# 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, 2021 at 17:56

## 1 Answer

I think Piterbarg's "Mixture of Models: A Simple Recipe for a … Hangover?" article would interest you (including Appendix A. Can Barrier Options be Valued with the “Weighted Average” Formula?). I inserted its abstract below.

The idea of using a weighted average of derivative security prices computed using different “simple” models (the so-called “mixture of models”, or “ensemble of models”, approach) has been put forth recently by a number of authors. Some view it as a simple way to add stochastic volatility to virtually any model, and others advocate it on the grounds that it provides a simple and tractable method for capturing certain market characteristics, most importantly volatility smile. Ease of calibration to market prices of vanilla and exotic instruments is also cited as the approach’s redeeming quality. While not disputing the fact that such “models” are easy to calibrate, we explain that these models are under-specified (leading to multiple possible prices of derivatives). We also demonstrate that the “weighted average” valuation formula, the main selling point of the “mixture of models” approach, is self-inconsistent and cannot be used for valuation.

• I read the article and it does not point to how to retrieve the implied densities Aug 12, 2021 at 17:07
• 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.
– ir7
Aug 12, 2021 at 18:03