Valuation problems in incomplete markets

In my thesis, I study a class of stochastic optimization problems arising in valuation models in markets with frictions. These frictions are caused by “stochastic factors” that affect the dynamics of the underlying price process. They are assumed to be observable processes, and are, in general, correlated with the price process. Interesting examples include models with non-traded assets, stochastic volatility and models with nonlinear price dynamics.; I examine three valuation models arising in portfolio management and derivative pricing. In the first model, I investigate delta hedging with stochastic volatility. The goal is to specify the minimal required information about the volatility function in order to build the hedging strategy needed for the case of European derivatives. I accomplish this by bounding the delta-hedging problem with a stochastic control problem whose value function dominates the delta hedge. Numerical solutions are produced for a large class of derivatives, with closed form solutions shown for special cases.; In the second section, a valuation problem with multi-stochastic factors is examined. The goal is to provide the general solution to utility maximization problems that arise in the pricing of assets in incomplete markets. These problems are the cornerstone of studying pricing questions in markets with frictions. Specifically, I assume that besides the source of uncertainty for the underlying asset. there are two additional sources of randomness (such models of two factors arise often in practical applications). Using a novel transformation, I produce closed form solutions for the value function and the optimal policies which turn out to have a natural connection to coherent measures and other measures of risk appropriate for this type of incomplete market model.; The final problem considered is that of optimal investment with stopping. The goal is to investigate explicit solutions to the utility maximization problem in incomplete markets with discretionary stopping. The stochastic optimization problem turns out to be a free boundary problem with analytic solutions in certain cases. These models arise often in the pricing of contingent claims with early exercise via utility methods, as well as in optimal portfolio management problems with inhomogeneous financial characteristics.