Computing Optimal Portfolios under Partial Information

About 1970 Merton derived in continuous time optimal dynamic portfolio policies using stochastic control theory. For a utility maximization criterion using utility functions with constant relative risk aversion it is optimal to keep a constant fraction of the wealth (portfolio value) invested in each stock. But while the Black-Scholes option-pricing formula, derived in the same market model, was widely accepted in practice and is still an important benchmark, the Merton strategy never had such a success. For optimizing portfolios practitioners still prefer the static Nobel Prize winning Markowitz model. For option pricing the drift parameter of the stocks cancels out, but for the optimization it is of uttermost importance. One reason for the poor performance of the Merton strategey might be the assumption of a constant drift parameter which implies selling in a bull market and buying in a bear market. So a more realistic modelling of the drift as a suitable stochastic process might improve the performance. But then the investor can only observe the prices and not the underlying drift process, meaning that only partial information is available. A further improvement can be expected by the introduction of stochastic volatility models. In the last dozen years the subject of portfolio optimization under partial information has been studied widely. Besides some extensions of the models the emphasis of the project will be placed on the efficient computation and implementation of theses strategies (including parameter estimation). In the context of partial information the literature on the latter is very sparse. We plan (i) to extend the model to cover different models of stochastic volatility and convex constraints, (ii) to improve the parameter estimation by replacing the EM algorithm with specially designed Markov chain Monte Carlo methods and moment based methods, and (iii) to apply quasi-Monte Carlo methods to compute the optimal trading strategies more effectively. In this project methods of mathematical finance, probability theory, statistics and number theory are to be combined. Justified by the promising results of the previous work we hope in addition to the expected mathematical achievements that this project can be a step to make dynamic portfolio more attractive, even for practitioners.

About 1970 Merton derived in continuous time optimal dynamic portfolio policies using stochastic control theory. For a utility maximization criterion using utility functions with constant relative risk aversion it is optimal to keep a constant fraction of the wealth (portfolio value) invested in each stock. But while the Black-Scholes option-pricing formula, derived in the same market model, was widely accepted in practice and is still an important benchmark, the Merton strategy never had such a success. For optimizing portfolios practitioners still prefer the static Nobel Prize winning Markowitz model. For option pricing the drift parameter of the stocks cancels out, but for the optimization it is of uttermost importance. One reason for the poor performance of the Merton strategey might be the assumption of a constant drift parameter which implies selling in a bull market and buying in a bear market. So a more realistic modelling of the drift as a suitable stochastic process might improve the performance. But then the investor can only observe the prices and not the underlying drift process, meaning that only partial information is available. A further improvement can be expected by the introduction of stochastic volatility models. In the last dozen years the subject of portfolio optimization under partial information has been studied widely. Besides some extensions of the models the emphasis of the project will be placed on the efficient computation and implementation of theses strategies (including parameter estimation). In the context of partial information the literature on the latter is very sparse. We plan (i) to extend the model to cover different models of stochastic volatility and convex constraints, (ii) to improve the parameter estimation by replacing the EM algorithm with specially designed Markov chain Monte Carlo methods and moment based methods, and (iii) to apply quasi-Monte Carlo methods to compute the optimal trading strategies more effectively. In this project methods of mathematical finance, probability theory, statistics and number theory are to be combined. Justified by the promising results of the previous work we hope in addition to the expected mathematical achievements that this project can be a step to make dynamic portfolio more attractive, even for practitioners.