Without assuming any probabilistic price dynamics, we consider a frictionless financial market given by the Skorokhod space, on which some financial options are liquidly traded. In this model-free setting we show various pricing-hedging dualities and the analogue of the fundamental theorem of asset pricing. For this purpose we study the corresponding martingale optimal transport (MOT) problem: We obtain a dual representation of the Kantorovich functional (super-replication functional) defined for functions (financial derivatives) on the Skorokhod space using quotient sets (hedging sets). Our representation takes the form of a Choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets. The talk is based on a joint work with Patrick Cheridito, Matti Kiiski and H. Mete Soner.
MOT Duality and Robust Finance
From systemic risk to supercooling and back
I will explain how structural models of default cascades in the systemic risk literature naturally lead to the supercooled Stefan problem of mathematical physics. On the one hand, this connection allows us to uncover a notion of global solutions to the supercooled Stefan problem, which we analyze in detail. On the other hand, the supercooled Stefan problem formulation allows to provide a truly intrinsic definition of systemic crises and to characterize the fragile states of the economy. Time permitting, I will also explain the network and game extensions of the problem. Based on a series of works with Francois Delarue and Sergey Nadtochiy.
XXVIII European Workshop on Economic Theory
The XXVIII. European Workshop on Economic Theory (EWET 2019) is hosted by the Finance Group @ Humboldt. It will take place at the School of Business and Economics, Humboldt University, in the city center of Berlin from June 13 to June 15. The workshop is a forum for researchers interested in the latest developments in economic theory and mathematical economics. Participants present and discuss recent results in areas such as general equilibrium theory, decision theory, information economics, game theory, bargaining and matching, financial markets, and social choice. Here is a link to the official webpage.
N-player games and mean-field games with smooth dependence on past absorptions
Mean-field games with absorption is a class of games, that have been introduced in Campi and Fischer (2018) and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit some given boundary. In this paper, we push the study of such games further, extending their scope along two main directions. First, a direct dependence on past absorptions has been introduced in the drift of players' state dynamics. Second, the boundedness of coefficients and costs has been considerably relaxed including drift and costs with linear growth. Therefore, the mean-field interaction among the players takes place in two ways: via the empirical sub-probability measure of the surviving players and through a process representing the fraction of past absorptions over time. Moreover, relaxing the boundedness of the coefficients allows for more realistic dynamics for players' private states. We prove existence of solutions of the mean-field game in strict as well as relaxed feedback form. Finally, we show that such solutions induce approximate Nash equilibria for the N-player game with vanishing error in the mean-field limit as $N \to \infty$. This talk is based on a joint work with M. Ghio and G. Livieri (SNS Pisa).
4th Berlin-Princeton-Singapore Workshop on Quantitative Finance
The 4th 4th Berlin-Princeton-Singapore Workshop on Quantitative Finance takes place March 18-20, 2019 at NUS. The workshop is supported through the HU's profile partnership program with NUS and Princeton University. More information will be made available soon.
Dynamic Noisy Rational Expectations Equilibrium with Insider Information
In this talk, we study equilibria in multi-asset and multi-agent continuous-time economies with asymmetric information. We establish existence of two equilibria. First, a full communication one where the informed agents' signal is disclosed to the market, and static policies are optimal. Second, a partial communication one where the signal disclosed is ane in the informed and noise traders' signals. Here, information asymmetry creates demand for a dark pool with endogenous participation where private information trades can be implemented. Markets are endogenously complete and equilibrium prices have a three factor structure. Results are valid for multiple dimensions; constant absolute risk averse investors; fundamental processes following a general diffusion; non-linear terminal payoffs, and non-Gaussian noise trading. Asset price dynamics and public information flows are endogenous, and are established using multiple filtration enlargements, in conjunction with predictable representation theorems for random analytic maps. Rational expectations equilibria are special cases of the general results.
Beratung bei d-fine – analytisch. technologisch. quantitativ.
Die d-fine GmbH ist seit über 15 Jahre mit ihrem Konzept, Naturwissenschaftler und Mathematiker (m/w/d) in der Beratung für den Finanzdienstleistungssektor in Deutschland und Europa einzusetzen, sehr erfolgreich. Bekannt u.a. durch Werbung im Physik Journal, Fachliteratur und Stipendienförderungen für Nachwuchswissenschaftler, erfreut sich d-fine einem regen Mitarbeiterwachstum, das in naher Zukunft die 1.000-Personen-Marke überschreiten wird. Dieses Wachstum spiegelt den Erfolg von d-fine sowohl in seinen etablierten Geschäftsfeldern als auch in neuen Themenbereichen wie Autonomes Fahren, Health Care, Machine Learning, Künstliche Intelligenz und Blockchain-Technologie wider. In diesem Vortrag stehen der berufliche Werdegang und die praktische Erfahrung des Vortragenden Dr. rer. nat. Patrick Mack, Alumnus des Karlsruher Instituts für Technologie, als Manager bei d-fine im Vordergrund. Herr Dr. Mack hat unter seinem inzwischen an der HU Berlin forschenden Doktorvater Prof. Dr. Kurt Busch in der Photonics Group an der Physikfakultät des KIT in 2011 promoviert. Der Vortrag ist eine kurzweilige first-hand Schilderung der Projekt- und Reiseerfahrungen von Herrn Dr. Mack bei d-fine und steht beispielhaft für die vielen attraktiven Beschäftigungsmöglichkeiten, die d-fine Berufseinsteigern bietet.
A mean-field game approach to price formation
Here, we introduce a price-formation model where a large number of small players can store and trade an asset. Our model is a constrained mean-field game (MFG) where the price is a Lagrange multiplier for the supply vs. demand balance condition. We establish the existence of a unique solution using a fixed-point argument. In particular, we show that the price is well-defined and it is a Lipschitz function of time. Then, we study linear-quadratic models that can be solved explicitly.
Large Tournament Games
We consider a stochastic tournament game in which each player is rewarded based on her rank in terms of the completion time of her own task and is subject to cost of effort. When players are homogeneous and the rewards are purely rank dependent, the equilibrium has a surprisingly explicit characterization, which allows us to conduct comparative statics and obtain explicit solution to several optimal reward design problems. In the general case when the players are heterogenous and payoffs are not purely rank dependent, we prove the existence, uniqueness and stability of the Nash equilibrium of the associated mean field game, and the existence of an approximate Nash equilibrium of the finite-player game.
Joint work with Jakša Cvitanić and Yuchong Zhang.