Asset allocation and portfolio optimization with small transaction costs

This thesis is devoted to the asset allocation and portfolio optimization with small transaction costs. Three topics are studied. The first and second topics are on asset allocation problems with purely proportional transaction cost and strictly positive transaction cost, respectively. The fundamental objective is to keep the asset portfolio close to a target portfolio and in the meantime to reduce the trading cost in doing so. For each problem, we derive the quasi-variational inequality and prove a verification theorem giving sufficient conditions for a QVI solution to be the value function. The optimal strategy is a singular control in the first topic and an impulse control in the second. Furthermore, we provide a matrix exponential representation of the QVI solution for both problems and perform asymptotic analysis to characterize the optimal no transaction region when transaction costs are sufficiently small. Additionally, for both topics, we apply the asymptotic results for the boundary points and derive an expansion for the QVI solution, the optimality of which can be shown via verification theorem (up to leading orders). The third topic is on portfolio optimization with proportional transaction cost. We construct an efficient frontier problem (EFP) of maximizing expected terminal utility and minimizing terminal CVAR. We first solve the frictionless case by duality approach and nonsmooth analysis. For three representative utility functions, we obtain numerically the optimal trading strategy, optimal expected terminal utility and optimal CVAR. Our analysis of how these three quantities change with respect to different CVAR constraints provides flexibility for an investor to balance return and risk according to her own preference. We then include transaction cost to the EFP, which is equivalent to including transaction cost to a non-smooth utility maximization problem. Asymptotic analysis gives expansions for no transaction boundaries which are then applied to EFPs with different utility functions. This topic ends by numerical analysis on impact of introducing CVAR constraint and/or transaction cost for classical utility maximization problem.