Titles, Abstracts, and Slides

Umut Cetin (slides)

Financial equilibrium with asymmetric information and random horizon

I will describe and solve a version of the Kyle model with random horizon first introduced in a specific case by Back and Baruch, where the trading horizon is given by an independent exponential random variable. I will discuss the characterisation of the equilibrium value function and the pricing rule in terms of the potential theory of one-dimensional diffusions. I will also discuss how the random horizon can be endogenised by including a manager optimally choosing the announcement date of the dividend.

Xin Guo

Stochastic games and MFGs for fuel follower problems: NE and PO

In this talk we aim to understand how MFG and especially the aggregation in MFG formulation changes the nature of the solution for stochastic games. We start by investigating the classical fuel follower problems for N-player games as well as the MFG. In this singular control framework, the NE points are easier than the PO to analyze. The latter is derived through an auxillary multi-dimensional control problem which provides interesting insight for the nature of PO. The analysis of the Nash Equilibria (NE) and Pareto Optimality (PO) are linked with the Skorokhod problem with non-smooth boundary and unbounded domains. We then show connections and differences of N players games and MFGs in terms of both NE and PO.

Ioannis Karatzas (slides)

Arbitrage Theory via Numeraires: A Survey

We survey a mathematical theory for finance which is based on the following principle: that it should not be possible to fund, starting with arbitrarily small initial capital, a cumulative capital withdrawal stream which is non-trivial, that is, strictly positive with positive probability. In the context of continuous asset prices modeled by semimartingales, we show that proscribing such egregious forms of arbitrage (but allowing for the possibility that one portfolio might outperform another) turns out to be equivalent to any one of the following conditions:

(i) a portfolio with the local martingale numeraire property exists,

(ii) a growth-optimal portfolio exists,

(iii) a portfolio with the log-optimality property exists,

(iv) a strictly positive local martingale deflator exists,

(v) the market has locally finite maximal growth.

We give precise meaning to these terms, and show that the above five equivalent conditions are descriptive — in that they can be formulated entirely, in fact very simply, in terms of the local characteristics (the drifts and covariations) of the underlying asset prices. Full-fledged theories for hedging and for portfolio/consumption optimization can be developed in such a setting, as can the important notion of “market completeness”. The semimartingale property of asset prices turns out to be necessary for viability, when investment is constrained to be “long-only” — that is, to avoid negative (“short”) positions in stocks, and never to borrow from the money market. These notions and results admit extensions to settings with an arbitrary number of assets.

(Joint work—book of the same title—with Constantinos Kardaras.)

Johannes Muhle-Karbe (slides)

Liquidity in competitive dealer markets

We study the equilibrium price at which competitive dealers intermediate between clients with exogenous trading needs, and an end-user market with finite liquidity. (Joint work in progress with Peter Bank and Ibrahim Ekren.)

Miklos Rasonyi (slides)

On utility maximization without passing by the dual problem

We treat utility maximization from terminal wealth for an agent with utility function defined on the real line, who dynamically invests in a continuous-time financial market and receives a possibly unbounded random endowment. We prove the existence of an optimal investment without introducing the associated dual problem. We rely on a recent result of Orlicz space theory, due to Delbaen and Owari which leads to a simple and transparent proof. Constraints on the terminal wealth can also be incorporated.

As examples, we treat frictionless markets with finitely many assets, large financial markets and markets with transaction costs. In the latter case we can even handle the case of model ambiguity.

Based on joint work with Ngoc Huy Chau.