Pricing, Hedging and Measuring Risk in Markets with Transaction Costs

这个项目的目标是解决存在交易成本的市场中的套期保值和定价问题。与此密切相关的问题是如何衡量这些市场的风险。困难在于，交易成本以一种自然的方式导致定值建筑。研究了随机偿付能力锥的集值风险度量。一个例子是有交易费的多元债权的超级套期保值价格。这种联系使得从实际角度研究更合理的价格界限成为可能，比如好的交易界限和无差别价格，它们已经被证明是在无摩擦的不完全市场中的一种强大的定价机制。另一个目标是解决在有交易成本的市场中，当未能达到或有债权的风险通过集值风险度量进行评估时的对冲问题。上述问题是通过最近的凸对偶理论来解决的，该理论是专门为集值函数设计的。特别是建立了集值主问题及其子问题的对偶问题，并研究了强对偶的条件。所选择的模型本身、集值优化问题以及所提出的求解方法都是数理金融学中的新方法，该方法还扩展了最优化理论的概念，对于有交易成本的市场模型比无摩擦模型更现实。金融业受益于对冲、定价和风险管理技术的研究，因为有必要应对交易成本导致的交易资产的买卖价差。在复杂的市场情况下，拥有一种以上货币的灵活风险评估工具和相应的风险管理技术是一种优势。因此，本研究项目有助于更好地理解多变量风险。

The objective of this project is to solve problems of hedging and pricing in markets where transaction costs are present. Closely related is the question on how to measure risk in these markets. The difficulty lies in the fact that transaction costs lead in a natural way to set-valued constructions. Set-valued risk measures for random solvency cones are studied. An example is the superhedging price of a multivariate claim under transaction costs. This link makes it possible to study price bounds that are more reasonable from a practical point of view like good deal bounds and indifference prices, which have already proven to be a powerful pricing mechanism in frictionless incomplete markets. Another goal is to solve a hedging problem in markets with transaction costs when the risk of falling short a contingent claim is evaluated by a set-valued risk measure. The above problems are solved by means of a recent convex duality theory, particularly designed for set-valued functions. Especially, dual problems for the main set-valued problem and its subproblems are established and conditions for strong duality will be studied. The chosen model itself, a set-valued optimization problem, as well as the proposed solution methods are new in mathematical finance andbeyond: The approach also extends concepts from optimization theory.Models for markets with transaction costs are more realistic than frictionless models. The finance industry benefits from research in hedging, pricing and risk managing techniques as there is a need to cope with bid-ask-spreads for traded assets caused by transaction costs. In complex market situations it is an advantage to have flexible tools for risk evaluation in terms of more than one currency and corresponding risk managing techniques. Thus, this research project leads to a better understanding of multivariate risks.