Portfolio optimization in arbitrary dimensions in the presence of small bid-ask spreads
This thesis deals with the problem of maximizing the expected utility of terminal wealth in financial markets with an arbitrary number of risky assets in the presence of small bid-ask spreads. The goal is to determine an asymptotically optimal trading strategy and to quantify the asymptotic welfare impact of small proportional fees levied on investor's transactions. The approach taken in this study relies on the concept of a shadow price transforming the problem of portfolio optimization with proportional costs into a frictionless one. With the help of the shadow price, an asymptotically optimal trading strategy is shown to be a solution to a reflecting stochastic differential equation. The (stochastic) reflecting boundary is characterized as solution to a free-boundary problem. The boundary constrains the motion of the trading strategy to a domain known as the no-trade region. Instead of attempting to find exact solutions, we propose several simple domains as candidates for the no-trade region. Trading strategy to each of the domains is defined as solution to a stochastic Skorohod problem. By adapting the notion of the shadow price, we establish a duality relation between trading strategies and martingale measures for shadow-price processes. This allows us to derive an upper bound on the expected utility generated by each candidate strategy, which provides an estimate of the expected utility of the exact asymptotic optimizer. Expected utility of each trading strategy together with the associated upper bound are evaluated by means of numerical simulations. The simulations are run on the Black-Scholes model for portfolios of up to 30 risky assets.