**Schedule**

**Register here.**

09:00-09:30 – Registration & Refreshments

09:30-10:30 – Patrick Cheridito, ETH Zurich

**Title:** Deep splitting method for parabolic PDEs

**Abstract:** A numerical method for nonlinear parabolic PDEs is introduced which combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle very high-dimensional PDEs. We illustrate the method on a non-linear Black-Scholes equation and a Hamilton-Jacobi-Bellmann equation.

10:30-11:30 – Fayçal Drissi, OMI and Université Paris 1 Panthéon-Sorbonne

**Title: **Decentralised Finance and Automated Market Making: Predictable Loss and Optimal Liquidity Provision

**Abstract: **We introduce a new comprehensive metric of predictable loss (PL) for liquidity providers in constant function automated market makers and derive an optimal liquidity provision strategy. PL compares the value of the LP’s holdings in the liquidity pool (assuming no fee revenue) with that of a self-financing portfolio that replicates the LP’s holdings and invests in a risk-free account. We provide closed-form formulae for PL, and show that the losses stem from two sources: the convexity cost, which depends on liquidity taking activity and the convexity of the pool’s trading function; the opportunity cost, which is due to locking the LP’s assets in the pool. For LPs in constant product market makers with concentrated liquidity, we derive a closed-form strategy that dynamically adjusts the range around the exchange rate as a function of market trend, volatility, and liquidity taking activity in the pool. We prove that the profitability of liquidity provision depends on the tradeoff between PL and fee income. Finally, we use Uniswap v3 data to show that LPs have traded at a significant loss, and to show that the out-of-sample performance of our strategy is considerably superior to the historical performance of LPs in the pool we consider.

11:30-12:00 – Break – Tea & Coffee

12:00-13:00 – Lukas Szpruch, University of Edinburgh

**Title:**Convergence of Policy Gradient MDPs and applications to Constant Functions Markets in DeFi.

**Abstract: **In the first part of the talk, I will present recent results about the convergence of policy gradient for infinite-horizon, continuous state and action space, and entropy-regularized Markov decision processes (MDPs). We consider a softmax policy with (one-hidden layer) neural network approximation in a mean-field regime. Additional entropic regularization in the associated mean-field probability measure is added, and the corresponding gradient flow is studied in the 2-Wasserstein metric. We show that the objective function is increasing along the gradient flow. Further, we prove that if the regularization in terms of the mean-field measure is sufficient, the gradient flow converges exponentially fast to the unique stationary solution, which is the unique maximizer of the regularized MDP objective.

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In the second part of the talk, I will focus on the problem of liquidity provision in Constant Functions Markets (CFMs). It has been shown empirically and experimentally that liquidity providers (LP) in these markets, on average, lose money. By working in a usual mathematical finance setting, we derive a theoretical, arbitrage-free level of fee income that would make LP positions fair. We then run extensive reinforcement learning simulations in various market conditions that help to shed light on how the fee for liquidity provision should be set up so that these markets are viable.

13:00-14:00 – Lunch

**Title: **Brokers and Informed Traders: dealing with toxic flow and extracting trading signals.

**Abstract: **We derive closed-form strategies for a broker who provides liquidity to an informed trader and to a noise trader over a finite-time and infinite-time trading horizon. The flow of the noise trader is uninformative and the broker trades with the noise trader at a profit, on average. On the other hand, the informed trader has privileged information about the trend in the price of the asset, so the broker trades with the informed trader at a loss, on average. These losses are payment for toxic flow from which the broker extracts the trend signal. The signal is one of the key ingredients in the broker’s trading strategy to internalise (i.e., how much of the flow she keeps in her books), to externalise (i.e., how much she unwinds in a lit exchange), and to speculate in the lit market. The broker’s dynamic strategy is a linear combination of four variables: the broker’s inventory, the informed trader’s inventory, the trend signal, and the uninformed trader’s rate of trading with the broker. When the trading horizon is infinite (resp. finite), the coefficients of the four terms are constants (resp. deterministic functions of time). Finally, in the infinite horizon case, we show how the broker uses the flow of her clients to estimate the constant coefficients of the optimal strategy.

15:00-16:00 – Garud N. Iyengar, Columbia University

**Title: **Distributionally Robust End-to-End Portfolio Construction

**Abstract:** We propose an end-to-end distributionally robust system for portfolio construction that integrates the asset return prediction model with a distributionally robust portfolio optimization model. We also show how to learn the risk-tolerance parameter and the degree of robustness directly from data. End-to-end systems have an advantage in that information can be communicated between the prediction and decision layers during training, allowing the parameters to be trained for the final task rather than solely for predictive performance. However, existing end-to-end systems are not able to quantify and correct for the impact of model risk on the decision layer. Our proposed distributionally robust end-to-end portfolio selection system explicitly accounts for the impact of model risk. The decision layer chooses portfolios by solving a minimax problem where the distribution of the asset returns is assumed to belong to an ambiguity set centered around a nominal distribution. Using convex duality, we recast the minimax problem in a form that allows for efficient training of the end-to-end system.

16:00-16:30 – Break – Tea & Coffee

16:30-17:30 – David Parkes, Harvard University

**Title: **Credible Decentralized Exchange Design via Verifiable Sequencing Rules

**Abstract: **Trading on decentralized exchanges has been one of the primary use cases for permissionless blockchains with daily trading volume exceeding billions of U.S.~dollars. In the status quo, users broadcast transactions and miners are responsible for composing a block of transactions and picking an execution ordering — the order in which transactions execute in the exchange. Due to the lack of a regulatory framework, it is common to observe miners exploiting their privileged position by front-running transactions and obtaining risk-free profits. In this work, we propose verifiable sequencing rules, where miners commit to respecting a sequencing rule that constrains the execution ordering and is verifiable. We give a verifiable sequencing rule that, for any user transaction A, ensures that if the miner gains (or loses) from including A in the block, that the execution price is at least as good as if A was the only transaction in the block.

**17:30-18:30 – Drinks Reception – Oxford-Man Institute**

**18.45-1940 – Drinks in the Emily Wilding Davison Room**

**19:45-21:30 – Dinner – St Hugh’s College, The Boardroom (invitation only)**