Volatility surface

From smile and term structure to surface

Each expiry has its own smile, and each moneyness level has its own term structure. Stacking the smiles by expiry gives a surface on which any listed option can be located. Traders use it to price options that are not quoted, to compare the richness of different parts of the curve and to measure how the whole shape moves: a parallel shift, a steepening of the put wing, or a rise of the front expiries relative to the back ones.

No-arbitrage constraints

• Calendar: at a fixed moneyness, total implied variance σ²·τ must not decrease as expiry lengthens, otherwise a calendar spread would lock in a profit.
• Butterfly: for one expiry, call prices must be convex in strike, which bounds how sharply the smile can curve.
• Surfaces fitted to noisy quotes can violate these rules between strikes or expiries; a violation usually means bad data or a bad fit rather than free money.
• Example: 18% for 7 days and 15% for 30 days is consistent, since 0.18² × 7 = 0.227 is below 0.15² × 30 = 0.675 (in variance-days); 30% for 7 days and 12% for 30 days would not be (0.63 against 0.43).

Reading it as a futures trader

The surface summarises what the options market charges for movement at each horizon and in each direction. Changes in its shape can be read next to the futures tape: a front-end rise with an unchanged back end suggests a near-term event priced in; a steepening put wing suggests demand for downside protection. These are descriptions of prices, not forecasts. Gamma exposure also depends on the surface, because the gamma of every leg is computed from its own implied volatility.

In Senzoukria

The GEX module's Surface page draws every strike and every expiry of the loaded chain at once, and its Value plotted control includes Implied volatility alongside gamma, vanna and charm exposure and open interest. The implied volatility shown at each point is the out-of-the-money side as published by the chain. It can be viewed as a 3D surface, a flat heatmap, one expiry or an aggregated profile, and missing cells stay empty rather than interpolated, with a note stating how many cells are quoted. The snapshot age is displayed, because the surface is a photograph of one chain, not a live feed.