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Model Uncertainty and Expert Opinions in Continuous-Time Financial Markets

  • Model uncertainty is a challenge that is inherent in many applications of mathematical models in various areas, for instance in mathematical finance and stochastic control. Optimization procedures in general take place under a particular model. This model, however, might be misspecified due to statistical estimation errors and incomplete information. In that sense, any specified model must be understood as an approximation of the unknown "true" model. Difficulties arise since a strategy which is optimal under the approximating model might perform rather bad in the true model. A natural way to deal with model uncertainty is to consider worst-case optimization. The optimization problems that we are interested in are utility maximization problems in continuous-time financial markets. It is well known that drift parameters in such markets are notoriously difficult to estimate. To obtain strategies that are robust with respect to a possible misspecification of the drift we consider a worst-case utility maximization problem with ellipsoidal uncertainty sets for the drift parameter and with a constraint on the strategies that prevents a pure bond investment. By a dual approach we derive an explicit representation of the optimal strategy and prove a minimax theorem. This enables us to show that the optimal strategy converges to a generalized uniform diversification strategy as uncertainty increases. To come up with a reasonable uncertainty set, investors can use filtering techniques to estimate the drift of asset returns based on return observations as well as external sources of information, so-called expert opinions. In a Black-Scholes type financial market with a Gaussian drift process we investigate the asymptotic behavior of the filter as the frequency of expert opinions tends to infinity. We derive limit theorems stating that the information obtained from observing the discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process which can be interpreted as a continuous-time expert. Our convergence results carry over to convergence of the value function in a portfolio optimization problem with logarithmic utility. Lastly, we use our observations about how expert opinions improve drift estimates for our robust utility maximization problem. We show that our duality approach carries over to a financial market with non-constant drift and time-dependence in the uncertainty set. A time-dependent uncertainty set can then be defined based on a generic filter. We apply this to various investor filtrations and investigate which effect expert opinions have on the robust strategies.

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Author:Dorothee Westphal
URN (permanent link):urn:nbn:de:hbz:386-kluedo-58414
Publisher:Verlag Dr. Hut
Place of publication:München
Advisor:Jörn Sass
Document Type:Doctoral Thesis
Language of publication:English
Publication Date:2019/12/19
Year of Publication:2019
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2019/10/31
Date of the Publication (Server):2019/12/19
Number of page:VI, 183
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
Licence (German):Zweitveröffentlichung