Implied Volatility: The Number Backed Out of a Price

Implied volatility is whatever number, placed into a pricing formula, makes that formula return a price that has already been handed over. Everything else in this subject area puts a volatility in and takes a price out. Implied volatility runs the other way: the price is the input, the volatility is the output. The number is a price restated in different units, not a forecast of anything.

Here is the sentence to hold on to before any of the mathematics arrives. The number is a translation of a price, and a translation carries no information the price did not already carry. Nothing is added by the translation. Nothing is predicted by it. The only thing that changes is the scale the price is written on, and the reason anyone bothers is that the new scale is easier to compare across contracts than the old one.

Consider a journey quoted in minutes. Somebody says the drive is forty minutes. Forty minutes is not a measurement of the road; the road is thirty kilometres long and it stayed thirty kilometres long. The forty minutes is the thirty kilometres restated, under an assumed speed. If the assumed speed is wrong, the forty minutes is still the exactly correct restatement of thirty kilometres under that assumption, and it is still not a forecast of how long the drive will take. Change the assumed speed and the same road quotes as a different number of minutes. Implied volatility is that same restatement, worked through carefully.

The object being worked on throughout is the standard process, a single invented traded quantity written S with a time subscript. The process starts at Rs 100/-, it carries a volatility of 20 per cent a year, it is watched over one year, and the rate is 5 per cent a year continuously compounded.

What is implied volatility?

Implied volatilityThe volatility that, put into a pricing formula, returns the observed price. is defined by a demand rather than by a construction. A price arrives from outside. The pricing formula is held fixed, with the starting value, the strike, the rate and the horizon all pinned to what they already are. Then a single question is asked: which volatility, if it had been the volatility all along, would have made this formula produce exactly the observed price? The answer to that question is the implied volatility, and it exists for no other reason.

Notice what that definition does not say. The definition does not say the process has that volatility. It does not say anyone believes it does, or that the number will hold for the next hour or the next year. The definition says only that the formula, fed that number, returns the price. Implied volatility is defined as the solution of an equation, and a solution of an equation is not a claim about the world.

The equals sign is doing all the work. In every other reading order the formula was an instruction: take these five things, do this arithmetic, and here is a price. Here the formula is a constraint: four of the five things are nailed down, the output is nailed down, and the fifth thing has to move until the constraint is satisfied. Nothing about the formula changed; only which of its slots is empty changed.

Solving an equation is why the quantity is so often misdescribed. Everything in the language around implied volatility sounds like a measurement. The number has a unit, per cent a year, and a value that moves through the day, and the number sits beside quantities that really are measurements. But the thing it is derived from is a price, and the thing it is derived by is an assumption, and neither of those becomes a measurement by being combined with the other.

How is it obtained, and why does the direction run backwards?

Every treatment up to this one runs the same way round. The settings of the process are chosen, the contract is chosen, the handle is turned, and a price comes out. The volatility was something supplied. The volatility was an assumption, sitting at the front of the calculation with the other assumptions, and the price was the thing not known until the arithmetic finished.

The inversionRunning the formula backwards, with the price as input and the volatility as output. turns that around completely. Somebody quoted the price, so now the price is the thing known and the volatility is the thing not known. The formula has not changed by a single symbol. The change is which end of the formula the work starts from.

Written with everything except the volatility already frozen, the pricing formula becomes a function of one argument. Freezing the other four arguments is worth doing explicitly. A function of one argument is an object that can be reasoned about: whether it rises, whether it falls, what values it can take, and whether it ever visits the same height twice.

At the locked settings on the at-the-money contract the two intermediate quantities come out at exactly 0.350000 and 0.150000, the two normal probabilities at 0.636831 and 0.559618, and the price at Rs 10.450584/-. Both intermediate quantities and both probabilities carry over unchanged from the forward calculation, rather than being rederived.

Why there is no formula for the answer, and what is used instead

There is an awkward fact here that surprises people. The forward direction has a closed form: the price can be written down. The backward direction does not. The volatility sits inside a normal distribution function in two places at once and refuses to be untangled, so no expression takes the price and returns the volatility. The inversion has an exact answer and no formula for it. Mathematics is full of such cases, more of them than beginners expect.

So the answer is found by numerical inversionFinding the answer by search rather than by formula, which this inversion needs.: by searching. The word searching makes the answer sound approximate. It is not. A search that narrows a bracket by half at every step reaches any named accuracy in a fixed and countable number of steps, and the answer it converges on is the exact root, not a substitute for it. The everyday version is guessing a number between one and a thousand when the only reply is higher or lower. Ten questions is enough. There is no formula for the answer there either, and nobody thinks the answer is approximate.

The simplest search is bracketing. Pick a low volatility whose price is below the observed one and a high volatility whose price is above it. The answer must lie between them. Try the middle. Whichever half still straddles the observed price becomes the new bracket, and the other half is thrown away for good. Each step halves what is left.

Halving is reliable but slow. The faster search uses the slope. Once it is known how steeply the price rises with the volatility, the location of the answer can be guessed instead of merely which side of the middle it lies on: the gap between the observed price and the price at the current trial, divided by the slope, gives the size of the step. The slope-guided move is the classical root-finding step, and almost every real inversion routine uses it.

Units are the trap here, and they catch nearly everybody. The vega quoted in the locked figures, 0.375240, is the rise in the price for one percentage point of volatility. The derivative in the step above is with respect to the volatility written as a decimal, so it is one hundred times larger, 37.524035. Mixing those two makes the first step land a hundred times too far away.

Start the search deliberately badly, at a trial of 0.30, and watch it work. The first step overshoots downward and lands at 0.20035996. The second lands at 0.20000002. The third is 0.20000000 and there is nowhere further to go.

Three steps from a starting guess that was half as wrong again as the answer. A positive, well behaved slope buys that speed, and it is the practical reason the inversion is treated as a solved problem rather than as a hard one.

Why does the inversion have exactly one answer?

A search only makes sense if there is something definite to find. If two different volatilities returned the same price, the question would have two answers, the search would land on whichever it happened to reach first, and the number would not be well defined at all. So uniqueness is not a technicality. Uniqueness is what earns the quantity its right to exist.

The argument is short. The price rises whenever the volatility rises, and it never falls. A function that only ever rises visits every height at most once. Therefore no two volatilities can return the same price, and therefore the observed price picks out exactly one volatility. The whole of uniqueness comes from one fact, that the slope is positive, and that fact can be written down exactly rather than asserted.

Look at what the lower end is. Rs 4.877058/- is the starting value less the discounted strike, and it is the same 4.877058 that the locked put-call parity check produces on this contract. The two ends of the inversion are therefore not new figures. Both figures were already sitting in the case, and both turn out to be the boundaries of what can be inverted at all.

The two ends matter practically. No volatility produces a price below Rs 4.877058/-, so such a price has no implied volatility. A price at or above Rs 100/- has none either. The quantity exists for prices strictly inside that band, and outside it the inversion returns nothing rather than returning something wrong. An honest routine reports that; a careless one reports its starting guess.

Now, is the slope itself constant? No, and this is where people trip. The price is always rising, but not always at the same rate. Vega climbs steeply out of the very low volatilities, flattens out through the middle, peaks at a volatility of 0.316228 where it reaches 0.379486 for one point, and then eases gently down again. VegaThe rate at which a price rises with volatility, 0.375240 at the locked settings. being non-constant is fine. Uniqueness needs the slope to stay on one side of zero, not to stay still.

The everyday version is a staircase that gets shallower near the top. The steps are not equal. Some are barely a rise at all. But as long as every step goes up rather than down, no two positions on the staircase are at the same height, so a height, once stated, identifies exactly which step it came from. Uniqueness is a claim about direction, not about pace.

What does the round trip look like when it is done in both directions?

Here is the worked instance, and it is deliberately the smallest one possible: a single contract, taken forwards and then taken back. The at-the-money contract on the standard process, strike Rs 100/-, one year. Put a volatility of 0.200000 into the formula and the price that comes out is Rs 10.450584/-. Now throw the volatility away, keep only the price, and run the inversion. The search returns 0.200000.

Not approximately 0.200000. Exactly it, to every decimal place either side has. The round tripPricing forwards and inverting back, which returns the starting volatility exactly. is lossless. Losslessness is what makes the quantity well defined rather than merely useful. If pricing forwards and inverting back returned something a little different each time, the number would be an artefact of the routine rather than a property of the price, and nobody could compare two of them.

Reversibility is the test that separates a restatement from a calculation. Thirty kilometres converted into forty minutes and then converted back into kilometres under the same assumed speed returns thirty kilometres. Nothing was learned and nothing was lost. An estimate discards information on the way through. Had something been estimated instead, the trip back would not close. The inversion closes, therefore nothing was estimated.

What does the number actually measure?

Now the honest answer, and it is smaller than most people expect. Implied volatility measures the price. Nothing else. The formula used for the translation happens to have a volatility slot. The price fills that slot up, and the scale the slot is labelled in is volatility.

The number does not measure the process. The standard process has a volatility of 20 per cent by construction, and that fact was true before anybody quoted a price and stays true if nobody ever quotes one again. The implied figure would still be produced from any price handed over, including a price that had nothing to do with the process at all. The formula does not check.

The number does carry the formula, unavoidably. Every assumption that went into the pricing formula is baked into the figure that comes out of it. The formula assumed one constant volatility for the whole horizon, so the figure is the constant volatility that would have had to hold. The formula assumed the process moves continuously with no jumps, so the figure is the smooth-moving volatility that would have had to hold. Model dependentCarrying whatever the formula assumed, which every implied figure does. is not a criticism of the number. Model dependence is the definition of the number.

Change the formula and the figure changes even though the price did not. The dependence on the formula is the cleanest test of what kind of object the figure is. A measurement of the process would not care which formula was used to read it, in the same way that a length does not care which ruler is reached for. The inverted figure cares completely.

The quoting conventionA way of stating a price, which is what this quantity mostly is. reading also explains why anyone bothers with the translation at all. Two contracts at different strikes have prices that cannot be compared. The two are contracts on different things, so Rs 25.227000/- and Rs 2.745149/- are not dear and cheap versions of each other. Restate both as volatilities and they land on one scale, 24.0 and 18.5. Per-kilogram pricing does exactly that for a shelf of differently sized packets. Nobody thinks the per-kilogram figure is a measurement of the packet.

Why is it not a forecast?

The forecast reading is the one worth killing outright. The figure is read as a forecast constantly, in conversation, in commentary and in the heads of people who otherwise know exactly what they are doing. So the argument has to be firm, and it has to be firm without the escape hatch of calling it a rough forecast or an imperfect forecast. The figure is not a poor forecast. The figure is not a forecast at all.

The definition settles the question on its own, so start there. The figure is whatever makes a formula reproduce a price already seen. Nothing in that sentence points forward in time. The price already happened. The formula is fixed. The figure is determined the moment those two are in place, and it would be exactly the same figure if the future had already been written down and read out loud. A quantity fully determined by things that have already happened cannot be a statement about what has not.

Then comes the arithmetic argument, and the arithmetic argument closes the door. Take the five invented contracts on the standard process at one horizon. The five prices invert to 0.240000, 0.220000, 0.200000, 0.190000 and 0.185000. Suppose, for one paragraph, that each is a forecast of the volatility of the standard process over the coming year. Then five forecasts of one quantity, made at the same instant, on the same process, over the same horizon, disagree with each other. At most one of them can be right. Nothing in the formula, and nothing in the prices, says which.

What does it become when five strikes give five answers?

Run the inversion five times instead of once. Same process, same starting value, same rate, same horizon, same instant. The only thing that differs is the strike, and the price attached to it. Five prices go in and five volatilities come out, each one exact to six decimal places. The price is monotoneAlways moving one way, which the price does in volatility and which makes the answer unique. in the volatility for every strike, not just for the middle one, so each answer is the unique root of its own equation.

Every one of the five is correct. Not one of them is an approximation or a failure of the search. Rs 25.227000/- really does invert to 0.240000 and Rs 2.745149/- really does invert to 0.185000, and either round trip run back returns the price it started from. The five answers are all right, and that is precisely the problem.

The five answers rule out the reading that the figure describes the process. A process carries one volatility. The inversion has handed back five, all of them exact. The only way to keep every one of the five and stay consistent is to stop claiming that any of them is a property of the process, and to say instead what they really are: five prices, each written in the units that the constant volatility formula happens to supply.

The behaviour of that function across strikes is set out under the volatility smile, skew and surface. The narrower point is the one that settles matters: the moment the answers differ across strikes, the reading of any single one of them as a property of the process is finished. The figure survives as a quoting convention and only as a quoting convention.

How does somebody reading a screen of prices actually use it?

Given everything above, what is the number for? Three things, and all three are compatible with it being a restatement rather than a reading.

The first is comparison on one scale. The two prices are not the same size of thing, so an analyst looking at contracts at Rs 80/- and Rs 120/- cannot compare Rs 25.227000/- with Rs 2.745149/- in any useful way. Restated they are 24.0 and 18.5, and both then sit on one axis where they can be set beside each other. Unit conversion is doing what unit conversion is for, and a shopper reads the per-kilogram strip on a shelf for the same reason.

The second is stability over the day. Prices move with the level of the underlying process even when nothing about the contract has changed. The level of the process is one of the frozen arguments in the translation, so restating the price as a volatility takes most of that movement out. The remainder moves less, and a quantity that moves less is easier to watch. The steadiness is a bookkeeping convenience, not an insight, and the honest way to describe it is as a quieter set of coordinates for the same information.

The third is that it is the input the next stage needs. A researcher fitting a model does not fit to prices directly in every case; the loss function may be built on the restated figures instead. The choice of loss function is consequential in ways set out under the loss function, and it is one more reason to know that the figure is defined by a formula rather than observed.

None of the three amounts to a view about what will happen. An investor or a household reading such a figure and taking it as a prediction of how much the process will move has substituted a restatement for a forecast, and the two are not interchangeable no matter how the number is labelled.