Seminars are on Thursdays
Time: 4:10pm – 5:25pm
Location: Columbia University, 903 SSW (1255 Amsterdam Ave, between 121st and 122nd Street)
Organizers: Ioannis Karatzas, Philip Protter, Marcel Nutz, Yuchong Zhang
Beatrice Acciaio (LSE)
Causal optimal transport and its links to enlargement of filtrations and stochastic optimization problems
The martingale part in the semimartingale decomposition of a Brownian motion, with respect to an enlarged filtration, is an anticipative mapping of said Brownian motion. In analogy to optimal transport theory, I will define causal transport plans in the context of enlargement of filtrations, as the Kantorovich counterparts of the aforementioned non-adapted mappings. I will present a necessary and sufficient condition for a Brownian motion to remain a semimartingale in an enlarged filtration, in terms of certain minimization problems over sets of causal transport plans. The latter will be also used in order to give an estimate of the value of having additional information, for some classical stochastic optimization problems.
This talk is based on a joint work with Julio Backhoff and Anastasiia Zalashko.
Alfred Galichon (NYU)
A theory of decentralized matching markets without transfers,with application to surge pricing
Most of the literature on two-sided matching markets without transfers focuses on the case where a central planner (often an algorithm) clears the market, like in the case of school assignments, or medical residents. In contrast, we focus on decentralized matching markets without transfers, where prices are regulated and thus cannot clear the market, such as the taxi market. In these markets, time waited in line often plays the role of a numeraire. We investigate the properties of equilibrium in these markets (existence, uniqueness, and welfare). We use this analysis to set up the problem of surge pricing: given beliefs on random demand and supply, how should a market designer set prices to minimize expected market inefficiency. (Joint with Yu-Wei Hsieh, USC.)
Dan Pirjol (JP Morgan)
Numerical moment explosions in discrete-time stochastic processes
Certain stochastic processes in discrete time appearing in mathematical finance display numerical moment explosions. As the model parameters cross certain thresholds, the moments of the stochastic variable have a rapid increase which is observed in simulations as a numerical explosion. We illustrate this phenomenon on two examples: a) the bank account compounding interest in discrete time, assuming that the interest rates follow a geometric Brownian motion (Black-Derman-Toy model), and b) the uncorrelated Hull-White stochastic volatility model. The moment explosions are explained as phase transitions in the Lyapunov exponents of the moments. The Lyapunov exponents are computed exactly using large deviations theory, and explosion criteria are obtained.
The talk is based on work with Lingjiong Zhu (FSU).
Kevin Webster (Citadel)
“Leland Strategy with both market and limit orders”
The Leland strategy proposes a discretization of the Black and Scholes delta-hedging strategy for European options under the presence of transaction costs. The talk revisits this problem in a market where limit orders and market orders are used simultaneously to trade. Adverse selection of the limit orders by other market participants plays a crucial role. Adverse selection drives the profitability of limit orders down, while providing hedging benefits for negative gamma options. This insight leads to an explicit execution strategy for delta-hedging after buying options.
Sergey Nadtochiy (U Michigan)
“Endogenous Formation of Limit Order Books: Dynamics Between Trades”
In this talk, I present a continuous-time extension of the framework for modeling market microstructure, developed in our previous work. I use this extension to model the shape and dynamics of the Limit Order Book (LOB) between two consecutive trades. In this model, the LOB arises as an outcome of an equilibrium between multiple agents who have different beliefs about the future demand for the asset. These beliefs may change according to the information observed by the agents (e.g. represented by a relevant stochastic factor), implying a change in the shape of the LOB. This model allows one to see how changing the relevant information signal (which is given in a very general form in our model) affects the LOB. In particular, if the relevant signal is a function of the LOB itself, then, our approach allows one to model the “informational” impact of market events (as opposed to the “mechanical” impact that a market order makes, by eliminating certain limit orders instantaneously). On the mathematical side, we formulate the problem as a mixed control-stopping game, with a continuum of players. We manage to split the equilibrium problem into two parts, and represent one of them through a two-dimensional system of Reflected Backward Stochastic Differential Equations, and the other one with an infinite-dimensional fixed-point equation. Although both problems present mathematical challenges, we manage to prove existence of their solutions and show how they can be computed in a simple example.
|10/20/16||Rene Carmona (Princeton): Minerva Lecture Series on Mean Field Games|
Rene Carmona (Princeton): Minerva Lecture Series on Mean Field Games
Francois Delarue (Nice)
“Master equations for mean field games: classical solutions and convergence problem”
Abstract: Mean field games is a theory for describing asymptotic Nash equilibria in games involving a large number of players interacting with one another in a mean field way. First, I will explain how equilibria may be described by an infinite dimensional PDE set on the space of probability measures, which is called the master equation. This description includes the case when players are submitted to a systemic noise. Second, I will explain how to solve this equation in the classical sense whenever the so-called Lasry Lions monotonicity condition is in force. Third, I will show how classical solutions may be used to establish the convergence of games with finitely many players to mean field games.
Xunyu Zhou (Columbia and Oxford)
“Weighted discounting – On group diversity, time-inconsistency, and consequences for investment”
This paper studies groups whose members disagree about the method of discounting. In particular, we provide a comprehensive treatment of discount functions that are given by the weighted average of the group members’ discount functions. This class of “weighted discount functions” admits a natural notion of group diversity, which has consequences for behavior. We show that more diverse groups discount less heavily and make more patient decisions. Within a real options framework, for example, greater group diversity leads to delayed investment. Finally, we show that well-known behavioral discount functions can be written as a weighted discount function. Therefore, all results in this paper find a correspondence in a single agent setting with non-standard, behavioral time preferences. In particular, we provide an equilibrium optimal stopping result for individuals who are aware of their time-inconsistency, but lack commitment. This is a joint work with S. Ebert and W. Wei.
|11/17/16||No seminar (NEPS and SIAM Conference)|
No seminar (Thanksgiving)
Phillip Yam from (CUHK)
“A Paradox in Time Consistency in Mean-Variance Problem?”
In this talk, we introduce some new conditions, with respect to mean-variance objective functions, under which one can start off his/her constrained (time- consistent) equilibrium strategy at a certain time to beat the unconstrained counterpart. We claim further that the pure strategy of solely investing in bond can sometimes simultaneously dominate both constrained and unconstrained equilibrium strategies. With numerical experiments, the constrained strategy can dominate the unconstrained one at most of the commencement dates (even more than 90%) over of a prescribed planning horizon. The source of paradox is rooted from the nature of straightforward game theoretic approach on time consistency in the existing literature, which purposely seeks for equilibrium solution but not the ultimate maximizer. Our obtained results actually advocate that, to properly implement the concept of time consistency in various financial problems, all economic aspects should be critically taken into account at the same time.