Fuzzy measures and asset prices: accounting for information ambiguity

Abstract

A recent stream of literature has suggested that many market imperfections or ‘puzzles’ can be easily explained once information ambiguity, or knightian uncertainty is taken into account. Here we propose a parametric representation of this concept by means of a special class of fuzzy measures, known as g _λ-measures. The parameter λ may be considered an indicator of uncertainty. Starting with a distribution, a value λ in (0, ∞) and a benchmark utility function we obtain a sub-additive expected utility, representing uncertainty aversion. A dual value λ∗ in (−1, 0) defining a super-additive expected utility is also recovered, while the benchmark expected utility is obtained for λ = λ∗ = 0. The two measures may be considered as lower and upper bounds of expected utility with respect to a set of probability measures, in the spirit of Gilboa-Schmeidler MMEU theory and of Dempster probability interval approach. The parametrization may be used to determine the effect of information ambiguity on asset prices in a very straightforward way. As examples, we determine the price of a corporate debt contract and a ‘fuzzified’ version of the Black and Scholes model.

Keywords:

• Knightian Uncertainty
• Market Incompleteness
• Non-additive Measures
• Asset Pricing