272 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory
The norm on the spaceGcan be calculated using Theorem 13.3.17 above and
using the characterisation of the measures inMe. We find
2 ‖f+gh‖=sup
μEP
[∫
[− 1 ,+1]|f+xh|μω(dx)]
.
For givenωthe measureμω(dx)on[− 1 ,+1] that maximises
∫
[− 1 ,+1]|f+xh|
×μω(dx) and that satisfies
∫
[− 1 ,+1]xμω(dx) = 0 is according to balayage ar-
guments (repeated application of Jensen’s inequality) the measure that gives
mass^12 to both−1 and +1. This measure does not satisfy the requirements
since it is not equivalent to the measuremon [− 1 ,1]. But an easy approxi-
mation argument shows nevertheless that
2 ‖f+gh‖=EP[
|f+h|+|f−h|
2]
.
This can be rewritten as
2 ‖f+gh‖=EP[max(|f|,|h|)].This equality shows thatGis isomorphic to anL^1 -space.
on the SetMe.......................................... 13.5 The Value of Maximal Admissible Contingent Claims
Contingent Claims on the SetMe
As shown in Example 13.4.4, maximal contingent claimsfmay have different
expected values for different measures inMe. In Chap. 10 we showed that
under rather general conditions such a phenomenon is generic for incomplete
markets. More precisely we have:
Theorem 13.5.1. (Theorem 10.3.1)Suppose that S is a continuousd-
dimensional semi-martingale with the (NFLVR) property. If there is a contin-
uous local martingaleWsuch that〈W, S〉=0butd〈W, W〉is not singular to
d〈S, S〉, then for eachRinMe, there is a maximal contingent claimf∈K 1
such thatER[f]< 0.
The preceding theorem brings up the question whether for givenf∈Kmax,
the set of measuresQ∈Mesuch thatEQ[f]=0isbig.
Theorem 13.5.2.Suppose thatSis a locally bounded semi-martingale that
satisfies the (NFLVR) property. Iffis a maximal contingent claim i.e.f∈
Kmax, then the mapping
φ: M(S) −→ R
Q −→ EQ[f]