The Implied Volatility Surface: What It Is and Is Not

An implied volatility surface is one number for every pairing of a strike with an expiry, strikes running across and expiries running down, and each number is what one quoted premium says about how far the reference asset might travel per year. A surface is a set of prices restated in a shared unit. A surface is a record of what was paid, not a forecast, and one contract fills one cell.

A premiumThe amount that actually moves from the buyer to the writer at the start of an option contract. is a price, and a price is one number. A single figure is the only thing a buyer can accept or refuse, so everything the two sides of the contract disagreed about has to be compressed into that one number. The compression is what makes dealing possible. The compression is also what makes comparing impossible, and the impossibility of comparing is the reason a surface exists at all.

Take two contracts written on the same thing. One of them deals at a level far from where the price sits today and runs for a long time. The other deals close to today's price and ends soon. The two contracts are not the same kind of promise, so their premiums are two numbers on the same screen that say almost nothing when set side by side. Restating each premium as an implied volatilityA number backed out of a quoted premium by running a pricing model backwards until the premium it returns matches the premium being quoted. puts both into a unit that can be compared across contracts whose strikes and expiries differ. Comparing premiums directly cannot do that.

Carry that restatement out on every contract quoted on one reference asset at one moment, then lay the results on the two axes that tell those contracts apart, and the object that results is the surface. The surface is not a further calculation stacked on top of the prices. The surface is the prices, converted and arranged.

What is being measured when somebody says volatility, and over what?

Volatility is a measure of how far a price has moved, or is expected to move, per unit of time. Read that again and notice which half does the work. The measure is stated per period, and it means nothing at all until the period is attached, in exactly the way a rate means nothing until it is known whether it covers a year or a month. A figure handed over without its period is not a small omission to be patched later. The figure is unfinished.

The second thing to hold on to is what the measure refuses to say. The measure reports distance travelled and stays silent about direction. A price that fell a long way over a week and a price that rose the same distance over the same week produce the same reading. Silence about direction is not a defect a better measure would fix. Silence about direction is the definition. Anybody who reads a volatility figure and begins forming a view about which way something went has read a number that never contained the answer.

Here is the everyday version. Ten shops sit in one shopping centre and each one counts its takings at close. The question this measure asks is not whether takings rose over the month. The question is how much the daily figure jumps about from one day to the next. A shop that takes a steady amount every day and a shop that takes nothing on four days and a great deal on the fifth can finish the month level with each other, and the phrase from one day to the next is doing all the work of telling them apart.

Look at the two panels for a moment longer than feels necessary. The paths are mirror images. One of them would have made money for somebody and the other would have lost it. The measure was never built to tell the two apart. The brackets down each side are the same length on purpose: that length, divided by the period along the bottom, is the whole of what is being asked.

Why does the same word name two different numbers?

Two quite different objects wear this one word, and almost every muddle in this subject begins with somebody treating them as one thing seen from two angles. The two are not one thing. A measured volatilityA number computed from a series of prices that have already happened, over a stated window and at stated intervals. is computed from a series of prices that already happened. An implied volatility is backed out of a premium being quoted now. The two are different objects, not two estimates of one quantity.

Each of the two needs something different before it can exist. The measured number needs data: a run of prices, a window over which to look at them, and an interval at which to sample them. Changing the window changes the number. A measured volatility is a property of the record chosen as much as of the asset. Nothing about it is in dispute and nobody has to agree to it. The measured number is arithmetic performed on the past.

The implied number needs something else entirely. The implied number needs one premium being quoted right now for one particular contract, and it needs a model to run backwards. Changing the model changes the number. An implied volatility is a property of the model chosen as much as of the price. No history is involved at any stage. An implied number can be produced for a contract on an asset whose price series has never been seen.

A surface is built entirely out of the second kind, so every cell of it is a price wearing a different unit. Carry that sentence into what follows. Not one number on a surface was measured. Every one of them was quoted, by somebody, for a contract, and then converted.

What does an implied number actually say?

Less than most readers expect, and more than they think once they see it. A pricing model is a machine that takes a set of inputs and hands back a premium. Given the price of the thing, the level written into the contract, the time left to run, the cost of financing and one number describing how far the asset might travel, it returns a figure. Run in that direction, the model is pricing.

Now hold the premium fixed instead. Somebody is quoting the contract, so the premium is already known. The other inputs are facts about the contract and about financing, so they are known too. Exactly one input is unknown, so a value is tried, the premium the model returns is compared with the premium actually quoted, and the trial value is adjusted. An implied number is what a price says, once the price has been pushed backwards through a model.

The habit this needs already exists in a different subject. A default rate implied by the spread on a bond is worked out the same way: the price the bond is quoted at is pushed backwards through a set of assumptions, and the rate that reproduces it is read off. Nobody sensible then says the bond is forecasting the borrower. The implied rate describes the price and does not predict the thing the price refers to, and that habit transfers here whole and unchanged.

Why is there a surface rather than a single number?

Start with the observation. The object only makes sense once the observation is in hand. If every contract written on one reference asset implied the same number, one number would do. Nobody would have drawn a surface and nobody would have named its shapes. Contracts on one reference asset do not imply the same number, and the refusal of those numbers to agree is the whole observation.

So what distinguishes one contract from another when the reference asset is held fixed? Two things, and only two. The strikeThe fixed level written into the contract at which the buyer may deal if they choose. is the level written into the contract. The expiryThe date the contract ends and the choice is either taken or abandoned. is the date the contract ends. With strikes laid out across the grid and expiries down it, every contract on that reference asset has exactly one place to sit.

One crossing of a strike with an expiry is called a cellOne crossing of a strike and an expiry on the grid, holding one number for one contract., and one contract fills one cell. A surface is a grid of cells, one per contract, and it is nothing more mysterious than that. The word surface makes people picture something sculptural, and the picture is doing them harm. Think of a school timetable: rooms across, periods down, one class in each box. Nobody calls a timetable an exotic object, and nobody assumes it predicts anything about next term.

Which Volatility Surface Shapes can one grid show, and what is a shape made of?

Before naming a single shape, settle what a shape is. Getting that wrong makes the rest of the subject impossible. A shape is a statement about how the number differs between neighbouring cells. No single cell has a shape, and two cells are the fewest that can show one. Sit with that. A cell holds a value. Two cells can hold a difference. A shape is a description of differences, so it lives between cells and never inside one.

Fixing the expiry and moving across the strikes means reading one row. There are exactly two things that row can be. The row can be flat, meaning the number is the same at every strike. The simplest models quietly assume a flat row, and rows in the world generally decline to be flat. Or it is not flat, in which case it is either a smile or a skew, and separating those two is the next block.

Fixing the strike instead and moving down the expiries means reading one column, and what that column shows is the third shape. The three shapes together are the whole vocabulary: a row that curves, a row that leans, and a column that slopes. Everything else people say about a surface is built out of those three.

Volatility Smile vs Volatility Skew: which one curves and which one leans?

Smile and skew get used interchangeably by people who ought to know better, and the two words name different pictures. A volatility smileA row in which the number sits higher at strikes away from the money on both sides and lower in the middle. is a row where the number sits higher at strikes away from the money on both sides and lower in the middle. Drawn as a line, the row curves up at each end, and the curve is where the name came from. Both wings lift. The middle sags. Drawn out as a line, it is plain why somebody thought of a mouth.

A volatility skewA row in which the number sits higher on one side of the money than the other, so the row leans instead of curving. is a row where the number is higher on one side than the other, so the line leans rather than curves. A skew can lean steeply or gently, and it can lean either way. A leaning row does not come back up on the far side. A steep skew is still a skew, however dramatic it looks, and calling it a smile because it is dramatic is the mistake.

Here is the test, and it is one question. Is the row symmetric about the middle, or does it lean? Symmetric and lifting at both ends is a smile. Leaning, with one side high and the other low, is a skew. The test needs no value, no reading, no figure of any kind. Both words describe a pattern rather than a value, so both can be recognised without ever holding a number.

Neither of the two vertical axes carries a scale. A shape is recognisable without values, so a smile and a skew can be told apart with no reading of the quantity in hand. Anybody who put numbers on those axes would have invented them.

What is being read when reading down one column of expiries?

Hold the strike still and walk down the expiries instead. The way the number changes as the expiry gets further away is called the term structureThe way the same measure differs between contracts with different lengths of time left to run. of the surface, and the phrase means here exactly what it means for a rate: the same measure quoted for different lengths of time. The phrase applied to interest rates carries the same idea, and nothing about it changes on the way over.

A column can slope at all because a contract with longer to run is a contract whose reference asset has more time to travel, and a number stated per year does not have to be the same number when a year is a small part of the contract's life as when it is the whole of it. The mismatch between the period the number is stated in and the length of the contract is the entire reason the third shape exists. The slope is not a puzzle and it does not need a model to see.

There is a household version. Asked how much their electricity bill varies per month, somebody gives one answer. Asked how much it varies per month averaged over three years, they give a different answer. A three year window contains two summers and a change of tariff and a move to a new flat. Same measure, same unit, different lengths of time, different figures, and nobody is contradicting anybody.

What does the worked pair here actually give?

One invented pair of contracts, on one invented reference asset. The reference asset has a price of Rs 2,000.00/-, and the price is exposure the contract refers to rather than an amount anybody has paid or received. Financing costs 6.50 per cent a year. Both contracts run for one year. The reference asset pays nothing at all while it is held. A payout during the year would change every figure in the table, so the absence of one has to be said out loud.

A call and a put are both struck at Rs 2,000.00/-. The strike is the same number as the price on purpose. The strike and the price agree because this pair is struck at the moneyThe case where the strike and the price of the reference asset are the same number., and that is what at the money means, rather than one number having been copied into the other. The call premium is Rs 180.00/-, and it is given rather than worked out. Working the premium out needs a pricing model, and the table says so in the row where the figure sits.

Work the last row yourself. Rs 180.00/- less Rs 57.93/- is Rs 122.07/-, and exact parity is Rs 122.0657/-. The put was rounded to the paisa on the way through, so the two differ by forty-three hundredths of a paisa. The relationship holds to the paisa and it does not hold exactly, and claiming an exact equality that the rounded figures do not produce would teach a reader to stop checking.

Now the sharp end of the worked pair. The gap of Rs 122.0657/- between the two premiums is fixed by financing and by nothing else. The two premiums move together and the gap between them is arithmetic, so the difference is the same whether the reference asset barely moves or throws itself about. The difference between the premiums carries no information at all about how far the reference asset might travel. Only the level of the two premiums does, and reading a level needs a pricing model.

What is held here, and what shape does the rest of it have?

One strike and one expiry make a single cell, and a single cell is not a surface. Every shape named here is a statement about how the number differs from one cell to the next, so a smile needs a row, a skew needs a row, and a term structure needs a column. All three can be recognised on sight, and not one of them can be demonstrated with a single cell.

The single cell in hand is drawn with nothing written inside it. A premium is in hand and so are the other inputs, but the machine that turns them into a number is missing, so the cell stays empty.

At the first setting there is one cell, at the crossing of the strike of Rs 2,000.00/- and the one year expiry, and it is empty. The worked pair above holds exactly that cell. At the second setting a row appears across the strikes, and whether the row curves or leans becomes a question that can be asked. At the third a column appears down the expiries, and the term structure becomes askable instead. Only at the fourth setting can all three be asked at once, and every cell is still an outline.

What would have to exist before one cell could be filled?

Four things. Walk down them and mark each one present or missing.

1. A premium quoted in a market for that exact strike and that exact expiry. The worked pair has one: the call premium of Rs 180.00/-, on a contract struck at Rs 2,000.00/- with one year to run. Not a premium for a similar contract, and not an average of several. That one.
2. The other inputs the model wants. The worked pair supplies these too: a price of Rs 2,000.00/- of exposure, a strike of Rs 2,000.00/-, financing of 6.50 per cent a year and one year to run. All four are facts about the arrangement rather than opinions about it.
3. A pricing model to invert. No pricing model is present. How such a model is built is covered separately.
4. The assumptions that model makes, stated rather than buried. Every implied number is conditional on them, so a number handed over without them is a number whose meaning has been quietly withheld.

Three of the four are in hand, and the missing one is not a detail. The missing one is the machine. A reader who has three quarters of the ingredients naturally assumes the last quarter is a formality, and the last quarter is the opposite of a formality. The first two items are the raw material, the fourth is the fine print, and the third does the work.

How does somebody working with these contracts actually read a surface?

What would have to be known before acting on a shape?

Recognising a shape on a grid and deciding to take on an obligation are two different acts. The first is a reading skill. The second needs three things that no grid supplies, and none of the three appears above.

The first is a view on how far the reference asset might travel and how likely each move is. No reading of that quantity is held, in any unit, for any contract. The second is the circumstances of the person deciding. No reference work can see those, and none should pretend to. The third is what the arrangement would cost to hold all the way to the end and what it would cost to unwind before that, neither of which appears anywhere here.

A shape on a grid carries no outcome, no track record and no probability, so one shape cannot be ranked against another as good or bad. Recognising a shape on a grid is a reading skill, and a reading skill is not a reason to take on an obligation.

The reference asset, the price of Rs 2,000.00/-, the strike of Rs 2,000.00/-, the financing of 6.50 per cent a year, the call premium of Rs 180.00/- and the put premium of Rs 57.93/- are invented.
Educational material. Not advice on any investment, tax, budget or market position.