Mathematical Finance Seminar Series

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Schedule for Spring 2022

Seminars are on Thursdays
Time: 4:10pm - 5:25pm
Attention: All talks are available online, via Zoom. Select talks take place in hybrid mode. In-person participation is only available to Columbia affiliates with building access.

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Meeting ID: 946 7022 5225

Passcode: bachelier

Organizers: Ioannis Karatzas, Marcel Nutz, Philip Protter, Xiaofei Shi, Johannes Wiesel



Ariel Neufeld (NTU Singapore)

Title: Deep Learning based algorithm for nonlinear PDEs in finance and gradient descent type algorithm for non-convex stochastic optimization problems with ReLU neural networks


In this talk, we first present a deep-learning based algorithm which can solve nonlinear parabolic PDEs in up to 10’000 dimensions with short run times, and apply it to price high-dimensional financial derivatives under default risk.
Then, we discuss a general problem when training neural networks, namely that it typically involves non-convex stochastic optimization.
To that end, we present TUSLA, a gradient descent type algorithm (or more precisely : stochastic gradient Langevin dynamics algorithm) for which we can prove that it can solve non-convex stochastic optimization problems involving ReLU neural networks.
This talk is based on joint works with C. Beck, S. Becker, P. Cheridito, A. Jentzen, and D.-Y. Lim, S. Sabanis, Y. Zhang, respectively.

Xiaolu Tan (CUHK)

Title: An exit contract optimization problem

Abstract: We study an exit contract optimization problem, where one provides a universal exit contract to different and heterogeneous agents, so that each agent solves an optimal stopping problem w.r.t. the exit contract. The problem consists in optimizing the universal exit contract. Under a technical monotone condition, and by using Bank-El Karoui's representation of stochastic process, we are able to transform the initial contract optimization problem into an optimal control problem. The latter is also equivalent to an optimal multiple stopping problem and the existence of the optimal contract is obtained. We next show that the problem in the continuous time setting can be approximated by a sequence of discrete time ones, which would induce a natural numerical approximation method. We finally discuss the problem if one restricts to the class of all Markovian and/or continuous contracts.


Zhenjie Ren (Paris Dauphine)

Title: Entropic Fictitious Play

Abstract: The classical fictitious play is a common algorithm for solving games. However, once the cost functions of the players are non-convex, the method becomes hard to implement. In our study we add the entropic regulariser, a common strategy for non-convex optimisation, to the cost functions, and look into the analog of fictitious play in this context. We shall further see that the entropic fictitious play not only helps to solve non-convex game, but also can be used to solve optimisations on the space of probability measures, and thus can be applied to train neural networks.



Sigrid Källblad Nordin (KTH)

Title: Controlled measure-valued martingales: theory and applications

Abstract: We consider a class of stochastic control problems where the state process is a probability measure-valued process satisfying an additional martingale condition on its dynamics, called measure-valued martingales (MVMs). We show that the `classical results’ of stochastic control hold for these problems: namely, the value function for the problem can be characterised as the unique solution to an Hamilton-Jacobi-Bellman equation in the sense of viscosity solutions. We also discuss the related problem of existence of controlled MVMs and an appropriate version of Itô’s formula for such processes. Control problems of this form are motivated by applications in (robust) mathematical finance; we illustrate this and also discuss further applications of independent interest. The talk is based on joint work with A. Cox, M. Larsson and S. Svaluto-Ferro.



*This talk will take place in hybrid mode

Thibaut Mastrolia (Berkeley)

Title: Auction market design

Abstract: We model sequential auctions in financial markets during a given time period receiving orders of market participants. A clearing price of the auction is determined as the price maximizing the exchanged volume at the clearing time according to the supply and demand of each market participant. We then focus on the optimal duration of an auction to reduce the error between the clearing price and the efficient price of the stock considered. When investors are strategic, they minimize simultaneously their transaction costs by adapting their trading intensities to the market state. We thus provide the existence of a Nash equilibrium for this stochastic game reduced to the analysis of a system of PDE with discontinuities. We then compute the optimal duration of the auctions for some stocks traded on Euronext and compare the quality of price formation process under this optimal value to the case of a continuous limit order book. We then extend the study to a new market mechanism "ad hoc electronic auction design" (AHEAD) in which market participants have the opportunity to trigger the auction when necessary in addition to controlling their trading intensities. We prove in particular that this model is well-posed and we compare it with classical sequential auctions and limit order books.


*This talk will take place in hybrid mode

Kevin Webster (Garden Leave from Citadel)

Title: Closed form formulas for a generalized Obhizaeva and Wang model with local concavity

Abstract: With the use of functional Central Limit theorems from Jacod and Protter, we provide a reduced form model for the locally concave propagator models introduced by Bouchaud et al. We find that these models behave in the continuous limit as a slight extension to the Obhizaeva and Wang model. Leveraging Fruth, Shoneberg and Urusov, we find a closed form formula for the optimal execution problem and derive conditions for absence of price manipulation strategies in this model.


*This talk will take place in hybrid mode

Julien Guyon (Bloomberg)

Title: Dispersion-Constrained Martingale Schrödinger Problems and the Joint S&P 500/VIX Smile Calibration Puzzle


The very high liquidity of S&P 500 (SPX) and VIX derivatives requires that financial institutions price, hedge, and risk-manage their SPX and VIX options portfolios using models that perfectly fit market prices of both SPX and VIX futures and options, jointly. This is known to be a very difficult problem. Since VIX options started trading in 2006, many practitioners and researchers have tried to build such a model. So far the best attempts, which used parametric continuous-time jump-diffusion models on the SPX, could only produce approximate fits. In this talk we solve this longstanding puzzle for the first time using a completely different approach: a nonparametric discrete-time model. Given a VIX future maturity T1, we build a joint probability measure on the SPX at T1, the VIX at T1, and the SPX at T2 = T1 + 30 days which is perfectly calibrated to the SPX smiles at T1 and T2, and the VIX future and VIX smile at T1. Our model satisfies the martingality constraint on the SPX as well as the requirement that the VIX at T1 is the implied volatility of the 30-day log-contract on the SPX.

The model is cast as the unique solution of what we call a Dispersion-Constrained Martingale Schrödinger Problem which is solved by duality using an extension of the Sinkhorn algorithm, in the spirit of (De March and Henry-Labordère, Building arbitrage-free implied volatility: Sinkhorn's algorithm and variants, 2019). We prove that the existence of such a model means that the SPX and VIX markets are jointly arbitrage-free. The algorithm identifies joint SPX/VIX arbitrages should they arise. Our numerical experiments show that the algorithm performs very well in both low and high volatility environments. Finally, we discuss how to extend our results to continuous-time models, (i) by building a martingale interpolation of the discrete-time model, and (ii) by extending our Schrödinger approach to continuous-time stochastic volatility models, via what we dub VIX-Constrained Martingale Schrödinger Bridges, inspired by the classical Schrödinger bridge of statistical mechanics.


No Seminar


No seminar (Spring Recess)


*This talk will take place in hybrid mode

Gökçe Dayanikli (Princeton)

Title: Choosing Incentives in Large Population Games with Applications to Epidemic Control

Abstract: In this talk, we consider a Stackelberg mean field game model between a principal and a mean field of agents evolving on a finite state space, motivated by models of epidemic control in large populations. The agents play a non-cooperative game in which they can control their transition rates between states to minimize an individual cost. The principal can influence the resulting Nash equilibrium through incentives to optimize its own objective. Later, we propose an application to an epidemic model of SIR type in which the agents control their interaction rate and the principal is a regulator acting with non pharmaceutical interventions. To compute the solutions, we use an innovative numerical approach based on Monte Carlo simulations and machine learning tools for stochastic optimization. Finally, we briefly discuss another game formulation for a continuum of non-identical players evolving on a finite state space where their interactions are represented by a graphon.


*This talk will take place in hybrid mode

Speaker: Dominykas Norgilas (University of Michigan)

Title: The shadow measure and associated supermartingale couplings.
Abstract: Given two measures $\mu,\nu$ on $\mathbb{R}$ with $\mu(\R)\leq\nu(\R)$, and such that $\mu$ is smaller than $\nu$ in positive convex-decreasing order (i.e., $\mu\leq_{pcd}\nu$), there exists a two-period supermartingale $S=(S_1,S_2)$ that transports $\mu$ to $\nu$. For each such supermartingale, $S_1\sim\mu$, but there are many possible choices for the law of $S_2$. In this talk we study two canonical choices (the minimal and the maximal measures) with respect to convex-decreasing order. We show how these measures give rise to the so-called \textit{supermartingale shadow} couplings of $S_1$ and $S_2$.


*This talk will take place in hybrid mode

Stephan Eckstein (ETH)

Title: Causal variants of optimal transport and computational methods

Abstract: The topic of this talk are optimal transport problems where the transport plans are restricted to have a causal (temporal) structure. The focus is on computational aspects of this problem, like approximation by linear programs, (entropic) regularization and causal variants of Sinkhorn's algorithm. Further, the talk will give an outlook for causal variants of optimal transport beyond temporal settings, where the causal structure is dictated by a directed graph. Based on joint work with Gudmund Pammer.


Christa Cuchiero (Vienna)

Title: Optimal bailout strategies resulting from the drift controlled supercooled Stefan problem

Abstract: We consider the problem faced by a central bank which bails out distressed financial institutions that pose systemic risk to the banking sector. In a structural default model with mutual obligations, the central agent seeks to inject a minimum amount of cash to a subset of the entities in order to limit defaults to a given proportion of entities. We prove that the value of the agent's control problem converges as the number of defaultable agents goes to infinity, and that this mean-field limit satisfies a drift controlled version of the supercooled Stefan problem. We compute optimal strategies in feedback form by solving numerically a forward-backward coupled system of PDEs. Our simulations show that the agent's optimal strategy is to subsidize banks whose asset values lie in a non-trivial time-dependent region. We also study a linear-quadratic version of the model where instead of the terminal losses, the agent optimizes a terminal cost function of the equity values. In this case, we are able to give semi-analytic strategies, which we again illustrate numerically. The talk is based on joint work with Christoph Reisinger and Stefan Rigger.


*This talk will take place in hybrid mode

Nicholas Westray (AllianceBernstein and NYU)

Title: Deep Order Flow Imbalance: Extracting Alpha at Multiple Horizons from the Limit Order Book

Abstract: We describe how deep learning methods may be applied to forecast stock returns from high frequency order book states. I will review the literature in this area and describe a study where we evaluate return forecasts for several deep learning models for a large subset of symbols traded on the Nasdaq exchange. We investigate whether transformation of the order book states is necessary and we relate the performance of deep learning models for a symbol to its microstructural properties. We also provide some colour on hyperparameter sensitivity for the problem of high frequency return forecasting. This is based off joint work with Petter Kolm and Jeremy Turiel.


*This talk will take place online only,

Ruodu Wang (Waterloo)

Title: Simultaneous optimal transport

Abstract: We propose a general framework of mass transport between vector-valued measures, which will be called simultaneous mass transport. The new framework is motivated by the need to transport resources of different types simultaneously, i.e., in single trips, from specified origins to destinations. In terms of matching, one needs to couple two groups, e.g., buyers and sellers, by meeting supplies and demands of different goods at the same time. The mathematical structure of simultaneous transport is quite different from the classic setting of optimal transport, leading to many new challenges. The Monge and Kantorovich formulations are contrasted and connected. Existence, uniqueness, and duality theorems are established, and a notion of Wasserstein distance in this setting is introduced. We illustrate the theory with a few applications in economics and finance. This talk is based on joint work with Zhenyuan Zhang.


*This talk will take place in hybrid mode

Alexandre Pannier (Imperial)

Title: Rough multifactor volatility for SPX and VIX options

After a short review of VIX and rough volatility models, we provide explicit small-time formulae for the at-the-money implied volatility, skew and curvature in a large class of models. Our general setup encompasses both European options on a stock and VIX options, thereby providing new insights on their joint calibration. This framework also allows to consider rough volatility models and their multi-factor versions; in particular we develop a detailed analysis of the two-factor rough Bergomi model. The tools used are essentially based on Malliavin calculus for Gaussian processes. This is a joint work with A. Jacquier and A. Muguruza.


*This talk will take place in hybrid mode

Johannes Muhle-Karbe (Imperial)

Title: Managing Transaction Costs in Dynamic Trading

Abstract: We explicitly solve the lifetime investment-consumption problem of investors trading in an incomplete market where asset returns are partially predictable but trading is costly. The
solution is expressed in terms of the unique, global solution of a risk-sensitive Riccati system. We show that the optimal trading strategy targets a portfolio that is optimal for a frictionless
version of the model, where asset returns have been adjusted for costs. The "legacy portfolio" (inherited undesirable positions) are then traded away in line with a backward-looking optimal execution problem. Thus, the investment process is separated into an investment stage, where a target portfolio is designed using a model of time-varying predictable net returns, and a execution stage that
disposes of any unwanted or legacy assets as efficiently as possible assuming there are no-excess returns to any of these assets.

(Joint work in progress with James Sefton (Imperial) and Xiaofei Shi (Columbia))