Titolo della tesi: Discrete time models for bid-ask pricing under Dempster-Shafer uncertainty
As is well-known, real financial markets depart from simplifying hypotheses of classical no-arbitrage pricing theory. In particular, they show the presence of frictions in the form of bid-ask spread. For this reason, the aim of the thesis is to provide a model able to manage these situations, relying on a non-linear pricing rule defined as (discounted) Choquet integral with respect to a belief function. Under the partially resolving uncertainty principle, we generalize the first fundamental theorem of asset pricing in the context of belief functions. Furthermore, we show that a generalized arbitrage-free lower pricing rule can be characterized as a (discounted) Choquet expectation with respect to an equivalent inner approximating (one-step) Choquet martingale belief function. Then, we generalize the Choquet pricing rule dinamically: we characterize a reference belief function such that a multiplicative binomial process satisfies a suitable version of time-homogeneity and Markov properties and we derive the induced conditional Choquet expectation operator. In a multi-period market with a risky asset admitting bid-ask spread, we assume that its lower price process is modeled by the proposed time-homogeneous Markov
multiplicative binomial process. Here, we generalize the theorem of change of measure, proving the existence of an equivalent one-step Choquet martingale belief function. Then, we prove that the (discounted) lower price process of a European derivative is a one-step Choquet martingale and a k-step Choquet super-martingale, for k >= 2.