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Constructing a Volatility Smile from Put and Call Prices

Article Quant Q&A · Author: Gaurav Khare
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Summary

The document addresses how implied volatilities from puts and calls are represented in a volatility smile by strike. Under put–call parity, a put and call with the same strike and expiry imply the same volatility in theory, after accounting for intrinsic value and carry. Thus, a theoretical smile does not require averaging the two option types into a single volatility at each strike.

In practice, smile construction often uses out-of-the-money options: puts at strikes below the forward price and calls at strikes above it, because those options tend to be more liquid and have more recent trades. The forward price provides the reference for deciding which strikes are low or high. The responses also note that recent trades can be useful when estimating the curve. These are practical conventions rather than a complete data-cleaning method; the document does not address stale quotes, bid–ask noise, or how to handle cases where put and call observations diverge from parity.

Key ideas

  • Put and call implied volatilities at the same strike and expiry are equal in theory under put–call parity.
  • Smile construction often uses out-of-the-money puts below the forward and calls above it.
  • Out-of-the-money options are commonly preferred because they tend to be more liquid and recently traded.
  • The document does not specify how to reconcile noisy or inconsistent market quotes.

Tags

OptionsVolatilityDerivatives pricing
Full text
# Volatility smile shows individual Put and Call IV or combination


# Volatility smile shows individual Put and Call IV or combination












IV is calculated per strike AND option type basis(for example WTI 50 CALL its x and WTI 50 PUT its y). The question is when its shown in "Smile" its just shown on strike basis, so does that mean lower strikes just plots PUT IVs (y) and higher strike plot CALL IVs (x) OR does that mean for every strike we combine (using diff or average or whatever) for both CALL and PUT (x+y/2) and then project the "Smile" curve? Can you please elaborate here a bit.

## Answer by Chris Taylor (score 2, accepted)

https://quant.stackexchange.com/a/45332

IV for a put and call is the same so it doesn’t matter (in theory). In practice you use puts for low strikes and calls for high strikes, since the OTM are more liquid. Low/high is relative to the forward price.

## Answer by Steve Becker (score 1)

https://quant.stackexchange.com/a/45343

I am assuming you are asking this question as a programmer, not as a trader. From a traders perspective, due to put/call parity, we look at puts and calls at the same expiry and strike price as the same thing. Each option has an intrinsic value and a time value. After accounting for the cost of carry, the time value for the put and call are equal as is their volatility. As a programmer trying to get the best estimate of the IV curve, The most recent trades are the best data. As Chris said above, the OTM are usually the most recent trades.

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