Market Completion and Robust Utility Maximization

In this thesis we study two problems of financial mathematics that are closely related. The first part proposes a method to find prices and hedging strategies for risky claims exposed to a risk factor that is not hedgeable on a financial market. In the second part we calculate the maximal utility and optimal trading strategies on incomplete markets using Backward Stochastic Differential Equations. We consider agents with incomes exposed to a non–hedgeable external source of risk who complete the market by creating either a bond or by signing contracts. Another possibility is a risk bond issued by an insurance company. The sources of risk we think of may be insurance, weather or climate risk. Stock prices are seen as exogenuosly given. We calculate prices for the additional securities such that supply is equal to demand, the market clears partially. The preferences of the agents are described by expected utility. In Chapter 2 through Chapter 4 the agents use exponential utility functions, the model is placed in a Brownian filtration. In order to find the equilibrium price, we use Backward Stochastic Differential Equations. Chapter 5 provides a one–period model where the agents use utility functions satisfying the Inada condition. The second part of this thesis considers the robust utility maximization problem of a small agent on a incomplete financial market. The model is placed in a Brownian filtration. Either the probability measure or drift and volatility of the stock price process are uncertain. The trading strategies are constrained to closed convex sets. We apply a martingale argument and solve a saddle point problem. The solution of a Backward Stochastic Differential Equation describes the maximizing trading strategy as well as the probability measure that is used in the evaluation of the robust utility. We consider the exponential, the power and the logarithmic utility functions. For the exponential utility function we calculate utility indifference prices of not perfectly hedgeable claims. Finally, we apply those techniques to the maximization of the expected utility with respect to a single probability measure. We apply a martingale argument and solve maximization problems instead of saddle point problems. This allows us to consider closed, in general non–convex constraints on the values of trading strategies.