Mathematical Finance Seminar Series

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Schedule for Spring 2024

Seminars are on Thursdays
Time: 4:10pm - 5:25pm
Location: Room 903 SSW

Organizers:  Steven Campbell, Ioannis Karatzas, Marcel Nutz, Philip Protter



Krzysztof Ciosmak (Toronto)


Title: Localisation for constrained transports

Abstract:  Martingale optimal transport is a tool that allows for a model-free pricing of options. It turns out that any martingale transport between given two probabilities is constrained by certain convex sets, dubbed irreducible components. We investigate an analogue of the irreducible convex paving in the context of generalised convexity. Consider two Radon probability measures ordered with respect to a lattice cone F of functions on X. Under the assumption that any F-transport between the two measures is local, we establish the existence of the finest partitioning of X, depending only on the measures and the cone F, into F-convex sets, called irreducible components, such that any F-transport between the measures must adhere to this partitioning. Furthermore, we demonstrate that a set, whose sections are contained in the corresponding irreducible components, is a polar set with respect to all F-transports between the two measures if and only if it is a polar set with respect to all transports. This provides an affirmative answer to a generalisation of a conjecture proposed by Obłój and Siorpaes regarding polar sets in the martingale transport setting. Among our contributions is also a generalisation of the Strassen’s theorem to the setting of generalised convexity. We present applications to the localisation of the Monge—Kantorovich problem,  to the martingale transport problem and to the submartingale transport problem in the infinite-dimensional setting.




Moritz Voss (UCLA)

Title: Equilibrium in functional stochastic games with mean-field interaction

Abstract: We study a general class of finite-player stochastic games with mean-field interaction where the linear-quadratic cost functional includes linear operators acting on controls in L^2. We propose a new approach for deriving the Nash equilibrium of these games in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations which can be solved. Moreover, by deriving stability results for the system of Fredholm equations, we obtain the convergence of the finite-player Nash equilibrium to the mean-field equilibrium in the infinite player limit. Our general framework includes examples of stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact. This is joint work with Eduardo Abi Jaber (Ecole Polytechnique) and Eyal Neuman (Imperial College London). The paper is available at



Nizar Touzi (NYU)

Title: Mean field game of cross-holding

Abstract:  We consider the mean field game of cross-holding introduced in Djete and Touzi in the context where the equity value dynamics are affected by a  common noise. In this context, the problem exhibits the standard paradigm of mean-variance trade off. Our crucial observation is to search for equilibrium solutions of our mean field game among those models which satisfy an appropriate notion of no-arbitrage. Under this condition, it follows that the representative agent optimization step is reduced to a standard portfolio optimization problem with random endowment.







Graeme Baker (Columbia)

Title: Two approaches to mean-field systemic risk models with default cascades

Abstract: We consider a class of models for systemic risk where the assets of firms interact through the hitting times of a default level. In the mean-field limit, we obtain a free boundary problem for a representative firm, and the boundary can exhibit singularities where a marcoscopic proportion of firms default simultaneously. We study two notions of solution for this problem: minimal solutions which arise as the fixed point of a monotone operator, and physical solutions which are obtained as large-system limits satisfying an energy conservation rule. We show that physical solutions can be used to make sense of the mean-field problem when the interaction term is non-monotonic, where the default of some firms may be beneficial to others. And for the monotonic case, we prove that physical solutions are well-posed if and only if minimal solutions are well-posed.




Silvana Pesenti (Toronto)

Title: Optimal transport divergences based on scoring functions and their applications.


We employ scoring functions, used in statistics for eliciting risk functionals, as cost functions in the Monge-Kantorovich (MK) optimal transport problem. This gives raise to a rich variety of novel asymmetric MK divergences, which subsume the family of Bregman-Wasserstein divergences. We show that for distributions on the real line, the comonotonic coupling is optimal for the majority the new divergences. Specifically, we derive the optimal coupling of the MK divergences induced by functionals including the mean, generalised quantiles, expectiles, and shortfall measures. Furthermore, we show that while any elicitable law-invariant convex risk measure gives raise to infinitely many MK divergences, the comonotonic coupling is simultaneously optimal.

The novel MK divergences, which can be efficiently calculated, open an array of  applications in robust stochastic optimisation.  We derive sharp bounds on distortion risk measures under a Bregman-Wasserstein divergence constraint, and solve for cost-efficient payoffs under benchmark constraints.




Harvey Stein (Two Sigma)

Title: Functions representable by neural networks
Abstract: It's hard to express the extent to which deep neural networks have transformed machine learning.  Networks continue to get larger and more complex and find winder and wider application.  On the other hand, some of the classical approximation theorems show that one layer is sufficient to approximate continuous functions on a bounded region.  But these theorems are not constructive in nature.
Here, we give an explicit construction to show that all piecewise linear functions can be approximated by ReLU networks with two layers and can be exactly replicated by ReLU networks with three layers without the use of infinite parameters or with two layers if appropriate infinite parameters are available.



Philippe Bergault (Paris Dauphine)

Title: A Mean Field Game between Informed Traders and a Broker

Abstract:  We find closed-form solutions to the stochastic game between a broker and a mean-field of informed traders. In the finite player game,  the informed traders observe a common signal and a private signal. The broker, on the other hand, observes the trading speed of each of his clients and  provides liquidity to the informed traders. Each player in the game optimises wealth adjusted by inventory penalties. In the mean field version of the game, using a Gâteaux derivative approach, we characterise the solution to the game with a system of forward-backward stochastic differential equations that we solve explicitly. We find that the optimal trading strategy of the broker is linear on his own inventory,  on the average inventory among informed traders, and on the common signal or the average trading speed of the informed traders. The Nash equilibrium we find helps informed traders decide how to use private information, and helps brokers decide how much of the order flow they should externalise or internalise when facing a large number of clients. (Joint work with Leandro Sanchez-Betancourt)










Roger Lee (U Chicago)

Title: All AMMs are CFMMs. All DeFi markets have invariants.  A DeFi market is arbitrage-free if and only if it has an increasing invariant

Abstract: In a universal framework that expresses any market system in terms of state transition rules, we prove that every DeFi market system has an invariant function and is thus by definition a CFMM; indeed, all automated market makers (AMMs) are CFMMs.

Invariants connect directly to arbitrage and to completeness, according to two fundamental equivalences. First, a DeFi market system is, we prove, arbitrage-free if and only if it has a strictly increasing invariant, where "arbitrage-free" means that no state can be transformed into a dominated state by any sequence of transactions. Second, the invariant is, we prove, unique if and only if the market system is complete, meaning that it allows transitions between all pairs of states in the state space, in at least one direction.

Thus a necessary and sufficient condition for no-arbitrage (respectively, for completeness) is the existence of the increasing (respectively, the uniqueness of the) invariant, which, therefore, fulfills in nonlinear DeFi theory a foundational role parallel to the existence (respectively, uniqueness) of the pricing measure in the Fundamental Theorem of Asset Pricing for linear markets. Moreover, a market system is recoverable by its invariant if and only if it is complete.

Our examples illustrate (non)existence of various specific types of arbitrage in the context of various specific types of market systems -- with or without fees, with or without liquidity operations, and with or without coordination among multiple pools -- but the fundamental theorems have full generality, applicable to any DeFi market system and to any notion of arbitrage expressible as a strict partial order.


Luhao Zhang (Columbia)

Title: Decision Making under Costly Sequential Information Acquisition: the Paradigm of Reversible and Irreversible Decisions

Abstract: Decision-making in modern stochastic systems, including e-commerce platforms, financial markets, and healthcare systems, has evolved into a multifaceted process that involves information acquisition and adaptive information sources. This paper initiates a study on this integrated process, where these elements are not only fundamental but also interact in a complex and dynamically intertwined manner. We introduce a relatively simple model, which, however, captures the novel elements we consider. Specifically, a decision maker (DM) can choose between an established product A with a known value and a new product B with an unknown value. The DM can observe signals about the unknown value of product B and can also opt to exchange it for product A if B is initially chosen. Mathematically, the model gives rise to a sequential optimal stopping problem with two different informational regimes (before and after buying product B), differentiated by the initial, coarser signal and the subsequent, finer one. We analyze the underlying problems using predominantly viscosity solution techniques, differing from the existing literature on information acquisition which is based on traditional optimal stopping techniques. Additionally, our modeling approach offers a novel framework for developing more complex interactions among decisions, information sources, and information costs through a sequence of nested obstacles.



Michel Groppe (Goettingen)

Title: Lower Complexity Adaptation for Empirical Entropic Optimal

Abstract: Entropic optimal transport (EOT) presents an effective and
computationally viable alternative to unregularized optimal transport
(OT), offering diverse applications for large-scale data analysis. We
derive novel statistical bounds for empirical plug-in estimators of the
EOT cost and show that their statistical performance in the entropy
regularization parameter epsilon and the sample size n only depends on
the simpler of the two probability measures. For instance, under
sufficiently smooth costs this yields the parametric rate n^{-1/2} with
factor epsilon^{-d/2}, where d is the minimum dimension of the two
population measures. This confirms that empirical EOT also adheres to
the lower complexity adaptation principle, a hallmark feature only
recently identified for unregularized OT. Our techniques employ
empirical process theory and rely on a dual formulation of EOT over a
single function class. Crucial to our analysis is the observation that
the entropic cost-transformation of a function class does not increase
its uniform metric entropy by much.


Johannes Ruf (LSE)

Title: The numeraire e-variable and reverse information projection

Abstract: We consider testing a composite null hypothesis $\mathcal{P}$ against a point alternative $\mathsf{Q}$ using e-variables, which are nonnegative random variables $X$ such that $\mathbb{E}_\mathsf{P}[X] \leq 1$ for every $\mathsf{P} \in \mathcal{P}$. This paper establishes a fundamental result: under no conditions whatsoever on $\mathcal{P}$ or $\mathsf{Q}$, there exists a special e-variable $X^*$ that we call the numeraire, which is strictly positive and satisfies $\mathbb{E}_\mathsf{Q}[X/X^*] \leq 1$ for every other e-variable $X$. In particular, $X^*$ is log-optimal in the sense that $\mathbb{E}_\mathsf{Q}[\log(X/X^*)] \leq 0$. Moreover, $X^*$ identifies a particular sub-probability measure $\mathsf{P}^*$ via the density $d \mathsf{P}^*/d \mathsf{Q} = 1/X^*$. As a result, $X^*$ can be seen as a generalized likelihood ratio of $\mathsf{Q}$ against $\mathcal{P}$. We show that $\mathsf{P}^*$ coincides with the reverse information projection (RIPr) when additional assumptions are made that are required for the latter to exist. Thus $\mathsf{P}^*$ is a natural definition of the RIPr in the absence of any assumptions on $\mathcal{P}$ or $\mathsf{Q}$. In addition to the abstract theory, we provide several tools for finding the numeraire and RIPr in concrete cases. We discuss several nonparametric examples where we can indeed identify the numeraire and RIPr, despite not having a reference measure. Our results have interpretations outside of testing in that they yield the optimal Kelly bet against $\mathcal{P}$ if we believe reality follows $\mathsf{Q}$.

Joint work with Martin Larsson and Aaditya Ramdas


Alvaro Cartea (Oxford)

Title: Spoofing and Manipulating Order Books with Learning Algorithms

Abstract:  We propose a dynamic model of the limit order book to derive conditions to test if a trading algorithm will learn to manipulate the order book. Our results show that as a market maker becomes more tolerant to bearing inventory risk, the learning algorithm will find optimal strategies that manipulate the book more frequently. Manipulation occurs to induce mean reversion in inventory to an optimal level and to execute round-trip trades with limit orders at a higher probability than was otherwise likely to occur; spoofing is a special case when the market maker prefers that manipulative limit orders are not filled. The conditions are tested with order book data from Nasdaq and we show that market conditions are conducive for an algorithm to learn to manipulate the order book. Finally, when two market makers use learning algorithms to trade, their algorithms can learn to coordinate their manipulation. Preprint:





4:00 p.m. Wednesday, May 1st, 2024


Columbia-NYU Financial Engineering Colloquium

Talk 1:
Marcel Nutz (Columbia)
Title: Unwinding Stochastic Order Flow.”

Talk 2:
Julien Guyon (ParisTech)
Title: Volatility Is (Mostly) Path-Dependent