on capital requirements and optimal strategies to achieve acceptability

This thesis is concerned with an agent in a financial market who wishes to trade between day t = 0 and day t = τ . The agent starts with a specific objective in mind, which defines a class of acceptable (random) financial positions at the end of day τ . He wants to determine an initial amount and a self-financing trading strategy which will turn the discounted terminal value of his portfolio acceptable. By suitable choices of acceptable positions, this simple structure becomes flexible enough to encompass most of the common problems in mathematical finance. There are at least two distinct viewpoints of acceptability: one, which comes from the theory of measures of risk, stipulates ‘proper’ measures of risk as a real function evaluated on future random net worths. The set of acceptable net worths are then defined by their non-positive risks. The other is a generalization of arbitrage opportunities. It defines acceptable positions as what everybody finds as good deals, and defines prices which are fair for both the buyer and the seller, in the sense that it rules out such good deals for both of them. In this thesis, we propose results determining the minimum capital requirement and a self-financing trading strategy via which the aforementioned agent can lead his wealth towards acceptability. Such minimum capital requirement are generalizations of the classical superhedging prices, although the mathematical techniques we apply are novel. The trading strategies can also be interpreted as generalized hedging strategies, and we propose new theoretical and computational methods to evaluate them.