Lesson 9 of 20 · 7 min
Implied volatility is an output, not an input
The market gives you a price; implied volatility is what you get by running the formula backwards.
Four of the model's five inputs are observable: the underlying price, the strike, the time remaining, and the rate. Volatility is not. Nobody can look up how much a stock will move over the next thirty days.
So in practice the formula is used in reverse. Take the option's market price, plug in the four known inputs, and solve for the volatility figure that makes the model agree with the market. That figure is implied volatility. It is not a measurement of anything; it is the answer to the question, what would volatility have to be for this price to make sense.
This inversion is why implied volatility is quoted at all. It converts prices in dollars, which are not comparable across strikes and expiries, into a single number that is. A $3.02 premium tells you almost nothing on its own; a 25 percent implied volatility can be compared with the same underlying's implied volatility last week, with its realised volatility, and with other underlyings.
Two cautions follow from the fact that it is derived rather than observed. First, implied volatility inherits every assumption in the model, so it is only a clean statement about expected movement to the extent the model is right — and lesson 19 lists the ways it is not. Second, it is a market price expressed in different units, which means it reflects supply and demand for that specific contract as much as any consensus forecast. A strike that many participants want exposure to can carry a higher implied volatility for that reason alone.
Implied volatility is the market's price, restated. Treating it as the market's prediction imports a confidence the number does not carry.
Key terms
- Implied volatility
- The volatility input that makes the model's price equal the observed market price. Solved for numerically; there is no closed form.
- Inversion
- Running a pricing model backwards to recover an input from an observed output. Requires iteration because the formula cannot be rearranged for σ.
- Volatility as a quoting convention
- Using implied volatility rather than dollars to quote options, so contracts across strikes and expiries become comparable.
Worked example
Solving backwards from three market prices
Underlying 100, strike 100, 30 days, rate 4%. The market quotes three different prices on three different days.
- Market price $2.45
- Implied volatility 20.0%
- Market price $3.02
- Implied volatility 25.0%
- Market price $3.59
- Implied volatility 30.0%
- What changed in the four observable inputs
- Nothing
- Interpretation
- The market repriced expected movement, not the stock
The base case.
Same stock, same strike, same calendar, three prices. Implied volatility is the only variable that can absorb the difference.
Check yourself
The base-case call is quoted at $3.59 rather than $3.02, with the underlying, strike, days and rate unchanged. What does that tell you?
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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.