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there have been loads of papers on skews ATM / OTM, volatility premium and such. Lots of explanations for why iv is different on same stock with different strikes focused on preference of informed traders.

But is there a model-based, non-behavioral explanation for the difference of implied vol for different strikes? Could it be approached through state price distribution?

Thanks and happy holidays

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This is an interesting question. I guess the fundamental belief is that any model serves the purpose of capturing what happens as aggregated behaviors in the market. Believe it or not, model also influences the market, especially when most of the transactions are done by people who utilize models (banks, funds, etc).

The basic idea of vol smile is the uncertainty of (realized) vol, and thus the demand for additional risk premia. This is kinda non-behavioral, but not model-based. It's an assumption but it checks out with the empirical observation. There's definitely some evolvement in people's views on the market. For a while in history, the vol was actually flat.

If you believe this is true, then the vol smile is an natural derivation from the addition of vol of vol. But I'll try to explain from the model-perspective. Let's start from Black-Scholes model (with some behavioral explanation). By the way, This is model-based, but somewhat behavioral:

If you plot the vega-gamma (second derivative of option price w.r.t vol) against the (log) strike from the Black-Scholes model, you would see a two-hump-shape curve.

enter image description here

Now if you long an ITM/OTM option, you are effectively long both vega and vega-gamma (since it's positive). And then you can short appropriate amount of ATM option to make vega-neutral (since ATM option has zero vega-gamma in Black-Scholes). Now you are long pure vega gamma, and whenever vol moves, you make money (if you believe that vol does move).

In this case, we'd be happy to buy high/low strike options if vol is flat, effectively ATM/OTM option prices are bid up, therefore the implied vol. This works for both high/low strikes, leading to a smile in vol.

Similar approach can be used to explain the vol skew (risk-reversal) by looking at vega-dspot, but I'll leave it out for now since it's not exactly what OP was asking about.

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  • $\begingroup$ think your comment is to the point of OP's question. Although that rationale can be applied to all preference in terms of strikes and / or term structure for options, no? since shorter term options and strikes farther away from ATM are less exposed to change in vol. $\endgroup$ – user201706 Dec 27 '16 at 1:09

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