Utility Maximisation in Incomplete Financial Markets

A basic question in mathematics of finance is the problem of an economic agent who invests in a financial market so as to maximize the expected utility of intermediate consumption and/or her terminal wealth at the end of the planning horizon. The aim of this project is to explore the following open questions in this field: (1) Utility maximization when wealth may become negative. (2) Efficient hedging for American options. (3) Utility maximization in models with transaction costs. (4) Equilibrium restrictions on the range of option prices. These problems are closely connected. They are natural extensions of the results obtained in a paper by Kramkov and Schachermayer (1999). In this paper, the duality theory was developed in full generality for an agent whose utility function does not allow for negative wealth and who is not subject to transaction costs such as sales taxes or brokerage fees. If one passes to utility functions giving finite values also to negative wealth (e.g. exponential utility), one is led to problem (1), which was analyzed in a paper by Schachermayer (2001). In this situation, open questions pertain - among others - to the case of non-locally bounded price processes and to the existence of the optimal hedging strategy for an investor who has sold an option. Problem (2) corresponds to the problem of maximizing utility from terminal wealth where the terminal time is uncertain. This yields a problem which is of the maxi-min type; it is of relevance both from an economical and a mathematical point of view. Concerning market models with transaction costs, the first step towards a satisfactory mathematical duality theory is to establish the validity of a "Fundamental Theorem of Asset Pricing". In this context, important advances along this line have been obtained very recently. These reveal new paths to derive the duality theory for problem (3) in full generality. Problem (4) corresponds to an application of the results on utility maximization to the theory of option pricing in incomplete markets. One says that a market with several interacting agents is in equilibrium if the total demand clears the market and each agent achieves maximum utility. The basic idea in problem (4) is to derive equlibrium- restrictions on option prices which do not depend on the agents` preferences and which at the same time sharpen the no-arbitrage bounds.

A basic problem of finance which every investor faces is the issue of portfolio optimisation. Each investor has to balance the expected returns versus the involved risk in his/her decisions to form a portfolio of investments. The mathematical treatment of this problem goes back as far as Daniel Bernoulli who proposed in 1738 the use of utility functions in order to model the trade-off between expectation and risk. This allowed him to propose a solution to the "Petersburg paradox". The modern development of this basic subject may be dated with the 1936 paper by Franz Alt followed by J. v. Neumann, O. Morgenstern, J. Savage, H. Markowitz and many other important authors. Since the work of R. Merton (1971) the dynamic aspects of portfolio optimisation linked this topic with the theory of stochastic processes. In the present research project some recent issues concerning the dynamic portfolio optimisation problem have been analysed in a number of papers. They focus around the following topics: 1) the pricing of options and other derivatives by marginal utility considerations; 2) risk minimisation and utility optimisation of insurance companies; 3) utility optimisation under transaction costs.