We consider a price-maker company which generates electricity and sells it in the spot market. The company can increase its level of installed power by irreversible installations of solar panels. In absence of any actions of the company, the electricity's spot price evolves as an Ornstein-Uhlenbeck process, and therefore it has a mean-reverting behavior. The current level of the company's installed power has a permanent impact on the electricity's price and affects its mean-reversion level. The company aims at maximizing the total expected profits from selling electricity in the market, net of the total expected proportional costs of installation. This problem is modeled as a two-dimensional degenerate singular stochastic control problem in which the installation strategy is identified as the company's control variable. We follow a guess-and-verify approach to solve the problem. We find that the optimal installation strategy is triggered by a curve which separates the waiting region, where it is not optimal to install additional panels, and the installation region, where it is. Such a curve depends on the current level of the company's installed power, and is the unique strictly increasing function which solves a first-order ODE. While studying the ODE, we obtain so far unproved properties of a ratio involving a class of Hermite and parabolic cylinder functions.
Optimal Installation of Solar Panels with Price Impact: a Solvable Singular Stochastic Control Problem
MOT Duality and Robust Finance
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.
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.