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Date: 15th October 2021

Nazem Khan (University of Warwick): Sensitivity to large losses and rho-arbitrage for convex risk measures

In this talk, we revisit mean-risk portfolio selection in a one-period financial market, where risk is quantified by a superlinear risk measure rho. We introduce two new axioms: weak and strong sensitivity to large losses. We show that the first axiom is key to ensure the existence of optimal portfolios and the second one is key to ensure the absence of rho-arbitrage. This leads to a new class of risk measures that are suitable for portfolio selection. We show that rho belongs to this class if and only if rho is real-valued and the smallest positively homogeneous risk measure dominating rho is the worst-case risk measure. We then specialise to the case that rho is convex and admits a dual representation. We derive necessary and sufficient dual characterisations of (strong) rho-arbitrage as well as the property that rho is suitable for portfolio selection. Finally, we introduce the new risk measure of Loss Sensitive Expected Shortfall, which is similar to and not more complicated to compute than Expected Shortfall but suitable for portfolio selection - which Expected Shortfall is not.

Date: 4th November 2021

Matteo Burzoni (Università degli Studi di Milano): Mean Field Games with absorption and a model of bank run

We consider a MFG problem obtained as the limit of N-particles systems with an absorbing region. Once a particle hits such a region, it leaves the game and the rest of the system continues to play with N-1 particles. We study existence of equilibria for the limiting problem in a framework with common noise and establish the existence of epsilon Nash equilibria for the N-particles problems. These results are applied to a novel model of bank run. This is a joint work with L. Campi.

Date: 26th November 2021

Johannes Muhle-Karbe (Imperial College London): Hedging with Market and Limit Orders

Trading with limit orders allows to earn rather than pay bid-ask spreads. However, limit orders do not guarantee immediate execution and, moreover, are exposed to adverse selection by counterparties with superior information. We study the resulting tradeoff between market and limit orders for option hedging in a tractable extension of Leland’s model. This serves as a stylised model for the management of “central risk books”, where risk management and trade implementation are centralised. (Joint work in progress with Kevin Webster (Citadel LLC) and Zexin Wang (Imperial).)

Date:3rd December 2021

Roxana Dumitrescu (King's College London): Control and optimal stopping mean-field games: a linear programming approach

We develop the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via the controlled/stopped martingale approach, another relaxation method used in a few previous papers in the case when there is only control (joint work with M. Leutscher and P. Tankov)