Fuzzy option prices with different sources of information smartly averaged

In a previous proposal we were able to elicit membership functions through probability-possibility transformations induced by confidence intervals around the median of specific simulating distributions. Hence we got so called ``fuzzy numbers", i.e. unimodal membership functions with nested $\alpha$-cuts. It was left open the problem of merging such kind of fuzzy numbers whenever different, generally more than two, sources of information are considered.\\ We give now a proposal of two different weighted fuzzy averages between fuzzy numbers. Such operators profit from $\alpha$-cuts and LR representations of fuzzy numbers. One operator is intended to generalize, through specific deformations of standard fuzzy means, the disjunction and the other to generalize the conjunction. Generalizations emphasize agreement or not between different sources of information. Such conflicts, as well as agreements, are endogenously embedded inside the average weights of the two new operators by measuring distances or superimposition between $\alpha$-cuts. No exogenous elements are added, except for the choice of the parameters of the deformation that emphasizes conflicts.\\ The main novelty is the aggregations performed among several $\alpha$-cuts by considering full/partial overlapping and generalizing Marzullo's algorithm (designed to compute the ``relaxed'' intersections among different information sources).\\ The proposal is motivated by the practical problem of assessing the fuzzy volatility parameter via both the historical volatility and the VIX estimators. In particular, for each estimator, different scenarios are considered on the base of historical data and experts evaluations.\\ Emphasis is posed on the consequences of the new operators on the fuzzy option pricing both in multi-periodal binary and in Black and Scholes environments. Crisp bid-ask price intervals are compared with fuzzy prices obtained through both new operators and standard fuzzy mean. Such comparisons are based on proper similarity indexes: the Bhattacharya distance and usual fuzzy similarity.