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 and beyond: 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.
Risk measurement has gained a lot of interest in practice and academia over the last two decades. The standard approach is to use a (dynamic) scalar risk measure to describe the capital requirements necessary to account for randomness in the holdings of banks or other companies. But often the world is too complex to represent it by a scalar variable. The recent financial crisis showed that more sophisticated models and tools are necessary to adequately describe the financial risks in a dynamic, highly interconnected world. Illiquidity and other market frictions have an impact that can no longer be neglected. In the research done under NSF Grant "Pricing, Hedging and Measuring Risk in Markets with Transaction Costs" we proposed a genuine change of paradigm: Instead of relying on scalar measures, we consider multivariate risks and multivariate risk measures, so called set-valued risk measures. This enables to include models that take frictions (transaction costs, illiquidity, Tobin-taxes) into account, it allows to consider networks of banks and the systemic risk they inherit, and can deal with dynamic models and time consistency. The results from the NSF Grant include representation results of set-valued risk measures, and the definition and characterization of multi-portfolio time consistency for set-valued risk measures. Time consistency is important to ensure that decisions made over time do not contradict each other. We also studied possibilities how to calculate (dynamic) multivariate risk measures and developed algorithms to do so. These algorithms are also of general interest in multi criteria optimization as they allow to solve linear, respectively, convex vector optimization problems. Examples of set-valued risk measure are given which include set-valued shortfall and divergence risk measures (e.g. the Average Value at Risk and the entropic risk measure), but also the set of super hedging portfolios under transaction costs, which provide no-arbitrage price bounds for options in markets with transaction costs. For the latter one we showed that their calculation reduces to solving linear vector optimization problems backwards in time. The project provided research training to three Ph.D. students. One student has now graduated and assumed a tenure track faculty positions at a university. We have organized professional conferences in Princeton and Italy.
