Title: Filtration Reduction and Market Completeness

 

Abstract: We study how filtration reduction can be used to obtain a unique equivalent martingale measure in incomplete markets. We provide a novel solution to the long-standing problem of non-uniqueness of the pricing measure. Our treatment is essentially self-contained and does not involve the introduction of external criteria (i.e. utility functions, minimal martingale measures, minimal entropy measures etc). We compare and contrast two distinct markets - the original and a fictitious - each associated with a different filtration, and employ the first and second fundamental theorems of asset pricing in both of these two markets. Our approach relies on the concept of consistent uplifts to obtain a unique measure in the original incomplete market. We recast the problem of choosing a measure into a task of choosing what information is relevant to the trader and what information can be discarded, averaged out or aggregated in some other meaningful sense. We carry out this program in settings of increasing generality: discrete-time general state-space market models, Brownian motion models and jump-diffusion models. This is joint work with Professor Robert A. Jarrow.