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.
Max Nendel (Bielefeld University)
Titel: A semigroup approach to nonlinear Lévy processes
Abstract: Nonlinear expectations, as introduced by S. Peng, are closely related to monetary risk measures. Nonlinear expectations naturally appear in the context of pricing under model uncertainty, e.g. drift uncertainty (g-expectation) or volatility uncertainty (G-expectation). In this talk, we demonstrate how Lévy processes under nonlinear expectations arise from solutions to certain fully nonlinear PDEs, where the Knigthian uncertainty is in the Lévy triplet. This is done using nonlinear semigroups and a nonlinear version of Kolmogorov?s extension theorem. We provide a sufficient condition for families of Lévy tiplets that guarantees the solvability of the related fully nonlinear partial integro-differential equation, and show that the solution admits a representation by means of a nonlinear Lévy process.
Marco Frittelli (University of Milan)
Titel: On Fairness of Systemic Risk Measures
Abstract:In a previous paper, wehave introduced a general class of systemic risk measures that allow random allocations to individual banks before aggregation of their risks. In the present paper, we address the question of fairness of these allocations and we propose a fair allocation of the total risk to individual banks. We show that the dual problem of the minimization problem which identify the systemic risk measure, provides a valuation of the random allocations which is fair both from the point of view of the society/regulator and from the individual financial institutions. The case with exponential utilities which allows for explicit computation is treated in details.
Hanspeter Schmidli (University of Cologne)
Titel: Optimal Dividend and Capital Injection Problems in Non-Life Insurance
Abstract: The traditional risk measure in actuarial mathematics is the ruin probability. This concept has been criticised because it does not take into account the time to ruin and the deficit at ruin. An alternative measure has been suggested by de Finetti (1957). He proposed to consider the discounted value of dividends paid from the portfolio. However, under the optimal dividend strategy ruin becomes certain. Moreover, the deficit at ruin is not taken into account, either. As an alternative, we allow capital injections that have to keep the surplus positive. Ruin is not allowed in our model. In one model, we measure the risk as the value of the (discounted) capital injections. We look for the reinsurance strategy that minimises the value. A second model allows also dividend payments. Here, the value is the discounted dividends minus penalised capital injections. We show that the optimal dividend strategy is a barrier strategy. Discounting in these models has to be seen as a preference measure: dividends today are preferred to dividends tomorrow and injections tomorrow are preferred to injections today. Since the parameters of the surplus process are kept constant, the new measures are also to be considered as technical measures used for risk management. This talk is based on joint work with Julia Eisenberg and Natalie Scheer.
Peter Grandits (TU Wien)
Titel: Some applications of stochastic control for ruin problems in insurance mathematics
Abstract: The calculation and estimation of ruin probabilities is a classical theme in insurance mathematics, starting probably with the celebrated Lundberg inequality. We will generalize the classical model in two directions. On the one hand, we shall consider companies, which invest in the stock market, on the other hand two companies are considered, which are allowedto collaborate. We want to find estimates for the ruin probabilities, respectively some information about the optimal strategies.
Marie-Claire Quenez (University Paris Diderot)
Titel: Nonlinear pricing of European and American options in an imperfect market with default
Abstract: We study pricing and hedging for contingent claims in an imperfect market model with default, where the imperfections are taken into account via the nonlinearity of the wealth dynamics, expressed in terms of a nonlinear driver g(y; z; k). In this framework, the seller?s (resp. buyer?s) pricing rule for European options corresponds to the nonlinear g-expectation Eg (resp. ~g-expectation E~g) 1, induced by a BSDE with driver g (resp ~g). We also address the case of options which generate intermediate cashflows modeled via an optional finite variational process. We then study the pricing of American options in this framework. The payoff is given by an optional irregular process (t). We define the seller?s price of the American option as the minimum of the initial capitals which allow her/him to build up a superhedging portfolio strategy '. We also consider the buyer?s price, defined as the supremum of the initial prices which allow the buyer to select an exercise time and a portfolio strategy ' so that she/he is superhedged. We prove that the seller?s (resp. buyer?s) price coincides with the value function of an Eg-(resp. E~g-) optimal stopping problem, which corresponds to the solution of a reflected BSDE with obstacle (t) and driver g (resp. ~g). At last, we study the pricing of a game option with irregular payoffs. In this case, the seller?s (resp. buyer?s) price is shown to be equal to the value function of an Eg-(resp. E~g-) Dynkin game, which coincides with the solution of a nonlinear doubly reflected BSDEs with driver g (resp. ~g). We also consider the case of ambiguity on the model.