I often see the right part of the vol smile referred to as 'call skew'. However, due to put/call parity, this also represents skew for ITM puts. Is there a reason behind this convention?

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    $\begingroup$ To simplify and answer your question, the right part is referred as call skew because the implied vol comes from OTM call options. Similarly, the left part comes from OTM put options. @LocalVolatility has put a thorough answer already for why this is the case. As a side note, the lowest point of the smile is often not ATM or ATMf. That's why there is an additional term called ATM skew. $\endgroup$ – Will Gu Dec 9 '16 at 17:43

I am not familiar with the terms call and put skew but rather upside and downside skew. However, I can image why they are used by some people.

Short Answer

At-the-money and out-of-they-money options are usually more liquid than in-the-money options. I.e., on the upside (downside) calls (puts) have smaller spreads and give you a stronger signal of the mid market price at this strike. Consequently, you mainly use these instruments to construct your implied volatility smile.

Some Intuition

Here is a bit of (heavily simplified) intuition why this is the case. Assume you are a market maker quoting otherwise identical European call and put options. You have a calibrated fair/mid-market implied volatility smile and now need to determine the corresponding bid and offer prices you are willing to quote. Your spread should account for your risk exposures, and you arrive at the following formula

\begin{equation} s(T, K) = \alpha \frac{\partial V}{\partial \sigma}(T, K) + \beta \left| \frac{\partial V}{\partial S}(T, K) \right|. \end{equation}

Here, $\alpha$ is the spread you charge for one unit of vega and $\beta$ is the spread you charge for one unit of delta. Since the vega of European call and put options of the same strike are identical the spread difference stems from the delta. Since out-of-the-money options have a lower absolute delta, you are showing tighter quotes for them.

  • $\begingroup$ Your answer makes sense to me. Also explains why usually only vols for low delta options are provided by data providers. $\endgroup$ – Trent Dec 12 '16 at 7:04

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