Lower and Upper Bounds of Martingale Measure Densities in Continuous Time Markets

Abstract

In a continuous time market model we consider the problem of existence of an equivalent martingale measure with density lying within given lower and upper bounds and we characterize a necessary and sufficient condition for this. In this sense our main result can be regarded as a version of the fundamental theorem of asset pricing. In our approach we suggest an axiomatic description of prices on L p -spaces (with p ∈ [1, ∞)) and we rely on extension theorems for operators.

Dynamic no-good-deal pricing measures and extension theorems for linear operators on L ∞

“…In this way, the concept of tame operator defined in [15] (see also [1]) is directly embedded in the definition.…”

“…The approach suggested in [12] and [3] (see also [10] and [23]) is to rule out not only arbitrage opportunities, but also deals that are "too good to be true". In the same line, but with a different criterion, [1] and [15] suggest to restrict the set of equivalent martingale measures by choosing those with a density lying within pre-considered lower and upper bounds. This criterion is motivated by the observation that some form of control on the socalled tail events, i.e., crucial events appearing with small but positive probability, should be maintained when shifting from the physical measure (where statistical analysis is performed) to some equivalent martingale measure.…”

“…The prices are in fact described in axiomatic form via their properties. Note that the axiomatic presentation of a timeconsistent price system, already present in [6] and [15], is inspired by the literature on dynamic risk measures. See e.g.…”

“…Naturally, the representation in terms of a conditional expectation with respect to P 0 is connected to no-arbitrage pricing rules. This approach is already introduced in [1] and later developed in [15] to include the continuoustime case. However, the present paper differs from these works in several ways.…”

“…However, the present paper differs from these works in several ways. [1] and [15] study only applications to pricing measures with stochastic bounds on densities in the L p -setting for p ∈ [1, ∞). The present contribution is framed in an L ∞ -setting (recall that L ∞ -spaces are not separable for the topology induced by the norm).…”