Lesson 9 of 25 · 7 min

What implied volatility is the price of

What the premium pays for is a hedging programme, and the result settles in variance, not in quoted points.

Implied volatility is often described as the market's expectation of movement. Track 3 lesson 9 already narrowed that to a price. This lesson narrows it further, to the price of one specific thing.

Track 3 lesson 2 built the pricing model out of replication: the premium equals the cost of manufacturing the payoff by adjusting a stock position as the price moves. That cost is not hypothetical. A party who sells the contract and runs the hedge pays it, session by session, and the total depends on how far the underlying actually travelled. Implied volatility is the realised volatility at which that programme breaks even.

Here is the part that surprises people, and it is arithmetic rather than interpretation. The result is proportional to the difference of squares, not the difference of levels. Sell at 27.4 percent implied and the relevant quantity is 0.274 squared, or 0.075076. If the period realises 24.1 percent, that is 0.241 squared, or 0.058081. The gap is 0.016995, which is 169.95 variance points, where a variance point is 0.0001.

Now move realised the same 3.3 points the other way, to 30.7 percent. That squares to 0.094249, and the gap against implied is 0.019173, or 191.73 variance points. At an illustrative $250 per variance point, the favourable case is worth $42,487.50 and the unfavourable case costs $47,932.50. A symmetric move in volatility produced an asymmetric result, larger against the seller by a factor of 1.128.

The general point is that volatility points are a misleading unit for anything that settles in variance. Three points added to 24 percent is a change of 153 variance points; three points added to 44 percent is 273 variance points, nearly twice as much. A collateral model that widens a haircut by a fixed number of points per point of volatility is treating those two situations identically, because the model is linear in volatility and the underlying quantity is not.

Key terms

Variance point: One ten-thousandth of a variance, the unit volatility-linked results are usually settled in. Quoting in volatility points obscures how much the result moved.
Break-even realised volatility: The level of realised volatility at which a hedging programme's cost exactly equals the premium received. It is the implied figure, restated as a threshold.
Convexity in volatility: The property that a result depends on the square of volatility, so equal moves up and down produce unequal outcomes and moves from a high base matter more.

Worked example

A symmetric volatility move, an asymmetric result

A position sold at 27.4 percent implied. Two illustrative outcomes, each 3.3 volatility points from that level. Settled at an illustrative $250 per variance point.

Implied, squared: 0.075076
Realised 24.1%, squared: 0.058081
Realised 30.7%, squared: 0.094249
Favourable case: +$42,487.50
Unfavourable case: −$47,932.50
Ratio of the two: 1.128

Equal and opposite moves in the quoted figure produced results differing by $5,445. The asymmetry is in the squaring, not in anyone's view.

Check yourself

Volatility rises three points, from 24 to 27 percent. Somewhere else it also rises three points, from 44 to 47 percent. Are those the same size of event?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.