# Comparing implied volatility in 2 different correlated assets

The general idea here is that I am trying to compare the volatility surface of two different financial assets whose prices and returns time series exhibit a strong relationship/correlation :

The intuition here is that given how correlated the price and daily return time series are, it should be possible to compare the volatility surfaces of the two assets and determine whether one is mis priced vs the other (for example, is one volatility surface assigning far too much probability to a tail event vs the other, in which case we can sell tail options in one asset vs buy them in the other). My question is what would be the best way of going about doing that - or rather, given there probably isn't one "best" way, does anyone have any suggestions of how one could do this? I have had several thoughts which I will outline below, with some questions about each method.

1. The Theta-Gamma approach. Produce a model that gives us a realised beta between the two assets in delta 1 (something like a linear regression on the return time series, or some kind of model that allows for a time varying beta), and then compare this to the beta that is priced in between the two volatility surfaces. The way I guess you could compute this is too look at the theta gamma breakeven (i.e. the implied daily move) of the same expiry and equivalent delta options in each asset, take a ratio of these and then compare it to the realised beta from your model - this then gives you a sense of whether the relative daily moves priced in by the volatility surfaces are "off whack" versus the actual realised beta, leading to possible trading opportunities.

Questions - firstly does it make sense, secondly how would you compute comparable daily implied moves if the two assets use different options models (e.g the theta, gamma and ivol itself may not be directly comparable), and how would you do this for options away from At the Money?

1. Some kind of Stochastic Vol model. The idea here consists of 3 stages. Stage 1 is again model the delta 1 relationship using some kind of multivariate Stoch Vol model, multivariate GARCH model or some combination of univariate GARCH models and a copula (still thinking about what the "best" way would be, so any insight appreciated). Stage 2 is to back out "implied spot distributions" from the volatility surfaces of each asset - we do this by looking at Call Flies (e.g. How to derive the implied probability distribution from B-S volatilities?). Then finally Stage 3 is to run simulations from the model in Stage 1 which are "informed" by the one of the distributions from Stage 2 - e.g. we use the distribution of Asset Y (for a given expiry) from Stage 2 and use this information and our model from Stage 1 to run simulations. This should give us a future spot distribution of Asset X that is based on what the options market of Asset Y is pricing, which we can then compare to the actual distribution of Asset X that is implied from its volatility surface.

Questions - this idea seems a bit wild to me (I dont know how I came up with this to be honest), so the first question is does it make any sense, or is it completely silly? If it isnt silly, the second question would be how one would go about achieving Stage 3 in the above - i.e. how do we use the implied distribution from the vol surface of one of the assets "inform" the simulations from the model in Stage 1.

These were two random ideas I had - I havent read or seen much literature on this type of stuff so any other ideas would be appreciated! Thanks!