Lesson 10 of 20 · 7 min
The volatility smile, plainly
The model assumes one volatility per underlying; the market quotes a different one at every strike.
Black and Scholes takes a single volatility number for an underlying. If the model described the market exactly, inverting real prices at every strike would return the same figure each time. It does not.
What comes back instead is a curve. Plot implied volatility against strike and you typically get a shape that is higher at the extremes than in the middle — hence smile — and for equities usually tilted, with downside strikes carrying higher implied volatility than upside strikes. That tilt is called skew.
The plain interpretation is that the market does not believe the model's assumption about the shape of the distribution. A single-volatility model implies a specific, well-behaved distribution of outcomes. Market prices for far strikes are higher than that distribution justifies, which is the market saying: large moves, and especially large downward moves, happen more often than your bell curve claims. Inverting an inadequate model against those prices pushes the volatility figure up to compensate.
In the base case, the model at a flat 25 percent prices the 90-strike put at $0.19. A market price of $0.55 for that put implies 32.9 percent, while a market price of $0.26 for the 110-strike call implies 23.1 percent. Same stock, same expiry, same afternoon, a nine-point spread in the answer.
The practical consequence is that there is no such thing as the implied volatility of a stock. There is an implied volatility surface, varying by strike and by expiry, and any single number quoted for an underlying is a summary of it. Track 4's lessons on fat tails explain why the surface has the shape it does.
Key terms
- Volatility smile
- The pattern of implied volatility being higher at strikes far from the current price than at strikes near it.
- Skew
- The tilt in that pattern. In equities, downside strikes usually carry higher implied volatility than upside strikes.
- Volatility surface
- Implied volatility mapped across both strike and expiry. The full picture, of which any single quoted figure is a summary.
Worked example
One expiry, three answers
Underlying 100, 30 days, rate 4%. Market prices inverted for implied volatility at each strike.
- 90-strike put, market $0.55
- Implied volatility 32.9%
- 95-strike put, market $1.45
- Implied volatility 31.3%
- 100-strike call, market $3.02
- Implied volatility 25.0%
- 105-strike call, market $1.00
- Implied volatility 23.2%
- 110-strike call, market $0.26
- Implied volatility 23.1%
- Model price of the 90 put at a flat 25%
- $0.19
Against $0.55 in the market.
The market pays nearly three times the flat-model price for the downside strike. The smile is that gap, expressed in volatility units.
Check yourself
The 90-strike put implies 32.9% volatility while the 110-strike call implies 23.1%, on the same underlying and expiry. What is the market saying?
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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.