Conference Agenda

This study suggests a parsimonious stationary diffusion model for the dynamics of stock prices relative to the entire market. Its aim is to contribute to the question how to choose the relative weights in a diversified portfolio and, in particular, whether the market portfolio behaves close to optimally in terms of the long-term growth rate. Specifically, we introduce the elasticity bias as a measure of the market portfolio's suboptimality. We heavily rely on the observed long-term stability of the capital distribution curve, which also served as a starting point for the Stochastic Portfolio Theory in the sense of Fernholz.

Classical approaches to optimal portfolio selection problems are based on probabilistic models for the asset returns or prices. However, by now it is well observed that the performance of optimal portfolios is highly sensitive to model misspecifications. To account for various type of model risk, robust and model-free approaches have gained increasing importance in portfolio theory. In this talk, we develop a pathwise framework and methodology to analyze the stability of well-known 'optimal' portfolios in local volatility models under model uncertainty. In particular, we study the pathwise stability of the classical log-optimal portfolio with respect to the model parameters and investigate the pathwise error created by trading with respect to a time-discretized version of the log-optimal portfolio.

Traditional portfolio optimization methods often rely on precise probabilistic models, which may be inadequate for reflecting the full extent of uncertainty present in financial markets. In response, robust optimization approaches have emerged, focusing on worst-case scenarios by considering a set of plausible probability measures instead of relying on a single model. The objective of this research is to develop a general framework for robust utility maximization. We introduce a novel assumption on utility functions that guarantees certain desirable properties, which allow us to derive a minimax result in general settings. This assumption is satisfied for log and power utility functions. The minimax result guarantees the existence of optimal strategies, particularly in continuous-time financial markets with uncertainty in both drift and volatility, without the need for a dominating reference measure. We compare our setting with similar approaches from literature.

In financial markets simple portfolio strategies often outperform more sophisticated optimized ones. E.g., in a one-period setting the equal weight or $1/N$-strategy often provides more stable results than mean-variance-optimal strategies. This is due to the estimation error for the mean and can be rigorously explained by showing that for increasing uncertainty on the means the equal weight strategy becomes optimal, which is due to its robustness. In earlier work, we extended this result to continuous-time strategies in a multivariate Black-Scholes type market. To this end we derived optimal trading strategies for maximizing expected utility of terminal wealth under CRRA utility when we have Knightian uncertainty on the drift, meaning that the only information is that the drift parameter lies in an uncertainty set. The investor takes this into account by considering the worst possible drift within this set. We showed that indeed a uniform strategy is asymptotically optimal when uncertainty increases. After presenting new results on the convergence, we then focus on a financial market with a stochastic drift process in view of uncertainty. We combine the worst-case approach with filtering techniques by defining an ellipsoidal uncertainty set based on the filters. We demonstrate that investors need to choose a robust strategy to profit from additional information.