Humboldt Universität zu Berlin Naturwissenschaftliche Fakultät II Institut für Mathematik

Forschungsseminar
Stochastische Analysis und Stochastik der Finanzmärkte

Bereich für Stochastik
P. BANK, D. BECHERER, P.K. FRIZ, H. FöLLMER, U. HORST, P. IMKELLER, M. KELLER-RESSEL, U. KüCHLER, M. KUPPER, A. PAPAPANTOLEON

Ort: HU Berlin, Institut für Mathematik, Johann von Naumann - Haus, Rudower Chaussee 25, Hörsaal 1.115

Zeit: Donnerstag, 16 Uhr/17 c.t.

 

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19. Oktober 2011 (ACHTUNG: außerplanmäßiger Vortrag)

Frank Page (Indiana University, Bloomington)

Stationary Markov Equilibria in Discounted Stochastic Games

Abstract:
While the existence of Nash equilibria in stationary Markov strategies for m-player, non-zero sum, discounted stochastic games with countable state spaces and compact metric action spaces has long been established (e.g., see Federgruen, 1978), the existence of such equilibria for the uncountable case has remained an open question since the problem was first analyzed by Himmelberg, Parthasarathy, Raghavan, and Van Vleck (1976). Beginning with Fudenberg and Levine (1983), Harris (1985), and Forges (1986), one of the striking insights to emerge from the literature on the existence of subgame perfect equilibria (SPE) in non-Markov (i.e., partly history-dependent) strategies in stage games with uncountable state spaces concerns the fundamental role played by public randomization devices in resolving existence problems in such games. The importance of public randomization devices for existence was then confirmed in an infinite horizon, stochastic game setting by Nowak and Raghavan (1992) and Duffie, Geanakoplos, Mas-Colell, and McLennan (1994) who showed that m-player, non-zero sum, uncountable-compact discounted stochastic games naturally possess stationary Markov correlated equilibria. Our main contribution is to establish the existence of stationary Markov equilibria (i.e., SPE in Markov stationary strategies) for this class of stochastic games, thus showing for the stationary Markov case that public randomization devices are not required for existence - and thus providing a positive resolution to a long-standing open question in stochastic games.

19. Oktober 2011 (ACHTUNG: außerplanmäßiger Vortrag)

Sebastin Jaimungal (University of Toronto)

Self-Exciting Marked Point Processes for Algorithmic Trading

Abstract:
In this I will present a class of self-effecting processes as a promising approach to modeling trading activity at high frequencies. Our model neatly accounts for the clustering of intensity of trades and the feedback effect which trading induces on both market orders as well as the shape of the limit order book (LOB). Further, it allows for efficient calibration to market data based on pseudo-likelihood methods. As well, various probabilistic quantities of interest such as the probability that the next market order is a buy or sell, the distribution of the time of arrival of a buy or sell order, and the probability that the mid-price moves a given amount before a market order arrives are also easily computable. Finally, we study an optimal control problem for a trader who places immediate-or-cancel limit buy-and-sell orders to take advantage of the bid-ask spread. Asymptotic expansions in the level of risk-aversion lead to closed form and intuitive results which are also adapted to the state of the market. Some numerical experiments will be used to demonstrate the utility of the model and optimal strategies. This is joint work with Alvaro Cartea, U. Carlos III de Madrid and Jason Ricci, U. Toronto

03. November 2011

Igor Evstigneev (University of Manchester)

Von Neumann-Gale Dynamical Systems with Applications in Finance

Abstract:
Von Neumann-Gale dynamical systems are defined in terms of multivalued operators possessing properties of convexity and homogeneity. These operators assign to each element of a given cone a convex subset of the cone describing possible one-step transitions from one state of the system to another. The classical, deterministic theory of such dynamics was originally aimed at the modelling of economic growth (von Neumann 1937 and Gale 1956). First attempts to build a stochastic generalization of this theory were undertaken in the 1970s by Dynkin, Radner and their research groups. However, the initial attack on the problem left many questions unanswered. Substantial progress was made only in the late 1990s, and final solutions to the main open problems were obtained only in the last four or five years. Recently it has been observed that stochastic analogues of von Neumann-Gale systems provide a natural and convenient framework for financial modelling (asset pricing and hedging under transaction costs). This observation gave a new momentum to studies in the field and posed new interesting questions. The talk will give an introduction into the theory, review recent progress and discuss applications.

03. November 2011

N.N.

 

Abstract:

17. November 2011

Georg Mainik (ETH Zürich)

Risk Diversification for Extremal Events: General Properties, Estimation, and Model Comparison

Abstract:
The central topic of this talk is the diversification of catastrophic losses. Under the assumption of multivariate regular variation, the asymptotic portfolio loss distribution is characterized by a functional of the portfolio weights, the tail index, and the so-called spectral measure representing the dependence structure in the tail region. Further results encompass the general properties of the optimization problem, the estimation of the portfolio risk functional, and the ordering of models with respect to the asymptotic behaviour of portfolio losses. Particular interest is paid to the occurrence of negative diversification effects, compensation of gains and losses, uniform convergence of estimates, and the influence of dependence on model ordering.

17. November 2011

Frank Lehrbass (RWE Supply and Trading GmbH)

Credit Risk at RWE Supply and Trading - An Overview

Abstract:
The main concepts of credit risk are revisited from the perspective of a utility and a trading house. It will be made transparent how exposures to counterparties arise out of commodity business and what can be done to mitigate the credit risk. As concerns the remaining part of the credit risk it has to be priced. A discussion of the market for contingent Credit Default Swaps will be given. A glimpse at recent challenges of credit risk measurement and management will conclude.

01. Dezember 2011

Eva Lütkebohmert (Universität Freiberg)

A multi-period bank run model for liquidity risk

Abstract:
We present a new dynamic bank run model for liquidity risk where a financial institution finances its risky assets by a mixture of short- and long-term debt. The financial institution is exposed to insolvency risk at any time until maturity and to illiquidity risk at a finite number of rollover dates. We compute both insolvency and illiquidity default probabilities in this multi-period setting using a structural credit risk model approach. Firesale rates can be determined endogenously as expected debt value over current asset value. Numerical results illustrate the impact of various input parameters on the default probabilities.

01. Dezember 2011

Thorsten Schmidt (Technische Universität Chemnitz)

Dynamic Term Structure Models with Ratings

Abstract:
Empirical investigations about rating transitions show typically a non-Markovian behavior. We take this as a motivation to generalize existing models and determine conditions for absence of arbitrage in a general forward rate model. This is the starting point for explicit modeling approaches and we propose a semi-Markovian model and discuss open questions. This is joint work with J. Jakubowski and M. Nieweglowski.

 

08. Dezember 2011

Jens Winter (Allianz Lebensversicherung)

Hans Georg Freiermuth (Allianz Pension Consult)

Bewertung langfristiger Garantiezusagen

Abstract:
Lebensversicherungsverträge beinhalten in der Regel lebenslange Garantiezusagen. Daher kommt der Bewertung dieser langfristigen Garantien in der Lebensversicherung eine bedeutende Rolle zu. Im Vortrag werden Methoden hierzu vorgestellt und insbesondere das Kreditausfallrisiko und die Zinsmodellierung diskutiert. Im zweiten Teil der Veranstaltung erfolgt eine kurze Vorstellung von Arbeits- und Einsatzfeldern für Mathematiker bei der Allianz anhand von Praxisbeispielen. Im Anschluss besteht im Rahmen eines Umtrunks für Sie die Möglichkeit, mit Referenten persönlich ins Gespräch zu kommen.

15. Dezember 2011

Jan Werner (University of Minnesota)

Participation in Risk-Sharing under Ambiguity

Abstract:
Expected utility hypothesis together with (strict) risk aversion and common probability beliefs among multiple agents have strong implications on patterns of efficient risk sharing. First, agents' consumption plans are comonotone with the aggregate resources. Second, every agent participates in risk sharing by holding at least a small fraction of the aggregate risk. These results are at odds with empirical observations. Ambiguity of beliefs has been suggested as a way to reconcile the differences between the observed patterns of risk sharing and the rules of efficient risk sharing. In this paper we focus on (non) participation in risk sharing. Ambiguity of beliefs is described by the multiple-prior expected utility model of Gilboa and Schmeidler (1989). The question we ask is whether and how can multiple-prior expected utility give rise to some agents not participating in efficient risk sharing. The main result says that if the aggregate risk is relatively small, then the agents whose beliefs are the most ambiguous will not participate in risk sharing. The higher the ambiguity of those agents' beliefs, the more likely is their non-participation. Another factor making the non-participation more likely is low risk aversion of agents whose beliefs are less ambiguous.

15. Dezember 2011

Mitja Stadje (Tilburg University)

Robust Portfolio Selection

Abstract:
We study problems in robust in robust portfolio choice and indifference evaluation with constraints on the trading strategies. Using dynamic programming priniciples we characterize the optimal solution in terms of certain backward stochastic differential equations which admit convex driver functions. We prove new existence, uniqueness, and comparison results for the associated BSDEs and also provide some numerical examples using MC simulations. The talk is based on joint work with Roger Leaeven.

12. Januar 2012

Mathias Beiglböck (Universität Wien)

Martingale Mass Transport and Robust Option Pricing

Abstract:
Robust pricing of an exotic option $\Phi$ written on a financial asset can be viewed as the task of estimating ${\bf E}_{{\bf Q}} \Phi$, where ${\bf Q}$ run through a set of martingale measures satisfying marginal constraints. It is fruitful to relate this to the theory mass transportation. E.g. the abstract duality theorem from optimal transport translates almost directly to new superhedging results for stock price processes in discrete time.

 

12. Januar 2012

Yan Dolinsky (Universität Zürich)

Duality and Convergence for Binomial Markets with Friction

Abstract:
We prove limit theorems for the super-replication cost of European options in a Binomial model with friction. The examples covered are markets with proportional transaction costs and the illiquid markets. The dual representation for the super-replication cost in these models are obtained and used to prove the limit theorems. In particular, the existence of the liquidity premium for the continuous time limit of the model proposed in by Cetin Jarrow and Protter is proved. Hence, this paper extends the previous convergence result of Gokay and Soner to the general non-Markovian case. Moreover, the special case of small transaction costs yields, in the continuous limit, the $G$-expectation of Peng as earlier proved by Kusuoka . (Joint work with Mere Soner)

26. Januar 2012

Rüdiger Kiesel (Universität Duisburg-Essen)

Market Risk Premium in Power Markets

Abstract:
In this talk we provide frameworks to explain the market risk premium, defined as the difference between forward prices and spot forecasts. We show how it depends on the risk preferences of market players and what impact information differences may have. Our focus will be on an empirical investigation of the so-called information premium, which is defined as the influence of future information not incorporated in spot prices but taken into consideration when pricing forwards. We test for the existence of the premium using data from the German EEX at beginning of 2008 when CO2 certificates were introduced and in 2011 when several nuclear power plants were switched off. Additionally, we will provide an estimate of the value and an analysis of the properties of the information premium. (joint work with Fred Espen Benth and Richard Biegler-Koenig)

26. Januar 2012

Johannes Ruf (University of Oxford)

Föllmer's measure, Novikov's condition and options on exploding exchange rates

Abstract:
In the first part of this talk, I will present a proof of Novikov's condition by means of the Föllmer measure. In the second part, I will discuss an application of the Föllmer measure to Foreign Exchange options. Strict local martingale models have been suggested to model the underlying exchange rate. In such models, put-call parity does not hold if one assumes minimal superreplicating costs as contingent claim prices. I will illustrate how put-call parity can be restored by changing the definition of a contingent claim price. More precisely, I will discuss a change of numeraire technique when the underlying is only a local martingale. Then, the new (Föllmer) measure is not necessarily equivalent to the old measure. If one now defines the price of a contingent claim as the minimal superreplicating costs under both measures, then put-call parity holds. I will discuss properties of this new pricing operator.
This talk is based on joint work with Peter Carr and Travis Fisher.

09. Februar 2012

Christoph Reisinger (University of Oxford)

Penalty methods for the numerical valuation of American options in complete and incomplete markets

Abstract:
In this talk, we discuss properties of penalty approximations to early exercise options and highlight their efficiency as a computational tool. We start by considering a standard Black-Scholes finite difference setting, where we analyse Newton's method for the penalised and unpenalised problem, resulting in some sort of policy iteration for both. We also gain insight into the local structure of the penalisation error by means of matched asymptotic expansions. In an incomplete market setting, e.g. indifference valuation of options on an untraded asset under exponential utility preferences, we show that similar penalisation error bounds as before still hold, and that Newton's method has the expected convergence properties. Numerical results confirm that the overall computational complexity for American options in incomplete markets is similar to the standard case of European options in Black-Scholes markets.
This talk is based on joint work with Jan Witte and Sam Howison.

 

 

09. Februar 2012

Kathrin Glau (Technische Universität München)

PIDE and Fourier methods for pricing European options in Levy models

Abstract:
We concentrate on the relation between time-inhomogeneous Lévy processes and evolution problems that are associated with prices of options such as calls, puts and barrier options. A major concern is to shed light on the structural affinity between the PIDE and the Fourier transform based approach for European options.We characterize Lévy processes according to the solution spaces of associated parabolic equations. It turns out that for a wide class of processes these spaces are weighted Sobolev-Slobodeckii spaces with different indices. To classify the processes according to these spaces, we define the related Sobolev index of the process. Since it is the most convenient to work with the Fourier transform of Lévy processes, the classification is done according to the symbol i.e. the characteristic function of the process. In contrast to the criteria provided in the literature, our criteria based on the Sobolev index does not require differentiability conditions of the symbol or smoothness of the Lévy kernel, but purely translates the ellipticity condition on the infinitesimal generator to the symbol.We derive the Sobolev index for several classes of Lévy processes and compare it to the Blumenthal-Getoor index, which reveals a relation between the Sobolev index and path properties of the process. More precisely, we discuss the Sobolev index as an indicator of the smoothness of the distribution and of the unboundedness of the paths of the process.
The talk is based on joint work with Ernst Eberlein.

 

 

 

 

 

 

 

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Für Rückfragen wenden Sie sich bitte an: Frau Sabine Bergmann
bergmann@mathematik.hu-berlin.de
Telefon: 2093 5811
Telefax: 2093 5848