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

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, obtained as the strict convex combination of two necessity measures. 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 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.