Forschungsseminar
Stochastische Analysis und Stochastik der Finanzmärkte

Bereich für Stochastik
P. BANK, D. BECHERER, H. FÖLLMER, U. HORST, P. IMKELLER, U. KÜCHLER

Ort: TU Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Raum MA 041 im Erdgeschoß

Zeit: Donnerstag, 16 Uhr c.t.

Kaffee & Gebäck ab 15.45 Uhr s.t. im Raum MA 721

23. Oktober 2008, 16 Uhr c.t.

Santiago Moreno (Humboldt-Universität zu Berlin)

Risk Minimization & Variational Problems with Convexity Constraints

Abstract:
We study the problem of a monopolist (or principal) who is exposed to some non-hedgeable risk factor. His aim is to minimize his exposure, as evaluated via a coherent risk measure, by engaging in OTC-trading with a continuum of buyers. The latter are risk-averse, Mean-Variance maximizers with heterogeneous risk aversion coefficients that lie in a closed and bounded interval. The risk aversion coefficient of a buyer (or agent) represents his type and it is private information. The principal is only aware of the distribution of types and therefore he cannot expect to extract the indifference price from each agent. We present an existence result for the principal's problem, as well as an algorithm to estimate the optimal pricing rule for a particular choice of the risk measure.

23. Oktober 2008, 17.30 Uhr s.t.

Thorsten Schmidt (Universität Leipzig)

Dynamic CDO Term Structure Modelling

Abstract:
The goal of this talk is to introduce a unifying approach for valuing contingent claims on a portfolio of credits, such as collateralized debt obligations (CDOs). We introduce the defaultable (T,x)-bonds, which pay one if the aggregated loss process in the underlying pool of the CDO has not exceeded x at maturity T, and zero else. Necessary and sufficient conditions on the stochastic term structure movements for the absence of arbitrage are given. Background market risk as well as feedback contagion effects of the loss process are taken into account. Moreover, we show that any ex-ogenous specification of the volatility and contagion parameters actually yields a unique consistent loss process and thus an arbitrage-free family of (T,x)-bond prices. For the sake of analytical and computational efficiency we then develop a tractable class of doubly stochastic affine term structure models.

6. November 2008, 16 Uhr c.t.

Jörn Sass (TU Kaiserslautern)

The numeraire portfolio under transaction costs

Abstract:
We study the existence of a numeraire portfolio for a discrete time financial market with proportional transaction costs. In an incomplete market without frictions, consistent prices for derivative securities can be obtained by taking the expectation of the claim with respect to a certain probability measure under which the discounted asset prices become martingales. The numeraire portfolio (defined suitably) allows to replace this change of measure by a change of numeraire. For models with transaction costs, the concept of a martingale measure and thus the concept of a numeraire portfolio have to be modified. Without transaction costs, a well known approach is to find the growth optimal portfolio (but the numeraire portfolio might not exist). With some modifications and under quite general conditions, using methods of dynamic programming we show that the same approach works in our model. This is to be expected from results on consistent price systems under transaction costs. The relation to these results, to portfolio optimization under transaction costs, and possible extensions will be discussed.

6. November 2008, 17.30 Uhr s.t.

John Schoenmakers (WIAS Berlin)

Holomorphic transforms with application to affine processes

Abstract:
In a rather general setting of Itô-Lévy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multidimensional affine Itô-Lévy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.

20. November 2008, 17.30 Uhr s.t. (ACHTUNG: An diesem Tag findet nur ein Vortrag statt)

Michael Brockmann (Deutsche Bank AG)

Incremental Risk Charge

Abstract:
The rules for bank capital to cover market risks are currently under review by the Basel Committee. Internal models to calculate Value-at-Risk typically do not cover credit default risk and other event risks. A new capital charge for these types of risks has been proposed called "Incremental Risk Charge". The presentation will discuss the main properties and modelling challenges of the new charge and its potential impact on trading activities of banks.

11. Dezember 2008, 16 Uhr c.t.

Nizar Touzi (École Polytechnique Paris)

Dual Formulation of 2nd Order Target Problems

Abstract:
We introduce a class of target problems which encompasses at least three relevant hedging problems in finance: volatility uncertainty, gamma constraints, liquidity risk. The main result is a general dual formulation of these problems as the infimum of solutions to a parameterized family of backward stochastic differential equations.

11. Dezember 2008, 17.30 Uhr s.t.

Alfred Müller (Universität Siegen)

Modelling, measuring and comparing dependent risks

Abstract:
This talk gives an overview on models of dependence for multivariate risks including copulas and Levy copulas and their dependence properties, how to compare them using stochastic orderings, and what effect this has on the riskiness of aggregate claims. The talk is based on joint work with Nicole Bäuerle and Anja Blatter.

18. Dezember 2008, 17.30 Uhr s.t. (ACHTUNG: An diesem Tag findet nur ein Vortrag statt)

Nick Westray (Imperial College London)

Constrained Nonsmooth Utility Maximization

Abstract:
We discuss the utility maximization problem where there are cone constraints on the investors portfolio. In particular we show how the solution to the utility maximization can be related to an interesting question in stochastic analysis, namely whether pointwise properties are preserved under convergence in the semimartingale topology. Several motivating examples will be given as well as necessary and sufficient conditions to ensure a positive answer to the above question. In the second part of the talk we will provide an introduction to nonsmooth utility maximization and go on to provide a counterexample to a conjecture posed by Ocone appearing in Deelstra, Pham and Touzi (2001) on the optimality of certain subgradient valued random variables. This has important ramifications as it shows that a ”naive” solution is impossible and justifies the use of their complicated techniques. The talk is based on joint work with Harry Zheng and Christoph Czichowsky.

22. Januar 2009, 16 Uhr c.t.

Denis Belomestny (WIAS Berlin)

Regression methods for stochastic control problems and their convergence analysis

Abstract:
In this talk we present several regression algorithms for solving general stochastic optimal control problems via Monte Carlo. This type of algorithms is particulary useful for problems with a high-dimensional state space and complex dependence structure of the underlying Markov process with respect to some control. The main idea behind the algorithms is to simulate a set of trajectories under some reference measure and to use the Bellman principle combined with fast methods for approximating conditional expectations and functional optimization. Theoretical properties of the presented algorithms will be investigated and the convergence to the optimal solution will be proved under mild assumptions. Finally, we present numerical results for the problem of pricing a high-dimensional Bermudan basket option under transaction costs in a financial market with a large investor.

22. Januar 2009, 17.30 Uhr s.t.

Rene Carmona (Princeton University)

Equity Market Models with and without Jumps

Abstract:
The first part of the talk will concentrate on the difficult problem of the characterization of arbitrage free dynamic stochastic models for the equity markets. Routinely, market models are based on Ito stochastic differential equations modeling the dynamics of a set of basic instruments including, but not limited to, the option underliers. These market models are usually recast in the framework of the HJM philosophy, but the technical difficulties inherent to the use of implied volatility dynamics are hard to overcome. We shall review recent attempts to give a new life to the subject by the introduction of market models based on the dynamics of the local volatility surface, and if time permits, we shall present new results on the form of the no-arbitrage condition for pure jump market models.

29. Januar 2009, 16 Uhr c.t.

Vicky Henderson (University of Oxford)

Prospect Theory, Partial Liquidation and the Disposition Effect

Abstract:
We solve the problem of an agent with prospect theory preferences who seeks to liquidate a portfolio of (divisible) claims. Our methodology enables us to consider different formulations of prospect preferences in the literature (piecewise exponential or piecewise power) and various price processes. We find that these differences in specification matter - for instance, with piecewise power functions, the agent may liquidate at a loss relative to break-even, albeit the likelihood of liquidating at a gain is much higher than liquidating at a loss. This is consistent with the disposition effect documented in empirical and experimental studies. We find the agent does not choose to partially liquidate a position, but rather, if liquidation occurs, the entire position is sold. This is in contrast to partial liquidation when agents have standard concave utilities.

29. Januar 2009, 17.30 Uhr s.t.

David Hobson (University of Warwick)

Recovering a stock process from perpetual put prices

Abstract:
It is well-known how to determine the value of an optimal stopping problem if the underlying stochastic process is a time-homogeneous process. In the talk we consider the inverse problem, i.e. given the values of a collection of optimal stopping problems, can one find a stochastic process which would give those values. The motivation comes from consideration of American puts. This is a joint work with Erik Ekstrom.

12. Februar 2009, 16 Uhr c.t.

Etienne Pardoux (Université de Provence, Marseille)

Periodic Homegenization : on the homogenized diffusion matrix.

Abstract:
We shall give conditions under which we can homogenize a second order PDE operator with periodic and fast oscillating coefficients, which allow that operator to degenerate (the second order part might vanish on open sets). It then raises the question whether the homogenized diffusion matrix is or not invertible, or more generally what is the dimension of its image ? We give a precise answer to that question, which can be read on the approximate operator. This is a joint work with M. Hairer.

12. Februar 2009, 17.30 Uhr s.t.

Marie Morlais (Université du Maine, Le Mans)

Utility maximization and BSDEs with jumps

Abstract:
This talk is based on the paper [4]. In this paper, we study the utility maximization problem with portfolio constraints. More precisely, we consider a financial model equipped with a Wiener Poisson filtration and, in that context, we are able to solve dynamically the utility maximization problem. This consists in deriving the expression of the associated value process and it is based on the two following ingredients: the dynamic programming principle and BSDEs technics (this methodology is very similar as in the following related papers [1], [2] or [3]). The talk is divided into two main parts: in the first one, I introduce and explain the problem under study and the methodology and in the second one, I give the main result, a squetch of the proof and, if time allows, a recent extension of this work. References [1] Becherer, D., Bounded solutions to Backward SDE’s with jumps for utility optimization and indifference hedging, Ann. Appl. Probab., 16(4) : 2027– 2054, 2006. [2] Hu, Y., Imkeller, P. and Müller, M., Utility maximization in incomplete markets, Ann. Appl. Probab., 15(3) : 1691–1712, 2005. [3] Mania, M. and Schweizer, M., Dynamic exponential utility indifference valuation, Ann. Appl. Probab., 15(3) : 2113–2143, 2005. [4] Morlais, M. A., Utility maximization in a jump market model, To appear in Stochastics (February 2009) and available on arxiV arxiV:math.PR/0612181v4, 2008.