How do you go about pricing skew in a way to identify mispricings in skewed options? If ATM vol of a stock is 50, and say historical vol has been 55 (not that this would ever happen) you can have some basis for saying that vol is cheap. But how does this extend to skew? If ATM vol is 50, but the 30 delta put is trading 55, is it too expensive? Too cheap? I know that puts tend to trade above ATM, but how do you get a sense of what's cheap or expensive? Phrased another way, if I tell you a $50 stock has ATM vol is 50 and I ask you to price the $30 put and $80 call, how would you go about doing that? What other information would I want in this case? I know that you could in theory back out an implied distribution of sorts, but I'm not sure how common that is and whether that's typical? Just trying to develop some intuition for identifying when skew is expensive or cheap.
You asked a very good question. I think the answer lies in your first para : In that expensive/cheap is relative. If your ATM vol of 50 is cheap cos it was 55 before then it is 'cheap'. A value of 40 for VIX is traditionally seen as dangerously high, yet, today it is seen as low because VIX was 70+ a couple of weeks ago. All relative. Similarly, skew is either cheap/expensive depends on where it was recently. Having said this, I understand what you are asking - is there is mathematical, quantifiable way to express this? There are some very bright people here with quant level vol knowledge and hopefully one of them can provide a better response. Personally, and practically, I trade skew. I use an options modelling tool as a black box, so all that I am interested in is the final numbers for : break-even points, max profit, max loss, theta and vega figures etc. The model does the behind the scenes complex calcs, which obviously take into account any skew that exists. If the forecast risk/reward looks good, I will trade. In your example of 30 delta calls and put compared to ATM - this would lead to a butterfly trade, either buying or selling depending on the value of the skew. If the skew was more a steep U shape rather than a 'smile', then you'd sell the wings and buy the body, and vice versa. I tend to trade skew horizontally across time periods, rather than vertically, across deltas - so comparing the vol of the 30 delta put for this weeks expiry to next weeks (rather than comparing it with the ATM vol for this week).
So you trade skew personally, if I understand what you're saying is you look at the break even points mainly? Or your model? So like for risk reversals or butterflies? I really like the idea of ditching models and thinking in terms of prices or just breakevens? To clarify one other thing, the reason 50 vol seems cheap in a 55 vol historical vol is that we're basically assuming the future will have closer to a 55 realized vol. It's not so much about the past as it is about what it purports to tell us about the future. I wouldn't' think about the VIX in historical terms, but instead that vix at 40 implies daily moves of ~2.5% , does that seem expensive or cheap to me? That has a quantifiable meaning to me you know? I can say, nah I call BS 2.5% is too much or too little. It's harder to express that about skew because you're dealing with, well idk weirdness. Your last part about trading skew horizontally makes a lot of sense to me especially since you're comparing very similar options. If the 30 delta put in one month is trading at 40 volatility and in another is trading at 35, it seems like a decent spread to put on
Measuring skew can even be a type of heuristic, something a trader develops over time via market awareness. Just by glancing at an option chain a trader can pick off skew points of interest within the surface and term structure. using risk reversals is a good way to measure the vol skew as well
For big picture and longer term horizon strategies, use accurate constant maturity EOD (end of day) implied volatilities (IV) in different time series and different market regimes to determine whether intra-month or term structure skew is mispriced or trading at extreme levels. For day trading and short-term analysis use intraday IV market data recordings (high, low, average data) to monitor and detect statistically significant moves in skew and kurtosis, on an intraday basis.
There are many ways to measure skew,but for simplicity you could use (25 Delta put IV - 25 Delta call IV)/50 Delta IV You could simply subtract the IV's,but I like to normalize the values.. There are many other methods,but I like to keep it simple.. You will need historic values to put things into context..This may help https://www.ivolatility.com/doc/JOD-MIXON_(2011)_+_Appendix.pdf
I don't look at historical volatility at all when doing a calendar - simply the difference in vols between the short leg and the long. The greater this difference the more resilient the calendar.
Track the IVs on a per delta basis in Simply track each IV on a per delta basis for both EOD for big picture, and intraday to monitor and trade the extreme daily moves. Use historical expirations for assets that need seasonal adjustment, and constant maturity for apples-to-apples IV comparisons. You can either pay for this service, pay for the market data, or start doing it yourself. Once you've accumulated enough quality IV data, you can analyze any IV combination on the vol surface to determine what's rich/cheap, statistically signficant, etc. Sky's the limit.