The recent and on-going financial crisis motivates a scrutinised study in the field of Financial Mathematics. In order to obtain better models, imperfections and complexity of real financial markets must be taken into account. Rather than assuming that arbitrary quantities of assets can be traded without impacting the market, liquidity risk needs to be carefully analysed.

Facing imperfections, good models must be robust, placing less emphasis on particular model assumptions which tend to be unrealistic in practical applications. A better understanding of such issues is of strategic importance to maintain a healthy financial system, and is currently attracting considerable interest from researchers, industry practitioners, as well as regulators.

Any model of liquidity risk will be incomplete without a detailed analysis of the dynamics of supply and demand and the causes of their imbalance. In particular it is important to understand how this imbalance evolve in time in an equilibrium framework where strategic agents trade to maximise their utility. In order to analyse interacting, possibly heterogeneous, agents, one often needs a diverse set of tools from filtering theory, multi-dimensional backward stochastic differential equations and mean-field games.

Recent years have also witnessed substantial developments in path-wise stochastic analysis and martingale transport theory. These results have found applications in obtaining robust financial models for derivative pricing. The aim of this workshop is to bring together researchers to discuss the latest developments in three aforementioned themes: liquidity, mean field games, and robust finance.

Invited speakers

Pierre Collin-Dufresne (École Polytechnique Fédérale de Lausanne)
Alexander Cox (University of Bath)
François Delarue (Université Nice-Sophia Antipolis)
Romuald Elie (Université Paris-Est Créteil Val-de-Marne)
Ying Hu (Université de Rennes 1)
Martin Huesmann (University of Bonn)
Dmitry Kramkov (Carnegie Mellon University)
Martin Larsson (ETH Zürich)
Mathieu Rosenbaum (Université Pierre et Marie Curie - Paris VI)

Programme (including titles and abstracts)

Tuesday 25 August 2015
11.30am - 12.45pm
(Tea and coffee on arrival)

Talks by (60 minutes per session):

Ying Hu
Time-inconsistent stochastic linear-quadratic control: characterisation and uniqueness of equilibrium
Presentation slides

Abstract: In this talk, we continue our study on a general time-inconsistent stochastic linear-quadratic (LQ) control problem. We derive a necessary and sufficient condition for equilibrium controls via a flow of forward--backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we prove that the explicit equilibrium control constructed before is indeed unique. Our proof is based on the derived equivalent condition for equilibria as well as a stochastic version of the Lebesgue differentiation theorem. Finally, we show that the equilibrium strategy is unique for a mean-variance portfolio selection model in a complete financial market where the risk-free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes. We will also study the rank dependent utility case.

Wednesday 26 August 2015
10.30am - 1.45pm
(Tea, coffee, pastries, coissants and fruint available from 10am)

Talks by (60 minutes per session):

Dmitry Kramkov
Muckenhoupt's Ap condition and the existence of the optimal martingale measure
Presentation slides

Abstract: In the problem of optimal investment with utility function defined on positive real line, we formulate sufficient conditions for the dual optimizer to be a uniformly integrable martingale. Our key requirement consists of the existence of a martingale measure whose density process satisfies the probabilistic Muckenhoupt Ap condition for the power p=1/(1-a), where 0<a<1 is a lower bound on the relative risk-aversion of the utility function. We construct a counterexample showing that this Ap condition is sharp.

(Joint work with Kim Weston)

Pierre Collin-Dufresne
Informed trading, market liquidity and endogenous liquidation value

Abstract: We analyze a dynamic model of informed trading where an investor accumulates shares in an anonymous market and then expends costly effort to affect the firm value. We show that the optimal trading strategy of the informed investor is independent of the effort cost function. Equilibrium prices are affected by the position accumulated by the shareholder, because the level of effort undertaken is increasing in the size of his acquired position, which is known perfectly only to him. Thus the model links market liquidity (price efficiency) to the insider's effort level (economic efficiency) in a way that depends on properties of the cost function. This provides a more nuanced perspective on the debate whether corporate governance and liquidity are complements or substitutes. The model also has implications for position disclosure regulation. On the technical side, we derive a simple SDE for a Brownian bridge process that attains almost surely at a fixed date the convex combination of itself with an independent Gaussian random variable.

(Joint work with Kerry Back and Vyacheslav Fos)

(Brief tea and coffee break)

Romuald Elie
Optimal incentives for multiple agents in interaction

Abstract: We consider a model where a principal requires to contract separately with a large number of agents. In this framework, each agent will be in charge of one project, whose stochastic dynamics can be influenced by all the agents. More specifically, each agent can choose to make efforts towards managing his own project, but can also decide to impact (positively or negatively) the projects of the other agents. Considering agents with relative performance concerns, we look towards the optimal way for the principal to contract with the interacting agents and discuss in particular the role of competition in this framework.

(Joint work with Dylan Possama)

Thursday 27 August 2015
9.30am - 3.45pm
(Tea and coffee on arrival)

Talks by (60 minutes per session):

Mathieu Rosenbaum
The microstructual foundations of rough volatility models
Presentation slides

Abstract: It has been recently shown that rough volatility models reproduce very well the statistical properties of low frequency financial data. In such models, the volatility process is driven by a fractional Brownian motion with Hurst parameter of order 0.1. The goal of this talk is to explain how such fractional dynamics can be obtained from the behaviour of market participants at the microstructural scales. Using limit theorems for Hawkes processes, we show that a rough volatility naturally arises in the presence of high frequency trading combined with metaorders splitting.

(Joint work with Thibault Jaisson)

Alex Cox
Model-independent bounds for Asian options: a dynamic programming approach
Presentation slides

Abstract: We consider the problem of finding model-independent bounds on the price of an Asian option, when the call prices at the maturity date of the option are known. Our methods differ from most approaches to model-independent pricing in that we consider the problem as a dynamic programming problem, where the controlled process is the conditional distribution of the asset at the maturity date. By formulating the problem in this manner, we are able to determine the model-independent price through a PDE formulation. Notably, this approach does not require specific constraints on the payoff function (e.g. convexity), and would appear to be generalisable to many related problems.

(Joint work with S. Källblad)

(Brief tea and coffee break)

Martin Larsson
Robust pricing by informed investors

Abstract: Well-informed agents can hedge more efficiently than poorly informed agents, even if they have access to the same set of traded securities. In a model-independent framework with semi-static trading opportunities, we study super-hedging prices for agents with different filtrations. A crucial role is played by the notion of semi-static completeness, which is the natural extension in this context of the predictable representation property. Under structural assumptions, we find that informed agents price using only probability measures under which their filtration differs minimally from the one available to uninformed agents.

(Joint work with Beatrice Acciaio)

(Buffet lunch)

François Delarue
Mean-field games with controls submitted to the mean-field interaction

Abstract: The purpose of this talk is to revisit existing results on mean-field games in the more general case when both the states and the controls of the players are submitted to the mean-field interaction. We first discuss existence of equilibria and then switch to uniqueness. We also address the notion of master equation in that framework.
Presentation Slides

Martin Huesmann
A flexible continuous-time framework for model-independent finance

Abstract: In the last two decades model independent finance developed into a major and important field within mathematical finance. In this talk I will present a natural and flexible continuous-time framework based on a combination of ideas of optimal transport and Vovk's outer measure that is well suited to incorporate various constraints in the market and, thus, allows to model rather complicated setups in a model independent way. To illustrate the power of this approach I will present two examples: first a general multi-marginal duality result and then a model independent approach to insider trading.

(Joint work with B Acciaio, M Beiglboeck, ACox, N Perkowski and D Prömel)

Please send any questions to Ian Marshall.

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