13
The Banach Space of Workable Contingent
Claims in Arbitrage Theory (1997)
Abstract.For a locally bounded local martingaleS, we investigate the vector space
generated by the convex cone of maximal admissible contingent claims. By a max-
imal contingent claim we mean a random variable (H·S)∞, obtained as a final
result of applying the admissible trading strategyHto a price processSand which
is optimal in the sense that it cannot be dominated by another admissible trading
strategy. We show that there is a natural, measure-independent, norm on this space
and we give applications in Mathematical Finance.
R ́esum ́e.SiSest une martingale locale, localement born ́ee, on ́etudie l’espace vec-
toriel engendr ́eparlecˆone des actifs contingents maximaux. Une variable al ́eatoire
est un actif contingent maximal si elle peut s’ ́ecrire sous la forme (H·S)∞,o` u
la strat ́egieHest admissible et optimale dans le sense qu’elle n’est pas domin ́ee
par une autre strat ́egie admissible. Sur cet espace, on introduit une norme na-
turelle, invariante par changement de mesure, et on donne des applications en finance
math ́ematique.
13.1 Introduction
A basic problem in Mathematical Finance is to see under what conditions
the price of an asset, e.g. an option, is given by the expectation with respect
to a so-called risk neutral measure. The existence of such a measure follows
from no-arbitrage properties on the price processSof given assets, see [HK 79],
[HP 81], [K 81] for the first papers on the topic and see [DS 94] (Chap. 9 above)
for a general form of this theory and for references to earlier papers.
Investment strategiesHare described byS-integrable predictable pro-
cesses and the outcome of the strategy is described by the value at infinity
(H·S)∞. In order to avoid doubling strategies one has to introduce lower
bounds on the losses incurred by the economic agent. Mathematically this is
[DS 97] The Banach Space of Workable Contingent Claims in Arbitrage Theory.
Annales de l’IHP, vol. 33, no. 1, pp. 114–144, (1997).