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)




















Johannes Ruf (LSE)




Alvaro Cartea (Oxford)