Fatou property,representations, and extensions of law-invariant risk measures on general Orlicz spaces

We provide a variety of results for quasiconvex, law-invariant functionals defined on a general Orlicz space, which extend well-known results from the setting of bounded random variables. First, we show that Delbaen’s representation of convex functionals with the Fatou property, which fails in a general Orlicz space, can always be achieved under the assumption of law-invariance. Second, we identify the class of Orlicz spaces where the characterization of the Fatou property in terms of norm-lower semicontinuity by Jouini, Schachermayer and Touzi continues to hold. Third, we extend Kusuoka’s representation to a general Orlicz space. Finally, we prove a version of the extension result by Filipovi? and Svindland by replacing norm-lower semicontinuity with the (generally non-equivalent) Fatou property. Our results have natural applications to the theory of risk measures.     

Maximal submarkets that replicate any option

In this article we study the replication of options in security markets X with a finite number of states. Specifically, we study the existence of maximal submarkets (subspaces) Y of X so that any option written on the elements of Y is replicated by a marketed asset x of X. So inside these subspaces the pricing problem is simple because any option is priced by the replicating portfolio. Using the theory of lattice-subspaces and positive bases developed by Polyrakis (Trans Am Math Soc 348:2793–2810, 1996; 351:4183–4203, 1999), we identify the set of all maximal replicated subspaces. In particular, for any maximal replicated subspace we determine a positive basis of the subspace. Moreover we show that the union of all maximal replicated subspaces is the set of all marketed securities x ? X{xin X} so that any option written on x is replicated. So we determine also the set of securities with replicated options.     

NONREPLICATION OF OPTIONS

In this paper, we study the replication of options in security markets X with a finite number of states. Specifically, we prove that in security markets without binary vectors, for any portfolio, at most m ? 3 options can be replicated where m is the number of states. This is an essential improvement of the result of Baptista where it is proved that the set of replicated options is of measure zero. Additionally, we extend the results of Aliprantis and Tourky on the nonreplication of options by generalizing their condition that markets are strongly resolving. Our results are based on the theory of lattice‐subspaces and positive bases.     

On the C‐property and ‐representations of risk measures

We identify a large class of Orlicz spaces for which the topology fails the C‐property introduced by Biagini and Frittelli. We also establish a variant of the C‐property and use it to prove a ‐representation theorem for proper convex increasing functionals, satisfying a suitable version of Delbaen's Fatou property, on Orlicz spaces with . Our results apply, in particular, to risk measures on all Orlicz spaces other than .