On Utility-Based Investment, Pricing and Hedging in Incomplete Markets

This thesis deals with rational investors who maximize their expected utility in incomplete markets. In Part I, we consider models where incompleteness is induced by jumps and stochastic volatility. Using martingale methods we determine optimal investment strategies for power utility in a wide class of different models. Moreover, we show how first-order approximations of utility-based prices and hedging strategies can be computed by solving a quadratic hedging problem under a suitable measure. This representation result is then applied to affine stochastic volatility models leading to semi-explicit solutions. In Part II, we deal with incompleteness due to proportional transaction costs. In finite discrete time we establish that there always exists a shadow price process, which lies within the bid-ask bounds of the original market with transaction costs and leads to the same maximal expected utility. We then show that this idea can also be used in actual computations. This is done by reconsidering the classical Merton problem with transaction costs and solving it by computing the shadow price and the optimal strategy simultaneously.