Portfolio management has always been a key topic in finance research area. While many researchers have studied portfolio management problems, most of the work to date assumes trading is frictionless. This dissertation presents our investigation of the optimal trading policies and efforts of applying duality method based on information relaxations to portfolio problems where the investor manages multiple securities and confronts trading frictions, in particular capital gain taxes and execution cost. In Chapter 2, we consider dynamic asset allocation problems where the investor is required to pay capital gains taxes on her investment gains. This is a very challenging problem because the tax to be paid whenever a security is sold depends on the tax basis, i.e. the price(s) at which the security was originally purchased. This feature results in high-dimensional and path-dependent problems which cannot be solved exactly except in the case of very stylized problems with just one or two securities and relatively few time periods.

The asset allocation problem with taxes has several variations depending on: (i) whether we use the exact or average tax-basis and (ii) whether we allow the full use of losses (FUL) or the limited use of losses (LUL). We consider all of these variations in this chapter but focus mainly on the exact and average-cost tax-basis LUL cases since these problems are the most realistic and generally the most challenging. We develop several sub-optimal trading policies for these problems and use duality techniques based on information relaxations to assess their performances. Our numerical experiments consider problems with as many as 20 securities and 20 time periods. The principal contribution of this chapter is in demonstrating that much larger problems can now be tackled through the use of sophisticated optimization techniques and duality methods based on information-relaxations. We show in fact that the dual formulation of exact tax-basis problems are much easier to solve than the corresponding primal problems.

Indeed, we can easily solve dual problem instances where the number of securities and time periods is much larger than 20. We also note, however, that while the average tax-basis problem is relatively easier to solve in general, its corresponding dual problem instances are non-convex and more difficult to solve. We therefore propose an approach for the average tax-basis dual problem that enables valid dual bounds to still be obtained. In Chapter 3, we consider a portfolio execution problem where a possibly risk-averse agent needs to trade a fixed number of shares in multiple stocks over a short time horizon. Our price dynamics can capture linear but stochastic temporary and permanent price impacts as well as stochastic volatility. In general it's not possible to solve even numerically for the optimal policy in this model, however, and so we must instead search for good sub-optimal policies. Our principal policy is a variant of an open-loop feedback control (OLFC) policy and we show how the corresponding OLFC value function may be used to construct good primal and dual bounds on the optimal value function.

The dual bound is constructed using the recently developed duality methods based on information relaxations. One of the contributions of this chapter is the identification of sufficient conditions to guarantee convexity, and hence tractability, of the associated dual problem instances. That said, we do not claim that the only plausible models are those where all dual problem instances are convex. We also show that it is straightforward to include a non-linear temporary price impact as well as return predictability in our model. We demonstrate numerically that good dual bounds can be computed quickly even when nested Monte-Carlo simulations are required to estimate the so-called dual penalties. These results suggest that the dual methodology can be applied in many models where closed-form expressions for the dual penalties cannot be computed. In Chapter 4, we apply duality methods based on information relaxations to dynamic zero-sum games. We show these methods can easily be used to construct dual lower and upper bounds for the optimal value of these games.

In particular, these bounds can be used to evaluate sub-optimal policies for zero-sum games when calculating the optimal policies and game value is intractable.