I'm new to volatility modeling, I'm struggling to understand when to use a Local Vol model and when to use Stochastic Vol Model, Also now we use a hybrid model combining the two models ? Can someone please provide some explanation ?
First, what is the SLV? It combines LV (not really a model, just uses vanilla surface to get a grid) with SV (in a nutshell, BSM with a separate stochastic process for vol, hence multiple dynamic factors).
Shortcomings SLV tries to address?:
- BS does not price exotic option well.
LVcalibrates nicely to vanilla but the calibrated leverage surface is typically observed to flatten with maturity which means the forward volatility smile will be less convex than on the initial pricing date and you will not be pricing deals properly which are primarily sensitive to forward volatility skew and smile (cliquets and co).
SVprices for barriers and touches tend to be overvalued by SV (undervalued by LV).
In SLV, mostly ($\xi$) vol of vol and correlation ($\rho$) control the mixing of LV and SV. Hence, appropriate calibration of the mixing parameters will allow you to closely match market quotes for exotic options. LV and stochastic SV are simply degenerate cases where the mixing fraction is such that only one or the other is used (e.g. if $\xi = 1$, SLV becomes purely SV).
These questions/answers should also help