Volatility Smile vs Skew vs Surface: Reading the Shape

A smile, a skew and a surface are three shapes made by the same numbers on different axes. A smile turns up at both ends across strikes. A skew falls or rises across them without turning. A surface adds expiry as a second axis and holds both at once.

Three words, one object. A set of readings, one per contract, arranged against something, forms a shape. Change what they are arranged against and the picture changes without a single reading changing. The three names do not describe three phenomena; they describe three viewing decisions taken over one set of numbers. That is the whole of the distinction, and almost every muddle in this area comes from treating the words as though they named different things in the world.

Here is the everyday version. A staircase, a valley path and a contour map are three ways of talking about ground. The staircase is the description used when every step goes the same way. The valley is the description used when the ground drops and then climbs again. The contour map is the description used when no single line is picked and the whole hillside is drawn instead. Nobody thinks the hill changed between the three descriptions. The same discipline is what this guide asks for.

What are the three shapes, and how do they differ?

A smileA shape turning up at both ends across strikes. is a set of readings across strikes at one expiry that falls, reaches a low point, and then climbs again. The shape has a turn in it. Draw it and the picture has a bottom somewhere in the middle, with both ends higher than the middle. The word is a picture word and it earns its keep only when the picture is actually that.

A skewA shape falling or rising across strikes without turning. is a set of readings across strikes at one expiry that runs one way and keeps running. A skew may fall the whole distance or rise the whole distance. A skew never changes direction. There is no bottom anywhere except at one end, so there is no bottom in the middle.

A surfaceBoth axes at once, strike and expiry together. refuses to fix the expiry. A surface holds strike and expiry together and gives a reading for every pair. A smile and a skew are what appears when a surface is sliced at one expiry and the slice is looked at on its own. The surface is the thing being sliced.

So the distinction reduces to two questions: how many axes are being carried, and does the shape turn? One axis and a turn is a smile. One axis and no turn is a skew. Two axes is a surface, and on a surface turning is judged slice by slice rather than once for the whole object.

What is the axis in each case?

The readings themselves are inert. The axis they are laid against is what makes a shape, so the axis deserves to be named out loud rather than assumed. A reading belongs to a contract, and a contract is picked out by two things: which strike it is struck at, and when it expires. Strike and expiry are the only candidates for an axis. Exactly three shapes follow, and not thirty.

A smile and a skew both use strike as the horizontal axis and hold expiry fixed. A surface uses both. That is it. Everything else that appears on a horizontal axis is a relabelling of strike, and relabelling is worth understanding because it changes the picture without changing a single reading.

Now the relabelling. Strike is measured in rupees, so the horizontal axis depends on where the process happens to be sitting. MoneynessHow far a strike sits from the starting value, the usual axis for these shapes. replaces the rupee strike with a measure of how far the strike sits from where the process is, most often as a logarithm of the ratio. Standardised moneyness goes one step further and divides by the volatility and by the square root of time, so the axis is counted in standard deviations rather than in rupees or in log units.

The respacing matters more than it sounds. On the rupee axis the five invented strikes sit at equal intervals of ten rupees. On the log axis they do not: the gap from Rs 80/- to Rs 90/- is 0.117783 in log units while the gap from Rs 110/- to Rs 120/- is only 0.087011, so the left of the picture stretches and the right compresses. The same readings therefore make a visibly different picture depending on the axis, and the only property that survives every relabelling is the direction of travel.

Why is the locked shape a skew and not a smile?

The observation set this subject area works against is five invented readings on the standard process at one year: 0.240000 at the Rs 80/- strike, 0.220000 at Rs 90/-, 0.200000 at Rs 100/-, 0.190000 at Rs 110/- and 0.185000 at Rs 120/-. No market anywhere produced those five readings. They are fixed by hand so that the descriptions set out under the Heston model and the SABR model have something concrete to be compared against.

Take the four steps between consecutive strikes. The four steps are minus 0.020000, minus 0.020000, minus 0.010000 and minus 0.005000. Four steps, four falls, and not one rise. There is no turn anywhere in this shape, so the correct word for it is skew and the word smile would be wrong. That is not a quibble about vocabulary. A smile is a claim that both ends are high and the middle is low; this shape has one high end, one low end, and a middle that sits between them exactly as a slope demands.

There is a trap here, and the trap is what makes people reach for the wrong word. The steps are not equal. The steps run minus 0.020000, minus 0.020000, minus 0.010000 and minus 0.005000, so the fall gets gentler towards the right. The differences of the differences are 0.000000, 0.010000 and 0.005000, every one of them non-negative. The shape is curved, and curved upward at that.

The everyday version is a flight of stairs. Suppose the first two steps drop twenty centimetres each, the third drops ten and the fourth drops five. The staircase is easing off. The flight is still a staircase going down, and no reasonable person would call it a valley just because the last step was gentle. A valley needs a step that climbs, and there is no climbing step anywhere in the five locked readings.

The two rows side by side, and where they meet

Add a second expiry and the object stops being a line. At six months the same five strikes carry 0.250000, 0.225000, 0.200000, 0.185000 and 0.178000. Turn each reading into a price with the constant volatility formula and the six-month row gives Rs 22.541545/-, Rs 13.950305/-, Rs 6.888729/-, Rs 2.523885/- and Rs 0.687999/-. The one-year row gives Rs 25.227000/-, Rs 17.257579/-, Rs 10.450584/-, Rs 5.644765/- and Rs 2.745149/-. Every figure here is an educational illustration on an invented process.

Read the fourth column downward and something orderly happens. The shorter expiry reads higher at the two low strikes, exactly equal at the middle strike, and lower at the two high strikes. The two rows cross once, and they cross precisely at the moneyThe strike equal to the starting value, where both rows agree at 0.200000., where the strike equals the starting value of Rs 100/- and both rows read 0.200000. The crossing at the money is not a coincidence of the numbers; it is the anchor the whole two-axis object is built around, and everything else in the picture is measured away from it.

Notice also that the two rows agree on a reading at that strike and still disagree on the price. Both read 0.200000, and the one-year contract prices at Rs 10.450584/- against the six-month contract at Rs 6.888729/-. Same volatility number, different price. The expiry itself is an input to the formula, and a shorter horizon leaves less room for the process to travel. The reading is a restatement of a price at a stated expiry, never a price on its own.

What does the second axis add?

Measure each row end to end. The tiltThe difference from one end of the strike range to the other, 0.055000 at a year here. at one year is 0.240000 less 0.185000, or 0.055000. At six months the tilt is 0.250000 less 0.178000, or 0.072000. The shorter expiry is steeper, by 0.017000 across the same five strikes, and that single fact is the entire justification for carrying a second axis.

If the tilt were the same at every expiry, a surface would be an extravagance. One row could be measured, its tilt noted, and that single number carried everywhere. Because the tilt is not the same at every expiry, the object has genuine extent in the second direction, and that extent has its own name: the term structureHow the shape changes with expiry, the second axis. of the shape.

The everyday version is a tailor. Measuring one person once gives a set of numbers. Measuring the same person on five different days gives a second direction of variation, and if the numbers move between days then a single day's measurement was never the whole story. Nobody would call the extra visits wasted once the numbers move. The second axis earns its place exactly when the first axis fails to repeat itself.

What does any of these shapes actually establish?

Less than most readers expect, and the gap between what a shape shows and what a reader takes from it is where the damage happens. Here is the honest inventory.

A shape that is not flat establishes one thing cleanly. One constant number cannot reprice every contract in the set. The conclusion follows immediately. If one number could reprice every contract, every reading in the set would be that number and the shape would be a horizontal line. The five locked readings are not all equal, so no single constant reprices all five. The finding is genuine and worth having.

The shape establishes nothing about which process produced the prices. A set of readings that falls without turning could come from any of several sources: a volatility that depends on the level of the process, a volatility with its own randomness correlated negatively with the process, a process that occasionally jumps downward, a mixture of two constant volatility worlds with different weights. Every one of those classes can produce a falling set of readings. The shape is consistent with all of them, so it selects none of them.

The condition has a name, underdeterminedConsistent with many processes, which any of these shapes is.. There is a picture, and more than one explanation fits it. The picture has already given up everything it contains, so looking harder at it does not reduce the count of explanations.

The everyday version is a weighing scale that reads differently depending on which corner of the room it stands in. Five readings around the room establish beyond doubt that one number does not describe this floor. The same five readings establish nothing at all about whether the joists sag, the tiles are uneven, or the scale itself is tilted. Five readings, one clean negative finding, and no positive one. Ruling something out is not the same as ruling something in, and a shape only ever does the first.

Which model does each shape point toward?

If a shape cannot select a model, it can still narrow what kind of model would be convenient. The narrowing is a weaker claim and worth keeping weak. The narrowing turns on which axis the question sits on rather than on what the picture looks like.

A question that lives on one expiry and moves across strikes wants a description organised around the pattern itself. The stochastic alpha, beta, rho model, known as SABR and set out separately under the SABR model, is built that way: it describes one expiry's readings across strikes and gives a compact expression for that row. Ask it about a single row and it answers directly. Ask it to span several expiries at once and it was not built for that. SABR takes the expiry as given rather than as a direction to travel in.

A question that spans expiries wants a description organised around the process instead. The Heston model, from Heston in 1993 and set out separately under the Heston model, gives the variance its own equation with its own noise, and once the variance is a process the expiry becomes a direction the model already travels in. Ask it for the whole two-axis object and it produces one. A process run for longer or shorter is the same process.

So the shape does not choose the model, but the axis the question sits on does suggest which shelf to reach toward. That is all this guide claims, and the claim survives only because it is about convenience rather than about truth.

What does a reader actually do with the shape once it is drawn?

Three things, and it is worth being concrete because the honest inventory above can read as though the picture were useless. The picture is not useless. The picture is precise about a small thing.

First, it is a specification. Whoever builds a description of the process now has a list of five numbers that the description must return, and a sixth requirement that it return them at two expiries at once. A specification is not a hypothesis and does not pretend to be one. A specification says what must come out, and stays silent about what goes in.

Second, it is a consistency check on the reader's own arithmetic. The two rows must agree at the strike where they were constructed to agree, both at 0.200000, and any working that loses that agreement has a mistake in it rather than a discovery. The at-the-money prices of Rs 10.450584/- and Rs 6.888729/- are two numbers a reader can recompute from the formula and check by hand.

Third, it is a boundary marker for what a single number can be asked to do. Anyone who wants to summarise the whole object in one figure has to choose which one, and the tilt is one honest choice: 0.055000 at a year, 0.072000 at six months. Naming the summary chosen, and admitting that the choice discards the rest, is the whole of good practice here.

A careful reader will not do any of the following. Announce which process is at work. Treat the flattening of the fall as though it were a turn. Quote one row's tilt as though it held at every expiry. Present any of these invented readings as an observation of anything outside this guide.