Why Weighted Model Mixtures May Fail for Path-Dependent Options
Summary
The document asks whether a weighted mixture of two local-volatility models can provide an implied density for path-dependent options, including barrier and touch products. It presents the mixture as a weighted average of the component models’ vanilla option prices, with one weight governing each model’s contribution.
The response points to an analysis of mixture-of-model valuation and reports its central critique: a weighted average of prices from separate models does not fully specify a joint model for the underlying process. As a result, derivative prices can be ambiguous, and the weighted-average valuation rule may be internally inconsistent. This challenges using the formula as a general method for exotic options; the document does not derive an implied density or provide a replacement pricing procedure. It summarizes an article’s conclusions rather than presenting supporting calculations, and the excerpt does not establish how every possible mixture specification behaves.
Key ideas
- A weighted average of vanilla prices from local-volatility models does not by itself specify the underlying process.
- The cited analysis argues that underspecified mixtures can imply multiple prices for derivatives.
- The response warns that the weighted-average pricing rule can be self-inconsistent.
- The document offers no general density-recovery or path-dependent pricing method.
Tags
Full text
# Method to retrieve implied density for a mixture of local volatility model
# 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
## Answer by ir7 (score 2)
https://quant.stackexchange.com/a/66414
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.