Asset pricing in an imperfect world

Abstract

In a model with no given probability measure, we consider asset pricing in the presence of frictions and other imperfections and characterize the property of coherent pricing, a notion related to (but much weaker than) the no arbitrage property. We show that prices are coherent if and only if the set of pricing measures is non empty, i.e. if pricing by expectation is possible.We then obtain a decomposition of coherent prices highlighting the role of bubbles. Eventually we show that under very weak conditions the coherent pricing of options allows for a very clear representation which allows, as in Breeden and Litzenberger [7], to extract the implied probability.principle of modern financial theory, i.e. risk neutral pricing. Although many an author inclines to believe that this basic principle rests on the simple tenet asserting that markets populated by rational economic agents cannot admit arbitrage opportunities, the proof of this claim, the fundamental theorem of asset pricing, has long been a challenge for mathematical economists, from Kreps [26] to Delbaen and Schachermayer [14]. In fact it requires a much more stringent condition than absence of arbitrage in which probability is needed to induce an appropriate topology.

Complete and Competitive Financial Markets in a Complex World

“…In this paper, following the thread of [13], we treat monotonicity as a primitive economic notion represented by a further transitive, reflexive binary relation on F(Ω). To distinguish it from the pointwise order ≥, we us the symbol ≥ * .…”

“…In a recent paper, Burzoni et al [9] adopt an approach quite similar to the present one. In [13,Theorem 1] we show that ≥ * may arise from a cash sub additive risk measure.…”

“…In addition to market power, our model departs from traditional financial literature inasmuch as it lacks of any particular mathematical structure, topological or measure theoretic. In particular, following the thread of our previous papers [10] and [13], we do not assume the existence of any exogenously given probability measure. Although this choice implies giving up the powerful artillery of stochastic analysis, particularly in continuous time, it permits, we believe, a better understanding of how financial markets work in a context of unrestricted complexity.…”

“…Our starting point is rather a criterion of economic rationality embodied in a partial order which describes on what all agents agree when saying that "f is more than g". This modeling of economic rationality, first introduced in [13], is referred to as common order in [9].…”

“…See[13] for a short discussion of alternative definitions of arbitrage in an imperfect market 5. This property, defined in slightly different terms, is discussed at length relatively to risk measures in[19].…”