Positivity 13 (2009), 201–224
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010201-24, published online October 28, 2008
Geometry of polar wedges in Riesz spaces
and super-replication prices in incomplete
Frank Oertel and Mark P. Owen
Abstract. This paper is devoted to a further generalisation of the main results
in  including the representation of the weak super-replication price (cf. equa-
tion (1.6)). In addition to the already established weakening of the technical
assumptions in (cf.and), the main results in  can be still gener-
alised by considering the geometric structure of the underlying problem (based
on the properties of Riesz spaces and polar wedges therein). In Section 5 we
show under which geometric conditions of the relevant sets the results still
hold (cf. Theorem 5.3 and Corollary 5.5). In particular, we can completely
remove the restrictive admissibility assumption and carry forward equation
(1.4) to a larger class of wedges K ⊆ L
(cf. Corollary 5.5).
Mathematics Subject Classiﬁcation (2000). 1B16, 46N10, 60G44.
Keywords. Super-replication, Incomplete markets, Contingent claims, Duality
theory, Weak topologies, Riesz spaces.
Although this paper primarily is written for a mathematical audience who need not
have a detailed knowledge of mathematical ﬁnance (including the related terminol-
ogy of stochastic analysis), we occasionally have to use some speciﬁc terminology
which cannot be explained in detail here, due to the limitation of space. There-
fore, we would like to refer the reader to the introductory overview references
[4,13,29,31] and the further references therein.
Firstly, let us revisit the ideal, non-realistic case, namely the case of a com-
plete (ﬁnancial) market. In a complete market, there is a unique arbitrage-free
price of a given derivative security or contingent claim. Its payoﬀ (i. e., its terminal
value) is modelled as a random variable X ∈ L
(P), where P denotes the original
The authors gratefully acknowledge support from EPSRC grant no. GR/S80202/01.