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.
Hao Xing (London School of Economics)
Titel: Optimal contracting with unobservable managerial hedging
Abstract: We develop a continuous-time model where a risk-neutral principal contracts with a CARA agent to initiate a project. Protected by the limited liability, the agent can increase the expected return of the project by exerting costly hidden effort. In addition, the agent can trade the market portfolio and a risk-free bond with an unobservable private account. This unobservable managerial hedging behavior partially offsets the incentive that agent receives. The agent?s limited liability protection induces the possibility of inefficient liquidation, and generates endogenous risk aversion for the principal. In the optimal contract, the principal uses the absolute performance evaluation and the relative performance evaluation at the same time. In order to share the market risk optimally, the optimal contract does not filter out the market risk completely from the compensation. Our model provides support for empirical analysis on relative performance evaluation contracts. Joint work with Nengjiu Ju and Yu Huang.
Thorsten Schmidt (University of Freiburg)
Titel: Unbiased Estimation of Risk
Abstract: The estimation of risk measures recently gained a lot of attention, partly because of the backtesting issues of expected shortfall related to elicitability. In this work we shed a new and fundamental light on optimal estimation procedures of risk measures in terms of bias. We show that once the parameters of a model need to be estimated, one has to take additional care when estimating risks. The typical plug-in approach, for example, introduces a bias which leads to a systematic underestimation of risk. In this regard, we introduce a novel notion of unbiasedness to the estimation of risk which is motivated by economic principles. In general, the proposed concept does not coincide with the well-known statistical notion of unbiasedness. We show that an appropriate bias correction is available for many well-known estimators. In particular, we consider value-at-risk and expected shortfall (tail value-at-risk). In the special case of normal distributions, closed-formed solutions for unbiased estimators can be obtained. We present a number of motivating examples which show the outperformance of unbiased estimators in many circumstances. The unbiasedness has a direct impact on backtesting and therefore adds a further viewpoint to established statistical properties. This is joint work with Marcin Pitera.
Wolfgang Runggaldier (University of Padova)
Titel: Classical and Restricted Impulse Control for the Exchange Rate under Incomplete Knowledge of the Model
Abstract: We consider the problem faced by a Central Bank of optimally controlling the exchange assuming that it can use two non-excluding tools: controlling directly the exchange rate in the form of an impulse control; controlling it indirectly via the domestic exchange rate in the form of a continuously acting control. In line with the existing literature, we consider this as a mixed classical-impulse control problem and, on the basis of a quasi-variational inequality, search for an analytic solution within a specific class of value functions and controls. Besides the finite horizon, the main novelty here is the assumption that the drift in the exchange rate dynamics is not directly observable and has thus to be filter-estimated from observable data. The problem becomes thus time inhomogeneous and the Markovian state variables have to include also the filter of the drift. (Joint with K.Yasuda).
Ralf Korn (TU Kaiserslautern)
Titel: Save for the Bad Times or Consume as Long as You Have? - Worst-Case Portfolio Optimization and Applications
Abstract: Rare events and their consequences are hard - if not impossible - to estimate. The worst-case approach deals with this problem via distuinguishing randomness and uncertainty. As a consequence, portfolio optimization problems are split into optimization and indifference parts. In this talk, we give a survey on the worst-case approach, highlight a surprising application in optimal consumption and present new aspects. Further, we also highlight the use of the martingal optimality principle in solving the resulting problems.
Stefan Ankirchner (University of Jena)
Titel: Optimal (financial) position targeting via decoupling fields
Abstract: In the talk we consider a variant of the basic problem of the calculus of variations, where the Lagrangian is convex and subject to randomness adapted to a Brownian filtration. We solve the problem by reducing it, via a limiting argument, to an unconstrained control problem that consists in finding an absolutely continuous process minimizing the expected sum of the Lagrangian and the deviation of the terminal state from a given target position. Using the Pontryagin maximum principle one can characterize a solution of the unconstrained control problem in terms of a fully coupled forward-backward stochastic differential equation (FBSDE). We use the method of decoupling fields for proving that the FBSDE has a unique solution. The talk is based on joint work with Alexander Fromm, Thomas Kruse and Alexandre Popier.
Miryana Grigorova (Bielefeld University)
Titel: Optimal stopping and Dynkin games with non-linear ƒ-expectations: beyond right-continuity
Abstract: We consider the optimal stopping problem with non-linear ƒ-expectation (which is a non-linear pricing operator induced by a Backward Stochastic Differential Equation) and with payoff process ξ on which we do not make any regularity assumptions. We characterize the value process of this optimal stopping problem as the non-linear εf-Snell envelope of ξ. We also establish an innitesimal characterization of the value process in terms of a Reflected BSDE with ξ as the lower obstacle. To do this, we prove some useful results on Reflected BSDEs with completely irregular obstacles, in particular, an existence and uniqueness result and a comparison theorem. In the second part of the talk, we formulate a non-linear εƒ-Dynkin game, that is, a game problem over stopping times with non-linear ƒ-expectation, between two agents whose payoff processes ξ and ζ are only right upper-semicontinuous (but not necessarily right-continuous). Under a technical assumption (a Mokobodzki- type condition), we show that our εƒ-Dynkin game has a value and that the value is characterized in terms of the solution of a Doubly Reflected BSDE where ξ and ζ play the role of lower and upper obstacles. The case where the obstacles ξ and ζ are completely irregular is more involved: we will see that in this case the solution of the doubly reflected BSDE is related to the value of "an extension" of the previous non-linear εƒ-game problem over a larger set of "stopping strategies" than the set of stopping times. Based on joint works with P. Imkeller, Y. Ouknine, and M.-C. Quenez.