Dear experts, I am constructing some grpahs for analysing the volatility, and when i looked at some example made by other research institutes, one may plot both historical implied and realized volatility. I am not sure if i was thinking right, but usually i could get one ATM- Call and Put seperately by using Black-schole formula, but how to get the single representative value (may include call and put) which can represent the historical implied volatility of a day? so my question is what are the most common ways that quants calculate their historical implied volatility? the value is used to draw the trend of historical implied volatility. My underlying is index. thanks

For ATM IV use the original VIX (now VXO) calculation as set out in Whaley's 1993 paper. For skew (mentioned in your post title) use the 25 delta risk reversal expressed in vol terms. For curvature (smile, not mentioned in your title but useful anyway) use the 25 delta fly. For dispersion (implied correlation, also useful) divide the weighted average ATM IV of your top twenty index components by your ATM index IV and you'll come close enough.

thanks for your reply, Does that mean the representative implied volatility (including Call and Put together) is the value point on the VIX(original)?