I want to construct a volatility curve $\Gamma = \{(\Delta_i, \sigma_i)\}$ but notice that the call and put with the same delta have a different vol (which shouldn't be the case in theory). Is the standard approach to average the call and put vols with the same delta?
1 Answer
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Yes it is. However, you should normally do this (average of the call and put IVs) only for ATM options (I myself bound them using moneyness 0.95 < K/S < 1.05).
For options outside these bounds, they are considered as OTM or ITM. You should avoid ITM options and plot the volatility curve using OTM options. For example, the left (right) side of the curve should be OTM put (call) options.
Personally I do not like to use delta as an x-axis, as it produces a confounding effect, I prefer to use moneyness (like K/S), which does not contain the IV.
