SIPTA Seminar- B.Vantaggi: A dynamic Choquet pricing rule with bid-ask spreads under D–S uncertainty

ABSTRACT: Imprecise probabilities play a central role in many fields, and finance is surely one of the most promising sources of interesting problems, challenging the classical probabilistic setting. As a prototypical example, real markets show frictions, most evidently in the form of bid-ask prices for every asset. Therefore, addressing asset pricing in a real market requires to deal with a non-linear dynamic model of uncertainty which possesses, at the same time, a nice parameterization so as to allow for both a semantic interpretation and a computationally feasible market-consistent calibration.

With this target problem in mind, in this talk we introduce a particular imprecise stochastic process: a multiplicative binomial process under Dempster-Shafer uncertainty (briefly, a DS-multiplicative binomial process). This process relies on suitable notions of Markov and time-homogeneity and on the product rule of conditioning for belief functions. Moreover, we investigate to what extent such properties can be constrained by means of one-step time-homogeneity.

Such a process permits to model the price evolution of a stock, allowing for frictions in the form of bid-ask spreads. Furthermore, it gives automatically rise to a conditional Choquet expectation operator. Next, we consider a market formed by a frictionless risk-free bond (whose price is modeled by a deterministic process) and a non-dividend paying stock with frictions (whose lower price is modeled by a DS-multiplicative binomial process).

In this market we prove an analog of the classical theorem of change of measure relying on the notion of equivalent one-step Choquet martingale belief function. We then propose a dynamic Choquet pricing rule with bid-ask spreads showing that the discounted lower price process of a European derivative contract on the stock is a Choquet super-martingale. We also provide a normative justification in terms of a dynamic generalized no-arbitrage condition relying on the notion of partially resolving uncertainty due to Jaffray. Finally, we introduce a market consistent calibration procedure and show the use of the calibrated model in bid-ask option pricing.

[Work in collaboration with Andrea Cinfrignini (Sapienza University of Rome) and Davide Petturiti (University of Perugia)]

This talk is part of a series of seminars on imprecise probabilities that are organized by SIPTA, the "Society for Imprecise Probabilities: Theories and Applications". We also organize conferences and schools, provide documentation and maintain a mailing list and blog. More information is available at http://sipta.org. Info on the SIPTA seminars in particular is available at http://sipta.org/events/sipta-seminars

Contents
00:00 - Start
03:36 - Motivation
26:14 - A glimpse of Dempster-Shafer theory
34:22 - Theory of DS-multiplicative binomial processes
45:43 - A dynamic lower pricing rule
59:04 - Market consistent calibration and bid-ask option pricing
01:05:22 - Conclusion and future perspectives