Optimal Portfolio and Consumption with Transaction Costs

In Chapter 1, we study optimal portfolio and consumption with both fixed and proportional transaction costs. For a power utility function we find an explicit solution to the HJB equation governing the no-transaction region. Based on the explicit solution, we formulate a combined stochastic and impulse control problem as a quasi-variational inequality and find the transaction regions, the no-transaction region, and the boundary curves separating them. We show that the explicit solution we find satisfies the verification theorem and it is also a viscosity solution for the quasi-variational inequality. We present numerical results where we compare the various cases of the fixed and proportional transaction costs.;In Chapter 2, we discuss the optimal portfolio and consumption on multiple risky assets with both fixed and proportional transaction costs. Explicit solutions to the corresponding HJB equations are provided. The explicit solutions are viscosity solutions. Numerical results for two risky assets and N risky assets are given

On using shadow prices in portfolio optimization with transaction costs

In frictionless markets, utility maximization problems are typically solved either by stochastic control or by martingale methods. Beginning with the seminal paper of Davis and Norman [Math. Oper. Res. 15 (1990) 676--713], stochastic control theory has also been used to solve various problems of this type in the presence of proportional transaction costs. Martingale methods, on the other hand, have so far only been used to derive general structural results. These apply the duality theory for frictionless markets typically to a fictitious shadow price process lying within the bid-ask bounds of the real price process. In this paper, we show that this dual approach can actually be used for both deriving a candidate solution and verification in Merton's problem with logarithmic utility and proportional transaction costs. In particular, we determine the shadow price process.Comment: Published in at http://dx.doi.org/10.1214/09-AAP648 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Unified Approach to Portfolio Optimization with Linear Transaction Costs

In this paper we study the continuous time optimal portfolio selection problem for an investor with a finite horizon who maximizes expected utility of terminal wealth and faces transaction costs in the capital market. It is well known that, depending on a particular structure of transaction costs, such a problem is formulated and solved within either stochastic singular control or stochastic impulse control framework. In this paper we propose a unified framework, which generalizes the contemporary approaches and is capable to deal with any problem where transaction costs are a linear/piecewise-linear function of the volume of trade. We also discuss some methods for solving numerically the problem within our unified framework.portfolio choice, transaction costs, stochastic singular control, stochastic impulse control, computational methods

Optimal Investment with Random Endowments and Transaction Costs: Duality Theory and Shadow Prices

This paper studies the utility maximization on the terminal wealth with random endowments and proportional transaction costs. To deal with unbounded random payoffs from some illiquid claims, we propose to work with the acceptable portfolios defined via the consistent price system (CPS) such that the liquidation value processes stay above some stochastic thresholds. In the market consisting of one riskless bond and one risky asset, we obtain a type of super-hedging result. Based on this characterization of the primal space, the existence and uniqueness of the optimal solution for the utility maximization problem are established using the duality approach. As an important application of the duality theorem, we provide some sufficient conditions for the existence of a shadow price process with random endowments in a generalized form as well as in the usual sense using acceptable portfolios.Comment: Final version. To appear in Mathematics and Financial Economics. Keywords: Proportional Transaction Costs, Unbounded Random Endowments, Acceptable Portfolios, Super-hedging Theorem, Utility Maximization, Shadow Prices, Convex Dualit