Mathematical Finance Seminar – Spring 2020

Schedule for Spring 2020

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


Xiaofei Shi (CMU)

“Liquidity Risk and Asset Pricing”


We study how the price dynamics of an asset depends on its “liquidity” – the ease with which can be traded. An equilibrium is achieved through a system of coupled forward-backward SDEs, whose solution turns out to be amenable to an asymptotic analysis for the practically relevant regime of large liquidity. We also calibrate our model to time series data of market prices and trading volume, and discuss how to leverage deep-learning techniques to obtain numerical solutions. (Based on joint works in progress with Agostino Capponi, Lukas Gonon, Johannes Muhle-Karbe).


Carsten Chong (EPFL)

“High-frequency analysis of SPDEs (and how it relates to rough volatility estimation)”


We consider the problem of estimating stochastic volatility for a class of parabolic stochastic PDEs. Assuming that the solution is observed at high temporal frequency, we use limit theorems for power variations to construct consistent nonparametric estimators and asymptotic confidence bounds for the integrated volatility process. Special attention is given to the case of multiplicative noise. We explain how the involved methods relate to estimation of rough volatility.


Nizar Touzi (Polytechnique)

“Path-dependent mean field optimal planning.”

Max Reppen (Princeton)

“A Mean Field Games Model for Cryptocurrency Mining.”

We propose a mean field game model to study the question of how centralization of reward and computational power occur in Bitcoin-like cryptocurrencies.  Miners compete against each other for mining rewards by increasing their computational power.  This leads to a novel mean field game of jump intensity control, which we solve explicitly for miners maximizing exponential utility, and handle numerically in the case of miners with power utilities.  We show that the heterogeneity of their initial wealth distribution leads to greater imbalance of the reward distribution, or a “rich get richer” effect.  This concentration phenomenon is aggravated by a higher bitcoin price, and reduced by competition.  Additionally, an advantaged miner with cost advantages such as access to cheaper electricity contributes a significant amount of computational power in equilibrium, unaffected by competition.


No seminar (Berkeley-Columbia meeting)

Jose Scheinkman (Columbia)

“Menu costs and the volatility of inflation” (joint with Makoto Nirei, University of Tokyo)

We present a state-dependent equilibrium pricing model that generates inflation rate fluctuations from idiosyncratic shocks to the cost of price changes of individual firms.  A firm’s nominal price increase lowers other firms’ relative prices, thereby inducing further nominal price increases. We first study a mean-field limit where the equilibrium is characterized by a variational inequality and exhibits a constant rate of inflation. We use the limit model to show that in the presence of a large but finite number n of firms the snowball effect of repricing causes fluctuations to the aggregate price level  and these fluctuations converge to zero slowly as n grows. The fluctuations caused by this mechanism are larger when the density of firms at the repricing threshold is high, and the density at the threshold is high when the trend inflation level is high. However a calibration to US data shows that this mechanism is quantitatively important even at modest levels of trend inflation and  can account for the positive relationship between inflation level and volatility that has been observed empirically.


Attention: This seminar takes place online, via Zoom. Join URL:

Yerkin Kitapbayev (NCSU)

“Mortgage Contracts and Selective Default (with Scott Robertson)”
Abstract: We analyze recently proposed mortgage contracts which aim to eliminate selective borrower default when the loan balance exceeds the house price (the “underwater” effect).  We show that contracts which automatically reduce the outstanding balance in the event of local house price decline remove the default incentive, but may induce prepayment in low price states.  However, low state prepayments vanish if borrower utility from home ownership, or outside options such as rental costs, are too high. We also show that capital gain sharing features, through prepayment penalties in high house price states, are ineffective, as they virtually eliminate prepayment in such states. For typical foreclosure costs, we find that contracts with automatic balance adjustments become preferable to the traditional fixed rate mortgage at contract rate spreads of approximately $50-100$ basis points, depending on how far prices must fall before adjustments are made. Furthermore,  these spreads rapidly decrease with the borrower utility from home ownership. Our results are obtained using American options pricing methods, in a model with  diffusive home prices, and either diffusive or constant interest rates. We determine the contract, default and prepayment option values with optimal decision rules. We provide explicit solutions in the perpetual case with constant interest rates; and numerically compute the prepayment and default boundaries in the general case.  

No seminar (Spring Recess)


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Matteo Basei (EDF)

Nonzero-sum stochastic games with impulse controls, and applications.”
Abstract. We consider a general class of nonzero-sum stochastic games with impulse controls. By means of a suitable system of quasi-variational inequalities, we provide a verification theorem for the equilibrium strategies. We then present some examples and applications. Finally, we consider some extensions and future research directions.
* Cancelled – Xunyu Zhou (Columbia)

* Cancelled – Jaksa Cvitanic (Caltech)


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Jim Gatheral (Baruch)

“Diamond trees, forests, cumulants, and martingales.”
We define a diamond operator on the space of continuous semimartingales. Compositions of diamond operations lead to trees and forests of trees.  We show how forests of trees can be identified with cumulants, giving explicit expressions for cumulants.  As an application, we show how to compute all terms in an expansion of the Lévy area.  By reordering the trees in our cumulant expansion according to number of leaves, we retrieve our earlier exponentiation theorem, applications of which include the extension of the Bergomi-Guyon expansion to all orders in volatility of volatility. Finally, we compute exact expressions under rough Heston, in particular a closed-form expression for the leverage swap.

*Cancelled – Semyon Malamud (EPFL)


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Giovanni Conforti (Ecole Polytechnique)

“An invitation to the Schrodinger problem.”

This talk is meant as an introduction to the Schroedinger problem. Initially posed by E. Schrödinger as the problem of finding the most likely evolution of a cloud of independent Brownian particles in between two given configurations, it is nowadays formulated as an entropy minimisation problem under marginal constraints. In the recent years, it has become clear that the study of the Schrodinger problem can bring new insights on other research fields in mathematics, machine learning and engineering. For example, it motivates the use of Sinkhorn’s algorithm and entropic regularisation techniques for the computation of optimal transport plans and it has led to the discovery to a novel class of geometric and functional inequalities. The aim of this talk is to elucidate some of these connections and to discuss interesting generalisations of the original problem that are nowadays emerging as interesting problems on their own.