Mathematical Finance Seminar – Spring 2019

Schedule for Spring 2019

Seminars are on Thursdays
Time: 4:10pm – 5:25pm
Location: Columbia University, 903 SSW (1255 Amsterdam Ave, between 121st and 122nd Street)

Organizers: Ruimeng Hu, Ioannis Karatzas, Marcel Nutz, Philip Protter

MAFN Seminar Archive


Martin Larsson (ETH)

“Short- and long-term relative arbitrage in stochastic portfolio theory”

Stochastic Portfolio Theory is a mathematical framework for studying large equity markets, especially the performance of long-term investments. An important focus are universal features that only depend weakly on specific modeling assumptions. A basic result of this kind states that a mild nondegeneracy condition suffices to guarantee long-term relative arbitrage, that is, the possibility to outperform the market over sufficiently long time horizons. A longstanding open question has been whether short-term relative arbitrage is also implied. A qualitative answer, in the negative, was recently given by Fernholz, Karatzas & Ruf. In this work we settle the question by characterizing and explicitly computing the critical time horizon beyond which relative arbitrage always exists. The key tool is a previously unknown connection between existence of relative arbitrage and certain geometric PDE describing mean curvature flow in high co-dimension.


No Seminar

Pierre Cardaliaguet (Paris Dauphine)

“Mean Field Games with a major player”

Mean field games with a major agent study optimal control problems with infinitely many small controllers facing a major controller. The “value function” of the agents then satisfy a so-called system of master equations, which is a nonlinear nonlocal system of partial differential equations stated in the space of measures. In this joint work with Marco Cirant (U. Padova) and A. Porretta (U. Rome Tor Vergata) we explain how to build short time a classical solution for this system and use the solution to prove the mean field limit of the associated N player game as the number N of the players tends to infinity.



Room: 1025 SSW

Time: 3:30pm

Daniel Bartl (Konstanz)

“Model uncertainty in mathematical finance via Wasserstein distances”


In this talk we model uncertainty through neighborhoods in Wasserstein distance within a one-period framework. After a short discussion on the choice of distance, we show (semi-)explicit formulas for some robust risk measures. We then conduct a sensitivity analysis (of e.g. utility maximization) and finally study a scaling limit in continuous time of Wasserstein neighborhoods. If time permits, we shortly elaborate why Wasserstein distances are not suited for a general multi-period analysis and introduce an adapted modification. Based on joint works with J.Backhoff, M.Beiglboeck, S.Drapeau, M.Eder, M.Kupper, J.Obloj, L.Tangpi, J.Wiesel.


Asaf Cohen (Haifa)

“Fluctuations in finite state many player games”

We consider an n-player symmetric stochastic game with weak interactions between the players. Time is continuous and the horizon and the number of states are finite. We show that the value function of each of the players can be approximated by the solution of a differential equation called the master equation. Moreover, we analyze the fluctuations of the empirical measure of the states of the players in the game and show that it is governed by a solution to a stochastic differential equation. (Joint work with Erhan Bayraktar)


Jianfeng Zhang (USC)

“Weak Solutions of  Mean Field Game Master Equations”
In this talk we study master equations arising from mean field game problems, under the crucial monotonicity conditions.
Classical solutions of such equations require very strong technical conditions. Moreover, unlike the master equations arising from mean field control problems, the mean field game master equations are non-local and even classical solutions typically do not satisfy the comparison principle, so the standard viscosity solution approach seems infeasible. We shall propose a notion of weak solution for such equations and establish its wellposedness. Our approach relies on a new smooth mollifier for functions of measures, which unfortunately does not keep the monotonicity property, and the stability result of master equations. The talk is based on an ongoing joint work with Chenchen Mou.


Benjamin Moll (Princeton)
Mean Field Games in Macroeconomics

I will discuss some examples of Mean Field Games (MFGs) that naturally arise in macroeconomics. These MFGs come from models designed to study some of the most important questions in economics. At the same time, they are highly interesting for mathematicians because their structure is often quite difficult. My talk will mostly focus on macroeconomic models with heterogeneous individuals that are used to model the joint distribution of income and wealth, i.e. inequality, and its interaction with the macroeconomy. While these models are MFGs they typically do not satisfy the restrictions imposed in the mathematics MFG literature to obtain theoretical characterizations (in particular basic results like existence and uniqueness). The situation is even more difficult for variants of these models with common noise. Therefore, new approaches are needed to analyze this class of theories and my hope is to get academics with backgrounds in mathematics, statistics and operations research etc interested in studying them.

Background reading:


Jean Jacod (Paris 6)

“Testing for the Markov Property in a High-Frequency Setting”

The aim is to present a test for the homogeneous Markov property of a one-dimensional process X observed at regularly spaced times over a finite time interval. The frequency goes to infinity, and we test the null hypothesis according to which the spot volatility takes the form f(X(t)) for some smooth enough non-vanishing function f. The test relies on some Central Limit Theorems related to the local times of a semimartingale. We allow the process X to have jumps, restricted to finite activity. We will mostly consider the case when the process is observed without error, and if time permits we will give a method covering the case where microstructure noise is present. (Joint work with Yacine Ait-Sahalia)

3/21/19 No seminar (Spring Break)

Mathieu Lauriere (Princeton)

“On the optimal control of conditioned processes”

In this talk, we consider an optimal control problem for a conditioned process. This model was first introduced by P.-L. Lions in his lectures at College de France. When the optimization is done over controls of feedback type (i.e., depending only on the state of the process), the optimal solution can be characterized by a system of two partial differential equations of mean field type: a forward (Fokker-Planck) equation and a backward (Hamilton-Jacobi-Bellman) equation, both with Dirichlet boundary conditions. They describe respectively the evolution of the distribution and of the value function. We also consider a problem arising in the long time asymptotics. This is a control problem driven by the principal eigenvalue problem associated with a Fokker-Planck equation with Dirichlet condition. We study in details particular aspects of the theory and discuss numerical results. This is based on lectures by P.-L. Lions and joint work with Y. Achdou.


“Robust XVA”

Abstract: We introduce an arbitrage-free framework for robust valuation adjustments. An investor trades a credit default swap portfolio with a defaultable counterparty. The investor does not know the expected rate of return of the counterparty bond, but he is confident that it lies within an uncertainty interval. We derive both upper and lower bounds for the XVA process of the portfolio, and show that these bounds may be recovered as solutions of nonlinear ordinary differential equations.

The presence of collateralization and closeout payoffs leads to fundamental differences with respect to classical credit risk valuation. The value of the super-replicating portfolio cannot be directly obtained by plugging one of the extremes of the uncertainty interval in the valuation equation, but rather depends on the relation between the XVA replicating portfolio and the close-out value throughout the life of the transaction. This is a joint work with

Agostino Capponi and Stephan Sturm.

Patrick Cheridito (ETH)

“Deep optimal stopping”

I present a deep learning method for optimal stopping problems which directly learns the optimal stopping rule from Monte Carlo samples. As such, it is broadly applicable in situations where the underlying randomness can efficiently be simulated. The approach is tested on three problems: the pricing of a Bermudan max-call option, the pricing of a callable multi barrier reverse convertible and the problem of optimally stopping a fractional Brownian motion. In all three cases it produces very accurate results in high-dimensional situations with short computing times.

Johannes Ruf (LSE)

“Filtration shrinkage, the structure of deflators, and the failure of market completeness”

We analyse the structure of stochastic discount factors (SDFs) projected on smaller filtrations. Via use of a Bayesian filtering approach, we demonstrate the exact mechanism of how updates on the possible class of models under less information result in the strict supermartingale property of projections of SDFs. In a general continuous-path setting, we show that the local martingale part in the multiplicative Doob-Meyer decomposition of projected SDFs are themselves SDFs in the smaller information market. Finally, we demonstrate that these projections are unable to span all possible SDFs in the smaller information market, by means of an interesting example where market completeness is not retained under filtration shrinkage. (Joint work with Kostas Kardaras)