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Why Option Prices Do Not Preserve the Shape of a Volatility Smile

Article Quant Q&A · Author: Danny
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Summary

The document addresses why converting smile data from implied volatilities into Black model option prices does not produce a curve with the same visual shape. Its answer explains that when puts are converted to equivalent calls using put-call parity, call prices across strikes should decrease monotonically and be convex. These price-shape constraints differ from the familiar smile or skew shape shown by implied volatility.

The key lesson is that implied volatility and option price are different representations: the smile describes how the volatility implied by each strike varies, while prices must satisfy structural relationships across strikes. The answer is brief and offers no numerical example, data, or details about the underlying instrument, expiry, or conventions. Its statement concerns parity-adjusted call prices and should not be read as saying that every plotted set of raw option prices will visually resemble a volatility smile.

Key ideas

  • Converting implied volatilities into prices does not preserve the visual shape of a volatility smile.
  • Put-call parity can express puts as equivalent calls for comparison across strikes.
  • Call prices should decrease with strike and form a convex curve under the stated setup.
  • Price-shape constraints and implied-volatility smile shape describe different features of option markets.

Tags

OptionsDerivatives pricingVolatility
Full text
# Shape of smile after converting to prices


# Shape of smile after converting to prices












I have smile data which looks like this,

but after converting the vol points to prices using black's model, i got something like this,

is this expected?

I was expecting the shape of the smile to be retained. What am I missing?

## Answer by Mark Joshi (score 2, accepted)

https://quant.stackexchange.com/a/34648

if you use put-call parity to make them all calls, they should be monotone decreasing, and convex. I wouldn't expect them to look like a market smile.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.