Envelopes of equivalent martingale measures and a generalized no-arbitrage principle in a finite setting

• Published in: Annals of operations research Vol. 321; no. 1-2; pp. 103 - 137
• Main Authors: Cinfrignini, Andrea, Petturiti, Davide, Vantaggi, Barbara
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2023; Springer; Springer Nature B.V
• Subjects: Arbitrage; Business and Management; Combinatorics; Decision theory; Equivalence; Financial markets; Martingales; Martingales (Mathematics); Operations research; Operations Research/Decision Theory; Original Research; Payoffs; Pricing; Risk; Theory of Computation; Uncertainty; Italy; Generalized no-arbitrage principle; Belief functions; Lower pricing rule; Equivalent martingale measures
• ISSN: 0254-5330; 1572-9338
• DOI: 10.1007/s10479-022-05126-z

We consider a one-period market model composed by a risk-free asset and a risky asset with n possible future values (namely, a n -nomial market model). We characterize the lower envelope of the class of equivalent martingale measures in such market model, showing that it is a belief function. Then, we reformulate a general one-period pricing problem in the framework of belief functions: this allows to model frictions in the market and can be justified in terms of partially resolving uncertainty according to Jaffray. We provide a generalized no-arbitrage condition for a generic one-period market model under partially resolving uncertainty and show that the “risk-neutral” belief function arising in the one-period n -nomial market model does not allow to satisfy such condition. Finally, we derive a generalized arbitrage-free lower pricing rule through an inner approximation of the “risk-neutral” belief function arising in the one-period n -nomial market model.