macrocephalopod

It seems to me that there's a deep relationship between the sticky strike vs sticky delta assumptions (in vol surface modelling) and Bayesian updating. A smile implies a probability distribution. So when spot moves, does your pdf stay the same (sticky strike) ...

or re-centre (sticky delta)? In reality it will be somewhere in between. The new spot price is a datapoint with which you can update your "prior" probability distribution. So the degree of delta-stickiness is related to how important you think the new spot price is.

the whole sticky strike / sticky delta convention always seems unnecessary and convoluted to me. You have a skew which is implying a certain relationship (ATM vols lower on spot up) and an actual realized measure of the same thing (covar or beta or whatever).

Can you explain how skew implies a relationship to spot? Im getting confused between spot and strike

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