The workshop is concerned with applications of stochastic analysis and related fields to problems from finance such as the pricing and hedging of derivatives and optimal portfolio choice. Both issues are well-understood in standard linear finite-dimensional models with complete markets and symmetric information. However, these models are at best a stylized picture of real markets, and the development of new, more realistic models leads to challenging mathematical questions. The two-day workshop will focus on two issues in this area, infinite-dimensional models on the one hand and models with incomplete markets and incomplete information on the other. Before the workshop there will be a one-day minicourse on stochastic analysis in infinite dimensions with applications to financial mathematics.

We provide an introduction to stochastic modelling of term structure problems (like interest rate term structures). We then discuss geometric properties of time evolutions such as finite dimensional realizations of term structure equations, or hypo-ellipticity. We mainly work with SDEs driven by (finitely or infinitly many) Brownian motions. Stochastic Calculus of Variations (Malliavin Calculus) will be introduced to analyse properties of the resulting processes.

The lecture will be concerned with stochastic evolution equations with white and coloured Levy noises. It will start from recalling basic facts about stochastic integration with respect to Poissonian random measures. Some old and new existence results will be presented. Asymptotic behaviour of the solutions will be discussed as well. Extended lecture notes on the topic are in preparation, in collaboration with Prof. S. Peszat, and should appear, on the web site of my institute, under Peszat's name. One can also find there a file of our joint paper "Stochastic heat and wave equations driven by an impulsive noise".

Uncertainty on the choice of an option pricing model can lead to "model risk" in the valuation of portfolios of options. After discussing some properties which a quantitative measure of model uncertainty should verify in order to be useful and relevant in the context of risk management of derivative instruments, we introduce a quantitative framework for measuring model uncertainty in the context of derivative pricing. Two methods are proposed: the first method is based on a coherent risk measure compatible with market prices of derivatives, while the second method is based on a convex risk measure. Our measures of model risk lead to a premium for model uncertainty which is comparable to other risk measures and compatible with observation of market prices of a set of benchmark derivatives. Finally, we discuss some implications for the management of "model risk".

It will be devoted to the problem of constructing impulsive modifications of the portfolios, introduced by Pliska and Suzuki. The problem is solved for models with general Markovian prices. More specific results are obtained in the case the prices satisfy equations with Levy noise. A file of the joint paper with J. Palczewski, "Portfolio diversification with Markovian prices", on which the talk is based, can be found on Palczewski's web page.

We show - by methods of Malliavin Calculus for Brownian motion - the existence and tractability of weigths for the Calculation of Greeks in the case of jump-diffusions. We do not need any further Calculus on Poisson space for this purpose. This extends works of Davis, Bismuth and Zhou in the direction of explicit formulas for the Greeks. We apply Hoermander conditions and invertibility of linking operators as assumptions. (joint work with Barbara Forster and Eva Luetkebohmert)

In this paper we study a fairly general Wiener driven model for the term structure of forward prices. The model, under a fixed martingale measure,$Q$, is described by using two infinite dimensional stochastic differential equations (SDEs). The first system is a standard HJM model for (forward) interest rates, driven by a multidimensional Wiener process $W$. The second system is an infinite SDE for the term structure of forward prices on some specified underlying asset driven by the same $W$. We are primarily interested in the forward prices. However, since for any fixed maturity $T$, the forward price process is a martingale under the $T$-forward neutral measure, the zero coupon bond volatilities will enter into the drift part of the SDE for these forward prices. The interest rate system is, thus, needed as input into the forward price system. Given this setup we use the Lie algebra methodology of Björk et al. to investigate under what conditions on the volatility structure of the forward prices and/or interst rates, the inherently (doubly) infinite dimensional SDE for forward prices can be realized by a finite dimensional Markovian state space model.

We introduce equity forward variance term-structure market models and derive HJM-type arbitrage conditions. We then discuss finite-dimensional Markovian representations of the infinite-dimensional fixed time-to-maturity forward variance swap curve and analyse examples of such Variance Curve Functionals. We extend these results to newly introduced "Entropy Swaps", which can be used to aid the calibration of such models. As an application, the results are then applied to show that if Heston's model is recalibrated on a daily basis, mean-reversion and the product correlation and volofvol must be kept constant to avoid dynamic arbitrage.

In the utility maximization theory under model uncertainty one usually aims at maximizing a robust expected utility functional over affordable contingent claims Here we want to add the constraint of bounded robust shortfall risk induced by a convex loss function and a set of subjective measures which may differ from the set of measures in the definition of the robust expected utility functional. We will give a characterization of the solution to this constrained utility maximization problem by means of its dual problem.

This talks considers a generalization of the CIR model to infinite dimensions and considers its applications to Credit and Mortgage Risk. First, we discuss the existence of solutions in several Hilbert spaces which leads to stochastic processes which are positive. Second, we use this results to construct a model for credit spreads which is proven to be free of arbitrage. Third, we consider models for Mortgage Risk where the default intensity of an obligor depends on the location. The generalized CIR model will be used to model the intensity. In simulation results we show that wavelet methods can be successfully used to infer the default intensity from data.

We develop a white noise framework and the theory of stochastic test function and distribution spaces for Hilbert space valued Levy processes in order to study stochastic evolution equations in these spaces driven by Levy white noise. This allows for a generalization of the classical solution concepts to solutions in the sense of stochastic distributions. For exapmle, the Heat equation driven by Levy white noise always has a solution in stochastic distribution spaces while it is easy to see that a classical solution only exists in special cases.

We plan to describe concepts and solution methods for stochastic control under incomplete information as applied to portfolio optimization and hedging. The concepts and methods are first illustrated in a discrete time setting and then carried over to continuous time thereby pointing at similarities and differences. Time permitting, we shall also consider an intermediate setting, namely a pure jump market model.

We determine the minimal entropy martingale measure in a general class of stochastic volatility models where both price and volatility process contain jump terms which are correlated. This generalizes previous studies which have treated either the geometric Lévy case or continuous price processes with an orthogonal volatility process. We proceed by linking the entropy measure to a certain semi-linear Integro-PDE for which we prove the existence of a classical solution.

Our aim is to examine the PDE approach to the valuation and hedging of a defaultable claim in various settings; this allows us to emphasize the importance of the choice of the traded assets. We start with a general model for the dynamics of the traded primary assets. Subsequently, we specify some particular models and we deal with particular defaultable claims such as, for instance, survival claims. In a first part, we examine the no-arbitrage property of a model in terms of a martingale measure. The following part is devoted to the study of the PDE approach to valuation of defaultable claims and we give the hedging strategies of a contingent claim under the assumption that prices of primary assets are strictly positive. Then, we study the particular case when one of the primary assets is a defaultable security with zero recovery, so that its price vanishes after default.

We extend the results of Kramkov and Schachermayer to the problem of robust utility maximization in an incomplete market setting. The robust utility functionals we are considering arise as numerical representations of investor preferences when the investor is uncertain about the correct probabilistic model and averse against both risk and model uncertainty. The usual approach is to reduce the problem to a standard problem under a least favorable model. An interesting feature of incomplete market models is that the least favorable model may admit arbitrage oportunities.

This paper derives a unified framework for portfolio optimization, derivative pricing, financial modeling and risk measurement. It is based on the natural assumption that investors prefer more rather than less, in the sense that given two portfolios with the same diffusion coefficient value, the one with the higher drift is preferred. Each such investor is shown to hold an efficient portfolio in the sense of Markowitz with units in the market portfolio and the savings account. The market portfolio of investable wealth is shown to equal a combination of the growth optimal portfolio (GOP) and the savings account. In this setup the capital asset pricing model follows without the use of expected utility functions, Markovianity or equilibrium assumptions.
The expected increase of the discounted value of the GOP is shown to coincide with the expected increase of its discounted underlying value. The discounted GOP has the dynamics of a time transformed squared Bessel process of dimension four. The time transformation is given by the discounted underlying value of the GOP. The squared volatility of the GOP equals the discounted GOP drift, when expressed in units of the discounted GOP. Risk neutral derivative pricing and actuarial pricing are generalized by the fair pricing concept, which uses the GOP as numeraire and the real world probability measure as pricing measure. An equivalent risk neutral martingale measure does not exist under the derived minimal market model.

This talk is concerned with an hedging and valuation approach for incomplete markets that is based on utility indifference arguments. It is known that the solution can be described on a general but abstract level by duality methods. We discuss alternative constructive solutions, examples, and properties of the indifference approach under additional structural assumptions on the model, which do not fall into the well-studied class of Brownian models with correlation.

Participants

Francis Insorh Akolbila

Elisa Alòs

Hirbod Assa

Ezekiel Olusola Ayoola

Jochen Backhaus

Nicole Baeuerle

Dirk Becherer

Arne Becker

Denis Belomestny

Yuliya Bregman

Hans Bühler

Raquel Bujalance

Ricardo Castano-Bernard

Rama Cont

Markus Fischer

Uta Freiberg

Teresa García Muniesa

Raquel Gaspar

Dariusz Gatarek

Anne Gundel

Jacek Jakubowski

Mesrop Janunts

Monique Jeanblanc

Xu Ling

Carlo Marinelli

Thilo Meyer-Brandis

Agatha Murgoci

Mariusz Niewęgłowski

Frank Oertel

Eckhard Platen

Ulrike Polte

Monika Popp

Thorsten Rheinländer

Ulrich Rieder

Wolfgang J. Runggaldier

Matthias Scherer

Alexander Schied

Thorsten Schmidt

Stephan Sturm

Josef Teichmann

Marco Tolotti

Constanze Wäldrich

Ubbo Wiersema

Christoph Winter

Johannes Wissel

Wiebke Wittmüß

Ralf Wunderlich

Jerzy Zabczyk

Carsten Zecher

Scientific Organizers

Rüdiger Frey

Universität Leipzig

Thorsten Schmidt

Universität Leipzig

Stefan Müller

Max Planck Institute for Mathematics in the Sciences

Administrative Contact

Katja Bieling

Max Planck Institute for Mathematics in the Sciences, Leipzig
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