Das Institut für Mathematische Wirtschaftsforschung veranstaltet im Rahmen des Bielefeld Stochastic Afternoon regelmäßig Seminare zum Thema Finanzmathematik. Das Programm des aktuellen Semesters finden Sie hier.
Christoph Kühn (University of Frankfurt am Main)
Titel: Portfolio optimization in models with capital gains taxes
Abstract: In most countries, trading gains have to be taxed. The modeling is complicated by the rule that gains on assets are taxed when assets are sold and not when gains actually accrue. Thus, for positive interest rates, there is an incentive to realize losses immediately and to defer the realization of profits. We discuss the duality theory for the utility maximization problem with a linear tax in discrete time. We show under what conditions there exists a system of shadow prices. These are price processes in a fictitious frictionless market a tax-exempt investor is faced with. For every transaction, the fictitious market is at least as favorable, but the optimal fictitious strategy trades at the same costs as in the original market, which results in optimality there. In addition, we show how the model can be extended to continuous time. We construct the tax payment stream for all adapted trading strategies with caglad paths, that means beyond strategies of finite variation. The talk concludes with open problems and conjectures (parts of the talk are based on joint work with Björn Ulbricht).
Fausto Gozzi (LUISS Guido Carli University of Rome)
Titel: Optimal portfolio choice with path dependent labor income: The infinite horizon case
Abstract: We consider an infinite horizon portfolio choice problem with borrowing constraints where an agent receives labor income that adjusts slowly to financial market shocks. The novelty of the model is the path-dependence of the wage income process, which leads to an infinite dimensional stochastic optimal control problem. We solve completely the problem, and find explicitly the optimal controls in feedback form. This is possible because we are able to find an explicit solution to the associated infinite dimensional Hamilton-Jacobi-Bellman (HJB) equation, even if state constraints are present. To the best of our knowledge, this is the first infinite dimensional generalization of Merton's optimal portfolio problem where explicit solutions can be found. The explicit solution allows us to study and discuss the behavior of optimal solutions. We conclude, if there is time, by showing how the solution strategy used here can be deployed to solve other problems, such as the finite horizon version of the model. (Joint work with Enrico Biffis and Cecilia Prosdocimi).
Peter Bank (TU Berlin)
Titel: Representing Stochastic Processes and Irreversible Investment Problems - When Preferences Clash with Surprises
Abstract: We show how some stochastic singular control problems arising, e.g., when investments are irreversible, can be linked to certain stochastic representations of the underlying stochastic processes. In this work, we extend beyond earlier results with Nicole El Karoui in a number of ways, providing a maximality result for solutions to the representation problem, a way to interpolate between predictable and optional representation by use of Meyer sigma-fields as well as allowing for preferred habitat preferences in the target functional. The latter necessitates a careful analysis of generalized optimal stopping problems and allows for an understanding why optimal controls may be increasing, but neither right- nor left-continuous. (This is joint work with David Beßlich.)
Jan Obloj (University of Oxford)
Titel: Pointwise arbitrage pricing theory in discrete time
Abstract: We pursue robust approach to pricing and hedging in mathematical finance. We develop a general discrete time setting in which some underlying assets and options are available for dynamic trading and a further set of European options, possibly with varying maturities, is available for static trading. We include in our setup modelling beliefs by allowing to specify a set of paths to be considered, e.g. super-replication of a contingent claim is required only for paths falling in the given set. Our framework thus interpolates between model-independent and model-specific settings and allows to quantify the impact of making assumptions. We establish suitable FTAP and Pricing-Hedging duality results which include as special cases previous results of Acciaio et al. (2013), Bouchard and Nutz (2015), Burzoni et al. (2016) as well the Dalang-Morton-Willinger theorem. Finally, we explain how to treat further problems, such as insider trading (information quantification) or American options pricing.
The talk will cover a body of results developed in collaboration with A. Aksamit, M. Burzoni, S. Deng, M. Frittelli, Z. Hou, M. Maggis, X. Tan and J. Wiesel.