Nash Equilibrium

Broker-Trader Partial Information Nash-Equilibria ArXiv ID: 2412.17712 “View on arXiv” Authors: Unknown Abstract We study partial information Nash equilibrium between a broker and an informed trader. In this setting, the informed trader, who possesses knowledge of a trading signal, trades multiple assets with the broker in a dealer market. Simultaneously, the broker offloads these assets in a lit exchange where their actions impact the asset prices. The broker, however, only observes aggregate prices and cannot distinguish between underlying trends and volatility. Both the broker and the informed trader aim to maximize their penalized expected wealth. Using convex analysis, we characterize the Nash equilibrium and demonstrate its existence and uniqueness. Furthermore, we establish that this equilibrium corresponds to the solution of a nonstandard system of forward-backward stochastic differential equations (FBSDEs) that involves the two differing filtrations. For short enough time horizons, we prove that a unique solution of this system exists. Finally, under quite general assumptions, we show that the solution to the FBSDE system admits a polynomial approximation in the strength of the transient impact to arbitrary order, and prove that the error is controlled. ...

Nash Equilibrium between Brokers and Traders ArXiv ID: 2407.10561 “View on arXiv” Authors: Unknown Abstract We study the perfect information Nash equilibrium between a broker and her clients – an informed trader and an uniformed trader. In our model, the broker trades in the lit exchange where trades have instantaneous and transient price impact with exponential resilience, while both clients trade with the broker. The informed trader and the broker maximise expected wealth subject to inventory penalties, while the uninformed trader is not strategic and sends the broker random buy and sell orders. We characterise the Nash equilibrium of the trading strategies with the solution to a coupled system of forward-backward stochastic differential equations (FBSDEs). We solve this system explicitly and study the effect of information, profitability, and inventory control in the trading strategies of the broker and the informed trader. ...

Continuous-time Equilibrium Returns in Markets with Price Impact and Transaction Costs ArXiv ID: 2405.14418 “View on arXiv” Authors: Unknown Abstract We consider an Ito-financial market at which the risky assets’ returns are derived endogenously through a market-clearing condition amongst heterogeneous risk-averse investors with quadratic preferences and random endowments. Investors act strategically by taking into account the impact that their orders have on the assets’ drift. A frictionless market and an one with quadratic transaction costs are analysed and compared. In the former, we derive the unique Nash equilibrium at which investors’ demand processes reveal different hedging needs than their true ones, resulting in a deviation of the Nash equilibrium from its competitive counterpart. Under price impact and transaction costs, we characterize the Nash equilibrium as the (unique) solution of a system of FBSDEs and derive its closed-form expression. We furthermore show that under common risk aversion and absence of noise traders, transaction costs do not change the equilibrium returns. On the contrary, when noise traders are present, the effect of transaction costs on equilibrium returns is amplified due to price impact. ...

A Mean Field Game between Informed Traders and a Broker ArXiv ID: 2401.05257 “View on arXiv” Authors: Unknown Abstract We find closed-form solutions to the stochastic game between a broker and a mean-field of informed traders. In the finite player game, the informed traders observe a common signal and a private signal. The broker, on the other hand, observes the trading speed of each of his clients and provides liquidity to the informed traders. Each player in the game optimises wealth adjusted by inventory penalties. In the mean field version of the game, using a Gâteaux derivative approach, we characterise the solution to the game with a system of forward-backward stochastic differential equations that we solve explicitly. We find that the optimal trading strategy of the broker is linear on his own inventory, on the average inventory among informed traders, and on the common signal or the average trading speed of the informed traders. The Nash equilibrium we find helps informed traders decide how to use private information, and helps brokers decide how much of the order flow they should externalise or internalise when facing a large number of clients. ...

Non-linear non-zero-sum Dynkin games with Bermudan strategies ArXiv ID: 2311.01086 “View on arXiv” Authors: Unknown Abstract In this paper, we study a non-zero-sum game with two players, where each of the players plays what we call Bermudan strategies and optimizes a general non-linear assessment functional of the pay-off. By using a recursive construction, we show that the game has a Nash equilibrium point. Keywords: Non-Zero-Sum Game, Bermudan Strategies, Nash Equilibrium, Recursive Construction, Non-Linear Assessment Functional, Derivatives/Contingent Claims ...

Stochastic Delay Differential Games: Financial Modeling and Machine Learning Algorithms ArXiv ID: 2307.06450 “View on arXiv” Authors: Unknown Abstract In this paper, we propose a numerical methodology for finding the closed-loop Nash equilibrium of stochastic delay differential games through deep learning. These games are prevalent in finance and economics where multi-agent interaction and delayed effects are often desired features in a model, but are introduced at the expense of increased dimensionality of the problem. This increased dimensionality is especially significant as that arising from the number of players is coupled with the potential infinite dimensionality caused by the delay. Our approach involves parameterizing the controls of each player using distinct recurrent neural networks. These recurrent neural network-based controls are then trained using a modified version of Brown’s fictitious play, incorporating deep learning techniques. To evaluate the effectiveness of our methodology, we test it on finance-related problems with known solutions. Furthermore, we also develop new problems and derive their analytical Nash equilibrium solutions, which serve as additional benchmarks for assessing the performance of our proposed deep learning approach. ...

A Game of Competition for Risk ArXiv ID: 2305.18941 “View on arXiv” Authors: Unknown Abstract In this study, we present models where participants strategically select their risk levels and earn corresponding rewards, mirroring real-world competition across various sectors. Our analysis starts with a normal form game involving two players in a continuous action space, confirming the existence and uniqueness of a Nash equilibrium and providing an analytical solution. We then extend this analysis to multi-player scenarios, introducing a new numerical algorithm for its calculation. A key novelty of our work lies in using regret minimization algorithms to solve continuous games through discretization. This groundbreaking approach enables us to incorporate additional real-world factors like market frictions and risk correlations among firms. We also experimentally validate that the Nash equilibrium in our model also serves as a correlated equilibrium. Our findings illuminate how market frictions and risk correlations affect strategic risk-taking. We also explore how policy measures can impact risk-taking and its associated rewards, with our model providing broader applicability than the Diamond-Dybvig framework. We make our methodology and open-source code available at https://github.com/louisabraham/cfrgame Finally, we contribute methodologically by advocating the use of algorithms in economics, shifting focus from finite games to games with continuous action sets. Our study provides a solid framework for analyzing strategic interactions in continuous action games, emphasizing the importance of market frictions, risk correlations, and policy measures in strategic risk-taking dynamics. ...