# Method to retrieve implied density for a mixture of local volatility model

Given a mixture model of two local volatility models, the price for an option is given by:

$$V(K,T) = p V_{loc1}(K,T) + (1-p) V_{loc2}(K,T)$$

where $$V_{loc}(K,T)$$ is the price of the option given a dupire local volatility function and $$p$$ a weight.

Is there a way to retrieve the implied density for a path dependent option using this model? I am looking for a general solution which could be used on barrier options, double barriers and touch-type options.

Thanks

• Just use Breeden-Litzenberger formula applied to $V(K,T)$ where $V$ is the value of a call/put option. Aug 12 at 17:56

• It shows a way in the Appendix for barriers. Assuming (A2), one gets (A3) and (A4) (the PDE $f_{MM}$ solves). I think there are a few Brigo/Mercurio articles related out there too.