Mathematical Finance Seminar – Fall 2019

Schedule for Fall 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


Ioannis Karatzas (Columbia)

Title: Arbitrage Theory via Numeraires

We develop a mathematical theory for finance based on the following “viability” principle: That it should not be possible to fund a non-trivial liability starting with arbitrarily small initial capital. In the context of continuous asset prices modeled by semimartingales, we show that proscribing such egregious forms of what is commonly called “arbitrage” (but allowing for the possibility that one portfolio might outperform another), turns out to be equivalent to any one of the following conditions: (i) a portfolio with the local martingale numeraire property exists, (ii) a growth-optimal portfolio exists, (iii) a portfolio with the log-optimality property exists, (iv) a local martingale deflator exists, (v) the market has locally finite maximal growth.

We assign precise meaning to these terms, and show that the above equivalent conditions can be formulated entirely, in fact very simply, in terms of the local characteristics (the drifts and covariations) of the underlying asset prices. Full-fledged theories for hedging and for portfolio/consumption optimization can be developed in such a setting, as can the important notion of “market completeness”.

The semimartingale property of asset prices is shown to be necessary and sufficient for viability, when investment is constrained to occur only along a discrete-time schedule, and to be long-only. When a strictly positive martingale (as opposed to local martingale) deflator exists, so does an equivalent martingale measure (EMM) on each time-horizon of finite length. We show that EMMs need not exist in a viable market, and that this notion is highly normative: Two markets may have the exact same local characteristics, while one of them admits such an EMM and the other does not.

(Joint work—book in progress by the same title—with Constantinos Kardaras.)


No seminar (conference 9/20-9/21 –


Ludovic Tangpi (Princeton)

Title: On backward propagation of chaos.
In this talk we will present an extension of the theory of propagation of chaos to backward (weakly) interacting diffusions. The focus will be on cases allowing for explicit convergence rates and concentration inequalities for the empirical measures. Among other consequences, we derive the approximation of some non-local, second order PDEs on an infinite dimensional space by a sequence of parabolic PDEs on finite dimensional spaces.
The talk is based on join works with D. Bartl and M. Lauriere.

Dylan Possamai (Columbia)

Title: A general approach to non-Markovian time-inconsistent stochastic control for sophisticated players

This talk will present a first attempt at a general non-Markovian theory of time-inconsistent stochastic control problems in continuous-time. We consider sophisticated agents who are aware of their time-inconsistency and take it into account in future decisions. We prove here that equilibria in such a problem can be characterised through a new type of multi-dimensional system of backward SDEs, for which we obtain wellposedness. Unlike the existing literature, we can treat the case of non-Markovian dynamics, and our results go beyond verification type theorems, in the sense that we prove that any equilibrium must necessarily arise from our system of BSDEs. This is a joint work with Camilo Hernández, Columbia University

Mete Soner (Princeton)
Title:  Arbitrage under Knightian uncertainty

Abstract: Recently, a large and an increasing body of literature has focused on decisions, markets, and economic interactions under uncertainty.
Indeed, the classical papers of Kreps ’81 and Harrison & Kreps ’70, which lay the basis of arbitrage pricing in relation to economic theory,  prove that if the market is viable, or equivalently supports an equilibrium, if and only if  there is a risk neutral measure. However, Frank Knight’s pioneering work distinguishes risk– a situation that allows for an objective  probabilistic  description  –  from uncertainty–  a  situation  that cannot be modelled by a single probability distribution.  In particular, markets modelled by these two seminal papers must necessarily be under risk and as such cannot accommodate models with Knightian uncertainty.

In this talk, I revisit these classical papers and show how one can modify the notions of viability and arbitrage to cover uncertainty.  I will prove that this modified definition of viability is equivalent to no-arbitrage, which is defined through the super-replication functional as by Karatzas & Kardaras and by appropriate choices it is equivalent to definitions given in the literature, including the one in Bouchard & Nutz, Burzoni, Fritelli & Maggis and Accaio etal.  

This is joint work with Matteo Burzoni from Oxford and Frank Riedel from Bielefeld.



Michail Anthropelos (Piraeus/Boston U)
Title: Price Impact: Optimal Investment, Derivative Demand & Arbitrage.


This paper studies the optimal investment and the derivative pricing under an inventory-based price impact model with competitive market makers. We establish two effects due to price impact: constrained trading and non-linear hedging costs. To the former, investor’s wealth process in the impact model are identified with those in a model without impact, but with constrained trading, where the (random) constraint set is generically neither closed nor convex. On the other hand, the non-linearity of hedging costs heavily affects the notion of arbitrage-free pricing. We provide three such definitions, which coincide in the frictionless case, but which dramatically differ in the presence of price impact. Additionally, we show arbitrage opportunities, should they arise from claim prices, can be exploited only for limited position sizes, and may be optimally ignored if outweighed by hedging considerations. Furthermore, we point out that, in segmented markets, it is likely to have arbitrage-inducing prices arising endogenously as equilibrium prices. This is a joint work with S. Roberson and K. Spiliopoulos (BU).

No Seminar


Walter Schachermayer (Vienna)

Title: From discrete to continuous time models: some surprising news on an old topic


We reconsider the approximation of the Black-Scholes model by discrete-time models such as the binominal or the trinominal model. We show that for continuous and bounded claims one may approximate the replication in the Back-Scholes model by trading in the discrete time models. The approximation holds true in measure as well as „with bounded risk“, the latter assertion being the delicate issue. The remarkable aspect is that this result does not only apply to the well-known binominal model, but to a much wider class of discrete approximating models, including, e.g., the trinominal model. By an example we show that we cannot do the approximation with „vanishing risk“. We apply this result to portfolio optimization and show that, for utility functions with „reasonable asymptotic elasticity“, the solutions to the discrete time portfolio optimization converge to their continuous limit, again in a wide class of discretizations including the trinominal model. In the absence of „reasonable asymptotic elasticity“, however, surprising pathologies may occur. Joint work with David Kreps (Stanford University).


Ting-Kam Leonard Wong (Toronto)

Title: Portfolio selection in stochastic portfolio theory

Stochastic portfolio theory (SPT) is a framework of portfolio selection based on macroscopic behaviors of the equity market. After discussing some empirical properties of the capital distribution we consider functionally generated portfolio, a class of portfolios that capture market diversity and volatility. We formulate several versions of portfolio optimization problems motivated by analogies with nonparametric statistics. In particular, we study Cover’s universal portfolio in the context of SPT, as well as prior distributions on the space of functionaly generated portfolios based on random concave functions. Some empirical results will be shown. Joint works with Heng Kan (Toronto) and Peter Baxendale (USC).


Johannes Wiesel (University of Oxford)

Title: Statistical estimation of superhedging prices

Abstract: In this talk I introduce a framework for the data-driven dynamic risk assessement of financial positions. This framework is exemplified by the construction of several estimators for the superhedging price given historical stock returns. I first pursue a plugin approach in an ergodic environment in order to construct a simple consistent estimator. To overcome the lack of robustness of this plugin estimator I then introduce improved estimators based on convergence bounds in Wasserstein distance between the true and empirical measure. I also give results regarding the convergence of corresponding superhedging strategies and other risk assessment criteria. This talk is based on joint work with Jan Obloj.

Agostino Capponi (Columbia)
Title: Robo-Advising: Modeling, Learning, and Mathematical Challenges.

Abstract: Robo-advisors are automated investment platforms that are rapidly emerging as an alternative to traditional financial advisors. They currently manage over $440 billion in the United States, and are expected to reach an estimated $1.5 trillion by 2023. Their performance strongly depends on the mechanism used to elicit the risk preferences of investors. We first introduce a reinforcement learning framework through which the robo-advisor learns the investor’s preferences from her trading choices, and show that the algorithm’s value function converges to that of an omniscient robo-advisor polynomially in the state and action space. We subsequently propose a control problem where the robo-advisor dynamically adapts its model for the investor’s stochastic risk aversion based on information solicited throughout the investment horizon. We derive a closed-form solution for the resulting time-inconsistent control problem, and introduce novel regret measures to quantify the benefits of more tailored investment advice. Our model explains empirical findings according to which clients who place higher emphasis on delegation and clients with a more unstable risk profile benefit less from robo-advising.