Outperformance and Tracking: A Framework for Optimal Active and Passive Portfolio Management

Portfolio management problems can be broadly divided into two classes of differing investing styles: active and passive. There are a number of philosophical and operational distinctions between these two investment approaches including mandate, complexity, activity level, constraints, and goals. Both forms of portfolio management, however, can involve absolute or relative goals. These goals are distinguished by the involvement (or lack thereof) of external processes in measuring an investor's performance. This thesis presents a unified framework for solving portfolio selection problems arising in both active and passive portfolio management with a focus on relative goals. We are, in particular, interested in stochastic optimization problems related to outperforming a selected benchmark and/or tracking a given portfolio, both natural and essential questions in portfolio management. In the first part of the thesis, we lay the foundation for our framework by introducing a flexible stochastic control problem that captures the range of goals we are interested in. We solve the problem, first, in a simple setting using a dynamic programming approach and derive an explicit closed-form solution for the optimal portfolio. We uncover several important features of the optimal portfolio, most notably, a decomposition result where the optimal solution is expressed in terms of the underlying benchmarks, the growth optimal portfolio, and the minimum variance portfolio. We probe the empirical performance of the optimal portfolio using historical as well as simulated market data. The latter parts of the thesis are dedicated to cases where (i) assets are driven by latent factors and general martingale noise processes, (ii) the market model and preference parameters are stochastic, and (iii) the investor has non-linear utility. These extensions are approached using variational analysis techniques which exploit the convexity of the underlying performance functionals to easily incorporate these additional features. In some instances, the optimal portfolio is characterized via forward backward stochastic differential equations (FBSDEs), and, due to the variational characterization, we establish the existence and uniqueness of solutions to those FBSDEs. The approach we take is also useful for lending interpretability to the solution of the well-known Merton problem when model parameters are stochastic. Moreover, in some special cases, we are able to extend the decomposition result established in the Brownian setting. From a practical perspective, this thesis also presents a fully implementable procedure for computing optimal portfolios for a hidden Markov model of asset prices, including parameter estimation along with filtering results for the hidden state in a general semimartingale setting. The investment performance is demonstrated with out-of-sample backtesting using historical market data.Ph.D