Volatility Skew

In equity markets, out-of-the-money puts consistently trade at higher implied volatilities than equidistant calls. This reflects the market's persistent fear of sudden crashes and the leverage effect—as stock prices fall, corporate leverage ratios rise, increasing equity volatility.

• Crash-o-phobia: Investors pay a premium for downside protection, bidding up put prices
• Spot-vol correlation: Typically negative (ρ ≈ -0.7), meaning volatility rises as the underlying falls
• Real-world example: During the COVID-19 crash in March 2020, 25-delta put skew on the S&P 500 exceeded 15 volatility points

Practitioners quantify skew using standardized metrics that isolate the asymmetry from the overall volatility level. The most common is the 25-delta risk reversal, calculated as the implied volatility of a 25-delta call minus that of a 25-delta put.

• 25-Delta Risk Reversal: Negative values indicate put skew; positive values indicate call skew
• Butterfly Spread: Measures convexity independent of directional skew
• Skew Index: CBOE SKEW Index tracks tail risk, with values above 135 indicating elevated crash probability
• Vanna exposure: The sensitivity of delta to changes in implied volatility, critical for skew hedging

Unlike equities, commodity markets often exhibit reverse skew—out-of-the-money calls trade richer than puts. This reflects supply-shock fears where prices spike upward rapidly, creating a positive spot-vol correlation.

• Supply disruption premium: Buyers of calls hedge against price explosions (e.g., oil supply crises)
• Positive spot-vol correlation: Volatility increases as commodity prices rise
• Example: Crude oil options consistently show call skew, with 25-delta calls trading 2-4 volatility points above puts
• FX markets: Skew direction depends on which currency is the "safe haven," with pairs like USD/JPY showing yen call skew during risk-off periods

Skew is not static—it steepens dramatically ahead of binary events and flattens after resolution. Earnings announcements, elections, and central bank decisions create temporary skew distortions that options market makers must manage.

• Pre-earnings skew: Single-stock options develop extreme put skew 1-2 weeks before earnings as investors hedge adverse surprises
• Event vol premium: The implied volatility of options expiring just after an event exceeds those expiring before it
• Volatility crush: Post-event, skew collapses as uncertainty resolves, creating opportunities for volatility sellers
• Term structure interaction: Short-dated skew can diverge significantly from long-dated skew during event windows

These two volatility surface dynamics rules describe how skew evolves as the underlying price moves. Understanding which regime prevails is critical for delta-hedging and risk management.

• Sticky Strike: Implied volatility for a fixed strike remains constant as spot moves. The volatility smile shifts horizontally with the underlying. Common in equity index markets
• Sticky Delta: Implied volatility for a fixed moneyness (delta) remains constant. The smile is fixed in delta-space. More common in FX and commodity markets
• Practical impact: Under sticky strike, a rally reduces the implied vol of fixed-strike puts, creating P&L implications for delta-hedged portfolios
• Regime identification: Traders analyze historical skew dynamics to determine which rule dominates in their market

Sophisticated traders exploit relative value discrepancies in skew across correlated assets, expirations, or between implied and realized skew. These strategies require precise volatility surface modeling.

• Dispersion trading: Selling index skew (via puts) while buying single-stock skew on constituents, profiting from the implied correlation premium
• Skew calendar spreads: Buying skew in one expiration and selling it in another when the term structure of skew appears mispriced
• Cross-asset skew pairs: Trading skew differentials between highly correlated underlyings (e.g., XLF vs. BKX financial sector ETFs)
• Risk considerations: Skew arbitrage carries gap risk during regime shifts and requires robust margin management

A graphical pattern where out-of-the-money and in-the-money options exhibit higher implied volatility than at-the-money options, forming a U-shape across strike prices. This phenomenon contradicts the Black-Scholes assumption of constant volatility and reflects the market's anticipation of fat-tailed asset returns. The smile is particularly pronounced in equity markets post-1987 crash, where deep out-of-the-money puts command a significant premium due to crash risk.

• Origin: Post-1987 equity market crash
• Shape: Symmetric U-curve in FX markets
• Cause: Excess kurtosis in the risk-neutral density

The curve representing the relationship between implied volatility and time to expiration for a fixed moneyness level. In normal markets, the term structure is often in contango, where longer-dated options trade at a premium to reflect greater uncertainty over extended horizons. During market stress, the curve can invert into backwardation, with near-term volatility spiking above long-term expectations.

• Contango: Upward-sloping, normal market condition
• Backwardation: Downward-sloping, crisis condition
• Roll Yield: The profit or loss from the term structure slope

A three-dimensional plot of implied volatility across varying strike prices and expiration dates, serving as the foundational pricing map for exotic derivatives. The surface encapsulates both the volatility smile (strike dimension) and the volatility term structure (time dimension) into a single, coherent representation. Market makers use this surface to interpolate prices for illiquid options and to calibrate advanced pricing models.

• Axes: Moneyness, Time to Expiry, Implied Volatility
• Usage: Exotic derivative pricing and risk management
• Calibration: Fitting parametric models to liquid market quotes

A class of models where volatility itself follows a random process, such as a mean-reverting diffusion, rather than remaining constant. The Heston model is the industry standard, assuming variance follows a Cox-Ingersoll-Ross (CIR) process correlated with the underlying asset price. This correlation parameter, known as spot-vol correlation, directly controls the steepness of the volatility skew.

• Heston Model: Mean-reverting square-root variance process
• SABR Model: Captures smile dynamics in fixed income markets
• Key Parameter: Volatility of volatility (vol-of-vol)

A deterministic function σ(S,t) of the underlying price and time that is calibrated to exactly fit the current market prices of vanilla options. The Dupire Equation provides a forward partial differential equation that uniquely derives this local volatility surface from a continuum of traded option prices. Unlike stochastic volatility models, local volatility is a complete model that perfectly reproduces the observed smile but fails to capture forward volatility dynamics.

• Dupire Formula: Extracts local vol from option prices
• Completeness: Uniquely determined by vanilla market
• Limitation: Underpredicts forward smile dynamics

Mathematical constraints ensuring a volatility surface is free of static arbitrage, preventing risk-free profits from butterfly and calendar spreads. A valid surface must guarantee that butterfly spreads (non-negative risk-neutral density) and calendar spreads (monotonic total variance) cannot produce negative prices. These conditions are essential for market makers to avoid quoting prices that can be arbitraged by sophisticated counterparties.

• Butterfly Arbitrage: Requires convex option prices across strikes
• Calendar Arbitrage: Requires non-decreasing total variance
• Breeden-Litzenberger: Links option prices to risk-neutral density