We analyze novel portfolio liquidation games with self-exciting order flow. Both the $N$-player game and the mean-field game are considered. We assume that players' trading activities have an impact on the dynamics of future market order arrivals thereby generating an additional transient price impact. Given the strategies of her competitors each player solves a mean-field control problem. We characterize open-loop Nash equilibria in both games in terms of a novel mean-field FBSDE system with unknown terminal condition. Under a weak interaction condition we prove that the FBSDE systems have unique solutions. Using a novel sufficient maximum principle that does not require convexity of the cost function we finally prove that the solution of the FBSDE systems do indeed provide existence and uniqueness of open-loop Nash equilibria. The talk is based on joint work with Guanxing Fu and Ulrich Horst.

## Portfolio Liquidation Games with Self-Exciting Order Flow

## Set Values of Mean Field Games

When a mean field game satisfies certain monotonicity conditions, the mean field equilibrium is unique and the corresponding value function satisfies the so called master equation. In general, however, there can be multiple equilibriums, and in the literature one typically studies the asymptotic behaviors of individual equilibriums of the corresponding $N$-player game. We instead study the set of values over all (mean field) equilibriums, which we call the set value of the game. We shall establish two crucial properties of the set value: (i) the dynamic programming principle; (ii) the convergence of the set values from the $N$-player game to the mean field game. We emphasize that the set value is very sensitive to the choice of the admissible controls. For the dynamic programming principle, one needs to use closed loop controls (not open loop controls) and it involves some very subtle path dependence issue. For the convergence, one has to restrict to the same type of equilibriums for the $N$-player game and for the mean field game. The talk is based on a joint work with Zach Feinstein and Birgit Rudloff and another ongoing joint work with Melih Iseri.

## On Set-valued Backward SDEs and Related Issues in Set-valued Stochastic Analysis

In this talk we try to establish an analytic framework for studying Set-Valued Backward Stochastic Differential Equations (SVBSDE for short), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will be based on the notion of Hukuhara difference between sets, in order to compensate the lack of “inverse” operation of the traditional Minkowski addition, whence the vector space structure, in traditional set-valued analysis. We shall examine and establish a useful foundation of set-valued stochastic analysis under this algebraic framework, and identify the challenges that may arise in the study of SVBSDEs.

## A simple microstructural explanation of the concavity of price impact

I will present a simple model of market microstructure which explains the concavity of price impact. In the proposed model, the local relationship between the order flow and the fundamental price (i.e. the local price impact) is linear, with a constant slope, which makes the model dynamically consistent. Nevertheless, the expected impact on midprice from a large sequence of co-directional trades is nonlinear and asymptotically concave. The main practical conclusion of the model is that, throughout a meta-order, the volumes at the best bid and ask prices change (on average) in favor of the executor. This conclusion, in turn, relies on two more concrete predictions of the model, one of which can be tested using publicly available market data and does not require the (difficult to obtain) information about meta-orders. I will present the theoretical results and will support them with the empirical analysis.

## Principal Trading Arrangements: When Are Common Contracts Optimal?

Many financial arrangements reference market prices that are yet to be realized at the time of contracting and consequently susceptible to manipulation. Two of the most common such arrangements are: (i) market-on-close contracts, which reference the price prevailing at the end of an execution window, and (ii) guaranteed VWAP contracts, which reference the volume-weighted average price (VWAP) prevailing over the execution window. To study such situations, we introduce a stylized model of financial contracting between a client, who wishes to trade a large position, and her dealer. Market-on-close contracts are generally not optimal in this principal-agent problem. In contrast, we provide conditions under which guaranteed VWAP contracts are optimal. These results question the usage of market-on-close contracts in practice, explain the usage of guaranteed VWAP contracts, and also suggest considerations for the design of financial benchmarks. The presentation is based on joint work with Markus Baldauf (University of British Columbia) and Joshua Mollner (Northwestern University).

## Excursions in Math Finance

The risk and return profiles of a broad class of dynamic trading strategies, including pairs trading and other statistical arbitrage strategies, may be characterized in terms of excursions of the market price of a portfolio away from a reference level. We propose a mathematical framework for the risk analysis of such strategies, based on a description in terms of price excursions, first in a pathwise setting, without probabilistic assumptions, then in a Markovian setting.

We introduce the notion of $\delta$-excursion, defined as a path which deviates by $\delta$ from a reference level before returning to this level. We show that every continuous path has a unique decomposition into such $\delta$-excursions, which turns out to be useful for the scenario analysis of dynamic trading strategies, leading to simple expressions for the number of trades, realized profit, maximum loss, and drawdown. As $\delta$ is decreased to zero, properties of this decomposition relate to the local time of the path.

When the underlying asset follows a Markov process, we combine these results with Ito's excursion theory to obtain a tractable decomposition of the process as a concatenation of independent $\delta$-excursions, whose distribution is described in terms of Ito's excursion measure. We provide analytical results for linear diffusions and give new examples of stochastic processes for flexible and tractable modeling of excursions. Finally, we describe a non-parametric scenario simulation method for generating paths whose excursions match those observed in a data set.

This is based on joint work with Anna Ananova and Rama Cont (Oxford).

## 6th Berlin Workshop on Mathematical Finance for Young Researchers - CANCELED -

The workshop will take place August 25-28 in Berlin. More information will be posted here soon. The lead organiser is Dirk Becherer.