Fuzzy Esscher changes of measure and copula invariance in Lévy markets

Abstract

In the context of a multidimensional exponential Lévy market, we focus on the Esscher change of measure and suggest a more flexible tool allowing for a fuzzy version of the standard Esscher transform. Motivated both by the empirical incompatibility of market data and the analytical form of the standard Esscher transform (see [8]) and by the desire to introduce a pricing technique under incompleteness conditions, we detect the impact of fuzziness in terms of measure change function and in contingent claims' pricing. In a multidimensional setting the fuzzy Esscher transform is a copula whose invariance, under margins' transformations induced by a change of measure, is investigated and connected to the notion of the absence of arbitrage opportunities. We highlight how Esscher transform, primarily used in pricing techniques, preserves the invariance of the aggregation operator and it can be generalized to the fuzzy version assuming that the measurable functions defining the Choquet marginal integrals are increasing. Furthermore, the empirical evidence seems to suggest that a weaker concept of invariance may be more suitable, i.e. the ε-measure invariance, coherent with the Esscher fuzzy copula tool. An empirical experiment for our model will make clear how this blurring technique fits the market data.