Consider an index of the type:

$I(t)/I(t-1) = 1+ a(t) (S(t)/S(t-1)-1)+(1-a(t))r(t-(t-1))$

It is arbitrarily initialized. $r$ is the risk free rate.

a(t) is determined piecewise as:


Where $s_{target}$ is the contractual target vol, something that the client wants the index to be exposed to. $s_{estimated}$ is the historically estimated measure of the realized vol of $S$.

Now I have a vanilla option on $I(t)$. Using a local vol model does not feel sufficient because one needs to model the joint dynamics of $a$ (vol) and $S$ (spot). Also, if $a$ is taken to be a constant, this reduces to somewhat of a cliquet which is sensitive to stochastic volatility.

I am looking for information on the industry/academic standards on how products like these are usually modelled?


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