Worked Example 6 shows three strikes on one expiry carrying three different implied volatilities. That is a slice. Do the same across every listed strike and every listed expiry and the result is a surface: implied volatility as a function of two coordinates, moneyness and time to expiry. Desks quote, hedge and argue in terms of this surface rather than in terms of prices, so it is worth treating as an object with its own properties rather than as a table of BSM inputs.
What are the coordinates of a volatility surface?
Start with the coordinates, because the naive ones do not work. Plotting implied volatility against the raw strike makes two expiries incomparable, since a 7,100 strike is 4.05% out of the money on a one-month option and a different distance in standard deviations on a one-year one. The standard fix is to plot against log-moneyness, ln(K/F), where F is the forward for that expiry, and often to divide again by σ√T so that the horizontal axis is measured in standard deviations of the terminal distribution. On that axis the same nine quotes look like this:
| Expiry | K = 7,100 | K = 7,250 | K = 7,400 | skew, vol points per 1% below spot |
|---|---|---|---|---|
| 30 days | 16.50% (ln(K/F) = −0.0433) | 14.50% (−0.0224) | 13.00% (−0.0019) | 0.863 |
| 90 days | 15.20% (−0.0472) | 14.00% (−0.0263) | 13.20% (−0.0058) | 0.494 |
| 1 year | 14.50% (−0.0649) | 13.90% (−0.0440) | 13.50% (−0.0235) | 0.247 |
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Skew, term structure and vega across expiries
Three properties of that table are general, and each one has a reason.
Read the last column first, because the skew flattens with maturity. The short-dated slice falls 3.50 volatility points over roughly four percent of spot, a slope of 0.863 points per one percent down; the one-year slice falls 1.00 point over the same distance, 0.247 points per one percent. The mechanism is that a crash is a single event, and a single event of fixed size is an enormous move measured against one month of ordinary diffusion and an ordinary move measured against a year of it. Any model that generates skew from jumps produces this flattening automatically, which is one reason jump-diffusion models are the standard repair to BSM's second assumption.
The term structure of at-the-money volatility has its own shape, here rising from 13.00% at one month to 13.50% at one year. It is usually upward-sloping in calm markets and inverts in stress, when near-dated volatility spikes above long-dated. An inverted term structure is one of the few genuinely informative readings available from an option chain: the market is saying that whatever is frightening it is expected to resolve soon.
Vega, meanwhile, concentrates at the long end. The one-year 7,100 put has vega of 2,561.85 per 1.00 of volatility against 498.04 for the one-month, a factor of five, at the same strike and the same assumed volatility. A book that is short short-dated options and long long-dated ones can look vega-flat in aggregate while being enormously exposed to a change in the shape of the surface rather than its level. Aggregating vega across expiries into one number is therefore a reporting convenience that hides the risk it claims to summarise, and desks that care use bucketed vega by expiry instead.
Two rules for working with a volatility surface
Two disciplines follow, and both are cheap. First, never fit one volatility to a chain. Feeding a single at-the-money volatility into a pricer for every strike reproduces the error Worked Example 6 quantifies at seventeen dollars on one contract. Second, check the surface is arbitrage-free before using it, because an interpolated or model-fitted surface can easily fail conditions the raw quotes satisfy. Two conditions catch almost everything: call prices must be non-increasing and convex in strike within an expiry, and total variance σ²T must be non-decreasing in maturity at fixed log-moneyness. A surface that violates either one implies a butterfly or calendar spread with a negative cost and a non-negative payoff, and any pricer built on it will quietly produce free money that does not exist.
The surface is where DV1.03 begins, because a hedge is a position in the surface as much as in the underlying, and it is also where DV1.05 picks up volatility as an asset class in its own right. What this module owes you is the reflex that stops the error: implied volatility is an output of a price, one per contract, and the word "the" in front of "implied volatility" is nearly always wrong.