Which would be the most appropriate models to find volatility trading opportunities (i.e. plot a theoretical volatility smile I can rely on) for the following instruments:
I know there may not be a clear cut answer, but any insight or guidance is appreciated.
You are competing against thousands of firms, many of them doing this professionally and employing people like the ones you see in the Vola Dynamics link I provided in the comments.
So my answer is, no you will not find trading opportunities. I go even further and claim you probably never will (on your own). If you use vendors liker Bloomberg, SuperD, and whoever else you can think of, you will also not find trading opportunities based on their, arguably (fairly) sophisticated, Vol Surfaces.
Realized vol is a single number per period. How do you intend to construct a surface from this? Generally, Black Scholes assumes a lognormal distribution of returns, meaning that the logarithmic return is normally distributed. In a nutshell, the Vol Smile mainly exists because this is not true. FX actually directly quotes IVOL in a way that adjusts for skewness and kurtosis. Meaning you have different Vols for different moneyness (delta) levels you look at.
For equity, the Vola Dynamics surface is really interesting (I assume also for commodity but I do not know enough about this asset class). SVI is a frequently used tool as @nnob2 stated. There are also mixed lognormal approaches but SVI will almost certainly beat that. It is not just curve fitting though. A large junk of work will be getting the data sorted. How to compute implied dividends and forwards, how to de-Americanize options that are quoted as American, how to get a reliable interest rate curve, how to handle illiquid quotes, missing data, outliers, timing differences (some underlying assets and options on the asset trade during different times or on different exchanges).
For commodity, there are at least two additional problems; the so called Samuelson Effect and seasonality. It will all depend on what commodity you look at though as precious metals are generally always handled (quoted in IVOL) like FX. Crude oil is essentially non-seasonal. On the other hand, seasonality is very prominent in natural gas markets. You will find lots of noise and gaps.
Some food for thought can be found in Timothy Klassen's linkedin page.
