## Looking at the smile from Roger Lee's shoulders

## Submodular mean field games: Existence and approximation of solutions

We study mean field games with scalar Itô-type dynamics and costs that are submodular with respect to a suitable order relation on the state and measure space. The submodularity assumption has a number of interesting consequences. Firstly, it allows us to prove existence of solutions via an application of Tarski's fixed point theorem, covering cases with discontinuous dependence on the measure variable. Secondly, it ensures that the set of solutions enjoys a lattice structure: in particular, there exist a minimal and a maximal solution. Thirdly, it guarantees that those two solutions can be obtained through a simple learning procedure based on the iterations of the best-response-map. The mean field game is first defined over ordinary stochastic controls, then extended to relaxed controls. Our approach also allows to treat a class of submodular mean field games with common noise in which the representative player at equilibrium interacts with the (conditional) mean of its state's distribution.

This talks is based on a joint work together with Giorgio Ferrari, Markus Fischer and Max Nendel

## Hedging with market and limit orders

## Asset-pricing puzzles and price-impact

We solve in closed-form a continuous-time Nash equilibrium model in which a finite number of investors with exponential utilities continuously consume and trade strategically with price-impact. Compared to the analogous Pareto-efficient equilibrium model, price-impact has an amplification effect on risk-sharing distortions that helps resolve the interest rate puzzle. However, price impact has little quantitative effect on the equity premium and stock-return volatility puzzles. Joint work with Xiao Chen (Rutgers), Jin Hyuk Choi (UNIST), and Duane J. Seppi (CMU).

## Portfolio Optimisation within a Wasserstein Ball

We consider the problem of active portfolio management where a loss-averse and/or gain-seeking investor aims to outperform a benchmark strategy's risk profile while not deviating too much from it. Specifically, an investor considers alternative strategies that have a specified copula with the benchmark and whose terminal wealth lies within a Wasserstein ball surrounding it. The investor then chooses the alternative strategy that minimises a distortion risk measure. We prove that an optimal dynamic strategy exists and is unique, and provide its characterisation through the notion of isotonic projections. Finally, we illustrate how investors with different risk preferences invest and improve upon the benchmark using the Tail Value-at-Risk, inverse S-shaped distortion risk measures, and lower- and upper-tail risk measures as examples. We find that investors' optimal terminal wealth distribution has larger probability masses in regions that reduce their risk measure relative to the benchmark while preserving some aspects of the benchmark. This is joint work with Silvana Pesenti, U. Toronto.

## Cryptocurrencies, Mining & Mean Field Games

We present 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 mining reward, 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 from less efficient miners. Hence, cost efficiency can also result in the type of centralization seen among miners of cryptocurrencies.

## Optimal trade execution in an order book model with stochastic liquidity parameters

We analyze an optimal trade execution problem in a financial market with stochastic liquidity. To this end we set up a limit order book model in which both order book depth and resilience evolve randomly in time. Trading is allowed in both directions. In discrete time, we discuss an explicit recursion that, under certain structural assumptions, characterizes minimal execution costs and observe some qualitative differences with related models. In continuous time, due to the stochastic dynamics of the order book depth and resilience, optimal execution strategies are typically of infinite variation, and the first thing to be discussed it how to extend the state dynamics and the cost functional to allow for general semimartingale strategies. We then derive a quadratic BSDE that under appropriate assumptions characterizes minimal execution costs, identify conditions under which an optimal execution strategy exists and, finally, illustrate our findings in several examples. This is a joint work with Julia Ackermann and Thomas Kruse.

## A cross-border market model

On the XBID-market 13 European countries can trade electricity between each other. Like other intraday electricity markets, this is handled using a limit order book. However, cross-border trading is limited via the total amount of available transmission capacities during a trading session. We present a cross-border market model between two countries and want to give insight into the interactions on this market. We introduce a so-called reduced-form representation of the market and a capacity process which may restrict cross-border trades in each direction. Assuming that the capacity process is non-restricted, we are able to derive heavy traffic approximations of the standing volumes and the capacity process. We will further motivate a candidate for the heavy traffic approximation of the restricted market model.