Implied Volatility Surface

A topographic map doesn't report "the elevation of this country" as one number — elevation depends on where you stand, both north-south and east-west, and the honest answer is a whole landscape of hills and valleys. Implied volatility works the same way. Ask "what's the volatility of Apple stock" and there is no single honest answer, because the number you back out of an option price (see calibrating a model to market prices) depends on which option you asked about — which strike, and which expiry. Plot implied volatility against both of those at once and you get a surface: a landscape with two coordinates, strike and time, and volatility as the height.

The two directions of the surface

Slice the surface one way — fix the expiry, vary the strike — and you get the smile (or skew): for equities, out-of-the-money puts almost always imply higher volatility than at-the-money options, which imply higher volatility than out-of-the-money calls. The market is pricing in more room for a crash than a rally, and it prices that fear directly into the strike axis. Slice it the other way — fix the strike at-the-money, vary the expiry — and you get the term structure: short-dated volatility often sits below long-dated volatility in calm markets (there's more time for something to happen further out), but spikes above it around known near-term events like earnings, because the market knows something is scheduled to happen soon and prices that specific date richer than the ones around it.

$$\sigma_{implied} = \sigma(K, T) .$$

In plain English: implied volatility, $\sigma_{implied}$, is not one number — it's a function of two inputs, strike $K$ and time to expiry $T$. Every point $(K, T)$ where an option actually trades gives you one height on the landscape; a full model of the surface is a way of filling in every other point too.

Worked example 1 — reading the skew

A one-month expiry has three quoted strikes: 90-strike implied vol 22%, 100-strike (at-the-money) implied vol 18%, 110-strike implied vol 16%. The skew's slope between the two wings: $(16 - 22) / (110 - 90) = -6 / 20 = -0.3$ vol points per $1 of strike — for every dollar the strike rises, implied volatility falls by roughly 0.3 points in this range. Use that slope to estimate the implied vol at strike $95, halfway between 90 and 100: starting from 22% at strike 90 and moving $5 toward 100, $22\% - 5 \times 0.3\% = 22\% - 1.5\% = 20.5\%$. A trader without a direct quote at 95 uses exactly this kind of linear interpolation across the strike axis to get a usable number — crude, but grounded in real quoted points on either side.

Worked example 2 — reading the term structure, in dollars

Same stock, at-the-money, but now compare two expiries: 1-month implied vol 18%, 3-month implied vol 22% (the market is pricing in extra uncertainty further out — maybe an anticipated event). Stock at $100, strike $100, rate zero. Price an at-the-money straddle (one call plus one put) at each maturity using Black-Scholes; with $r=0$ and $K=S$, a straddle's price simplifies to $2S\big(N(d_1) - N(d_2)\big)$.

1-month ($T = 1/12$, $\sigma = 18\%$): $\sigma\sqrt{T} = 0.18 \times 0.2887 = 0.05196$. $d_1 = 0.02598$, $d_2 = -0.02598$. $N(d_1) \approx 0.5104$, $N(d_2) \approx 0.4896$. Straddle $= 200 \times (0.5104 - 0.4896) = 200 \times 0.0208 = 4.16$ = $4.16.

3-month ($T = 0.25$, $\sigma = 22\%$): $\sigma\sqrt{T} = 0.22 \times 0.5 = 0.11$. $d_1 = 0.055$, $d_2 = -0.055$. $N(d_1) \approx 0.5219$, $N(d_2) \approx 0.4781$. Straddle $= 200 \times (0.5219 - 0.4781) = 200 \times 0.0438 = 8.76$ = $8.76.

A calendar spread — sell the 1-month straddle, buy the 3-month straddle — costs $8.76 - 4.16 = 4.60$ = $4.60 net. That $4.60 is the market's price for the difference in implied volatility between two dates on the same stock, converted from vol points into a real, tradable dollar number. This is the term-structure axis of the surface doing exactly the same job the strike axis did in example 1 — turning a landscape of implied vols into a specific price for a specific trade.

Drag across the surface above: moving along the strike axis at a fixed maturity traces out the skew from worked example 1; moving along the maturity axis at a fixed strike traces out the term structure from worked example 2. Watch for how a bump near a single maturity (an earnings-style spike) looks like a ridge running across the strike axis at just that one date.

What this means in practice

Every option desk maintains a live implied volatility surface, not a single number, because it prices everything — from a vanilla option at a strike nobody quoted directly, to complex exotics whose value depends on the whole shape of the surface, not just one point on it. The surface is rebuilt continuously as trades happen, and its shape carries information: a steep skew says the market is paying up for crash protection, a term-structure spike says the market expects news on a specific date, and both move independently of where the stock itself is trading.

Practice

1. A three-month expiry quotes 95-strike vol 24%, 105-strike vol 19%. Using the linear skew method from worked example 1, estimate the implied vol at strike 100.
2. If the 1-month ATM vol in worked example 2 rose from 18% to 21% while the 3-month vol stayed at 22%, would the calendar spread (sell 1-month, buy 3-month) get more or less expensive to put on, and roughly why?