Volatility Is a Surface, Not a Number

On 11 September 2012, a session with a median VIX reading, SPX implied volatility read 16.94, 16.41, 18.29 and 21.23 across nine days, thirty days, three months and six months: a slight dip at the front and then the ordinary upward slope. On 8 April 2025 the same four points read 67.63, 52.33, 41.50 and 35.77, sloping steeply down. Both are term structures of the same market. A scalar that reports the 30-day point gives 16.41 for the first and 52.33 for the second, and says nothing about the sign of the slope in either.

Figure 1 measures that slope over fifteen years. It is inverted on 13.4 per cent of sessions, and its correlation with the VIX level is minus 0.56, which leaves about two-thirds of its variation unexplained by the level.

Section 01Implied and realised

Two quantities share the name volatility and answer different questions. Realised volatility measures a historical price path. Implied volatility is the parameter that reproduces an observed option price under a chosen pricing convention, and it is a statement about a risk-neutral distribution rather than a forecast.

An option market does not quote one volatility. It quotes prices across strikes and expiries, which can be expressed as a surface once a convention is fixed. The surface has two axes, and the arguments about what a single number loses are different along each.

Section 02The maturity axis, measured

Cboe publishes constant-maturity volatility indices at four horizons, computed from the same SPX option strips by the same methodology: nine days, thirty days, three months and six months. Taken together they trace the term structure every session, and they are official index values rather than a reconstruction.

The average shape is upward sloping, as theory expects: over the sample the nine-day index averages 0.95 times the VIX, the three-month 1.13, and the six-month 1.23. But the average shape is not the shape on any particular day. Three-month implied volatility sits below nine-day on 13.4 per cent of sessions, and six-month below thirty-day on 6.5 per cent.

Conditioning on the level barely helps. Holding the VIX between 14 and 16, the three-month minus nine-day spread still ranges from minus 4.6 to plus 9.1 index points. Knowing that the market is quiet tells you very little about whether the near horizon is priced above or below the far one, which is the quantity a calendar position lives or dies on.

Section 03The strike axis, and why it is not shown

Figure 2 is a conceptual plate, and it is retained because it carries the axis the measurement cannot reach. Reconstructing a real SPX surface across strikes requires licensed option quotes. A synthetic surface presented as an observation would be exactly the error this article is about, so the strike axis appears here as a diagram and is labelled as one.

What the diagram represents is well established. The Black-Scholes model assumes a single constant volatility, and if that assumption held every option on the same asset would imply the same number. They do not. Equity indices show a pronounced skew, with downside strikes implying higher volatility, which is the market pricing the model's own failure in volatility units. Gatheral's treatment is the standard practitioner account of how the two axes combine into one arbitrage-free object.

The skew carries information a level cannot. Its slope encodes the asymmetry of feared outcomes. A book that is flat to a headline volatility index can be badly exposed to a steepening skew, and the index will not move to warn it.

a VIX-style benchmark integrates across strikes and interpolates
across maturities to a fixed horizon:

    sigma^2(30d)  =  weighted integral of out-of-the-money option prices,
                     interpolated between two eligible expiries

the integral is a many-to-one map. Two surfaces with different skew
or term structure can produce the same scalar.

measured, Cboe constant-maturity indices, Jan 2011 to Sep 2026:

    mean ratio to VIX:   9D 0.947    3M 1.127    6M 1.226
    3M below 9D on                   13.4% of sessions
    6M below 30D on                   6.5% of sessions
    corr( VIX level, 3M minus 9D )   -0.563

    conditional on VIX in [14,16]:  3M minus 9D ranges -4.6 to +9.1

Section 04Why a number misleads

The practical danger is that a single number invites a single hedge, and a single hedge against a multi-dimensional exposure is usually mispriced. The two dated inversions in the lower panel of Figure 1 are the cases worth holding in mind: on 5 August 2024 and 8 April 2025 the curve did not merely rise, it turned over, so a position hedged on the assumption of an upward-sloping term structure was wrong about the direction of its own exposure rather than about its size.

A scalar can preserve the level of a surface. It cannot preserve the shape.

The dynamics are subtler still than the static surface. Gatheral, Jaisson and Rosenbaum presented evidence that volatility behaves statistically as a rough process, with a low Hurst exponent relative to standard models, which affects how a surface should be expected to move rather than how it looks at a point in time. That literature remains actively debated and is cited here as a caution about model risk rather than as a settled result.

What this article establishes is bounded and worth stating plainly. On the maturity axis, using official indices, the shape moves substantially independently of the level and inverts about one session in eight. On the strike axis, it establishes nothing, because the data required is not available to this journal, and the diagram is offered as a diagram.

This research is analysis and commentary for general information. It is not investment advice, an offer, or a solicitation, and it contains no price forecasts. Statistics in the text are the author's own calculations from the official index values cited.