Volatility Surface

A volatility surface is a map of implied volatility for the options on a single underlying asset, with strike price on one axis, time to expiry on the other and implied volatility as the height. Each traded option contributes one point. Fitting a smooth surface through the points gives an implied volatility for any strike and expiry, including those that are not quoted.

The surface exists because options on the same asset do not trade at one volatility, as the Black-Scholes-Merton model would require. It is the standard way in which options desks store the state of the market. Dealers, market makers and risk managers use it to price and revalue options, to measure exposure and to look for options that appear out of line with their neighbours.

A slice of the surface at one expiry shows how implied volatility varies with strike price. This is the volatility smile or, where it slopes in one direction, the volatility skew. The strike axis is often expressed as moneyness or as delta, so that surfaces for different assets and dates can be compared.

A slice at one level of moneyness shows how implied volatility varies with expiry. This is the term structure of volatility. In calm markets it usually slopes upward, with longer-dated options at higher volatilities. In stressed markets it inverts, as short-dated volatility jumps above long-dated levels. Scheduled events such as an earnings release or an election create a visible bump at the first expiry that covers the event.

Because variance accumulates over time, the surface implies a volatility for the period between any two expiries:

• $\sigma_1$ and $\sigma_2$ are the implied volatilities for the nearer and further expiries.
• $T_1$ and $T_2$ are the times to those expiries in years.
• $\sigma_{1,2}$ is the forward volatility between them.

Suppose three-month options trade at 20 percent and six-month options at 22 percent. The total variances are $0.20^2 \times 0.25 = 0.0100$ and $0.22^2 \times 0.50 = 0.0242$. The difference, 0.0142, divided by 0.25 is 0.0568, and its square root is about 23.8 percent. The market is pricing the second three months as more volatile than the first.

Construction starts with option quotes. Each price is converted to an implied volatility by inverting the pricing formula, which requires the forward price of the underlying, interest rates and expected dividends. Stale or one-sided quotes are removed, and out-of-the-money options are generally preferred because they are more actively traded.

The points are then fitted with a function, either a parametric form for each expiry or a model with a small number of parameters, and interpolated between expiries. A widely used parametric form was introduced by Jim Gatheral in 2004. The result must be free of arbitrage. Two conditions are checked: total variance must not decrease as expiry lengthens, or a calendar spread would have a negative price, and the curve at each expiry must not be so sharply bent that a butterfly spread would.

• Pricing and marking. An option at any strike and expiry is valued by reading its volatility from the surface. Positions are revalued daily in the same way, which matters especially for over-the-counter contracts with no quoted price.
• Risk measurement. Exposure to volatility, measured by vega, is reported by region of the surface, since different parts move differently.
• Exotic options. Models for path-dependent products are calibrated to the whole surface so that they reprice ordinary options correctly.
• Relative value. Traders compare points on the surface with each other and with their own forecasts, the starting point of volatility arbitrage.
• Volatility indices. Published indices of expected volatility are computed from a strip of option prices across strikes at the relevant expiries.

Interest rate markets use a comparable object with an extra dimension, since a swaption is defined by its expiry, the length of the underlying swap and the strike.

Most day-to-day change in a surface can be described by a few movements: a parallel shift in level, a change in the slope of the skew and a change in the slope of the term structure. Short expiries move far more than long ones. When the underlying price changes, traders must also decide whether each option's volatility stays fixed at its strike or travels with the price. The assumption affects the hedge ratios produced by the option Greeks.

A surface is only as reliable as the quotes behind it. Far from the money and at long expiries, options trade rarely and bid-ask spreads are wide, so much of the surface is extrapolated. Different fitting methods give different values in those regions, and firms can disagree materially on the worth of the same illiquid option. The surface is a snapshot of prices at one moment, expressed through a model known to be imperfect. It describes what options cost, including risk premiums, and is not a direct forecast of future volatility.