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Expand out pnl in terms of Greeks to include theta, gamma, vega, vanna and volga terms (assume delta is hedged). Take expectations and assume risk neutral so that E(pnl) = 0. Spot-vol covariance and vol-of-vol will be parameters. With some manipulation you can express implied …
… vol at each strike as a function of spot-vol covar (skewness) and vol-of-vol (convexity) depending on vanna/volga exposure at each strike.
This Peter Carr paper has some good details — https://poseidon01.ssrn.com/delivery.php?ID=887104096000094090107030026116070007059014046014006078009096019067075112006014090110126001100120033059005011077005074018001112016049064053067093018118113071001068088028055096108004065029105096024077124066116008093029119020107071097122088068006123022&EXT=pdf&INDEX=TRUE …
yeah i would strongly advise using the constrained model in this paper over a larger one, as a larger one will fit the data better but will sometimes assign loopy economic values to the vanna / volga pricing
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