Volatility Surface

The surface maps implied volatility on the z-axis against two independent variables: moneyness (strike/spot ratio) on the x-axis and time to expiration on the y-axis. This 3D representation reveals how the market's expectation of future volatility changes non-linearly across different option contracts. A single point on the surface corresponds to the implied volatility input required to match the market price of a specific vanilla option using a reference model like Black-Scholes.

A valid volatility surface must satisfy strict no-arbitrage conditions to prevent static arbitrage opportunities. Key constraints include:

• Butterfly arbitrage: The risk-neutral density implied by the surface must be non-negative, requiring the second derivative of call prices with respect to strike to be positive.
• Calendar arbitrage: The total implied variance must be monotonically increasing with time to expiration; longer-dated options cannot have lower total variance than shorter-dated ones.
• Absence of vertical spread arbitrage: Call option prices must be decreasing and convex with respect to strike.

Two prominent cross-sectional features define the surface's shape at a fixed expiration:

• Volatility Smile: A U-shaped pattern where both deep out-of-the-money (OTM) and in-the-money (ITM) options exhibit higher implied volatility than at-the-money (ATM) options. This is prevalent in foreign exchange markets and reflects the market's expectation of fat-tailed returns.
• Volatility Skew: An asymmetric pattern, dominant in equity markets, where OTM puts trade at a premium to OTM calls. This reflects the market pricing in a higher probability of sharp downward moves (crash risk) and the hedging demand for downside protection.

The term structure describes how implied volatility varies with time to expiration for a fixed moneyness level. Common shapes include:

• Contango (Upward Sloping): Longer-dated options have higher implied volatility, reflecting greater uncertainty over longer horizons. This is the typical state in calm markets.
• Backwardation (Downward Sloping): Near-term options trade at a premium to longer-dated ones, signaling immediate market stress or an anticipated event that will resolve. This structure is common during crises or ahead of binary events like earnings announcements.

The surface does not remain static; it evolves as the underlying asset price moves. Two primary dynamics models describe this behavior:

• Sticky Strike: The implied volatility for a specific strike price remains constant as the spot moves. The surface shifts horizontally, and the ATM volatility changes. This is often observed in fixed-income markets.
• Sticky Delta (Sticky Moneyness): The implied volatility for a specific delta or moneyness level remains constant. The surface moves vertically with the spot price, preserving the skew shape relative to the current price. This is more common in equity markets.

Applying PCA to historical volatility surface changes reveals that the vast majority of daily variation is captured by a few orthogonal factors:

• First Component (Parallel Shift/Level): A uniform increase or decrease in implied volatility across all strikes and maturities, accounting for typically 70-80% of the variance.
• Second Component (Slope/Tilt): A change in the steepness of the skew, where short-dated OTM put volatility moves inversely to long-dated ATM volatility.
• Third Component (Curvature/Butterfly): A change in the smile's convexity, affecting the wings relative to the belly of the surface.