markets.
The Volatility Smile
Plot implied volatility against moneyness (K/S or delta) and you observe a characteristic smile or skew:
- Equity skew (put skew). OTM puts trade at higher IV than OTM calls. Markets price in a crash risk premium — investors pay more for downside protection. This asymmetry grew dramatically after the 1987 crash.
- Currency smile. For FX options, the smile is more symmetric — both tails are fat, since either currency can depreciate.
- Commodity smile. Often a positive skew (call skew), as commodity markets fear upward supply shocks.
Term Structure of Volatility
Implied volatility also varies with time to expiry:
- Near-term IV spikes around earnings, central bank decisions, and geopolitical events.
- Long-dated IV is smoother and mean-reverts more slowly.
- Contango: short-term IV below long-term IV — typical in calm markets.
- Backwardation: short-term IV above long-term IV — occurs during stress (VIX spikes).
The VIX
The CBOE VIX is the 30-day model-free implied volatility of the S&P 500, computed from a strip of OTM options:
VIX² ≈ (2/T) · Σ [ΔK_i / K_i²] · e^(rT) · Q(K_i) − (1/T)[F/K₀ − 1]²
It is often called the "fear gauge." VIX above 30 historically signals elevated uncertainty; above 40 signals acute stress.
Local vs. Stochastic Volatility
Local Volatility (Dupire, 1994)
Dupire showed that any arbitrage-free surface uniquely determines a local volatility function σ_loc(S, t) such that a single-factor model perfectly fits the surface. The formula is:
σ_loc²(K,T) = [∂C/∂T + rK·∂C/∂K] / [½K²·∂²C/∂K²]
Local vol models reprice the current surface exactly but produce poor forward smile dynamics — future smiles flatten unrealistically.
Stochastic Volatility (Heston, SABR)
Stochastic volatility models introduce a second randomness: volatility itself follows a mean-reverting diffusion. The Heston model uses:
dS = r·S·dt + √v·S·dW₁ dv = κ(θ − v)dt + ξ√v·dW₂ with dW₁dW₂ = ρ·dt
The negative correlation ρ < 0 generates the equity skew. The semi-analytical Fourier inversion gives fast calibration. Stochastic vol models produce better forward smile dynamics but don't fit the current smile as well as local vol.
Practical Use Cases
- Hedging exotic options. A vanilla delta hedge ignores the vega risk from a changing smile. Traders use vanna and vomma hedges to neutralise skew and convexity exposure.
- Calendar spread trading. Flattening term structure (backwardation → contango) signals vol mean-reversion. Buy near-term vol, sell long-term vol.
- Risk reversal. The 25-delta risk reversal (cost of a 25Δ call minus 25Δ put) directly measures skew — a key input to FX and equity structured products.