Mean Field Game ε-Nash equilibria for partially observed optimal execution problems in finance

Firoozi, Dena; Caines, Peter E.

doi:10.1109/cdc.2016.7798281

Cited by 24 publications

(17 citation statements)

References 9 publications

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“…Generally, the performance of the agent is either strictly increasing with respect to initial inventory, or it is decreasing initially, but increases past a certain inventory level (around 10%). The trading behaviour is largely consistent with Firoozi and Caines (2016b). …”

Section: Impact Of Execution Quantity On Performancesupporting

confidence: 69%

“…A more realistic setting would involve traders whom have only partial information available to them, and attempt to optimize their performance criteria subject to this information. Some research in this direction for mean-field games include Şen and Caines (2014), Caines and Kizilkale (2016), Firoozi and Caines (2016b), Saldi et al (2017) and Buckdahn et al (2017). The framework of Şen and Caines (2014) has been applied to the problem in Chapter 2 in a limited manner by Firoozi and Caines (2016a), by assuming that the observed inventory levels of the major and minor agents are noisy.…”

Section: Optimal Executionmentioning

confidence: 99%

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Mean-Field Games and Ambiguity Aversion

Huang

2017

SSRN Journal

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This thesis focuses on incorporating the idea of ambiguity aversion into mean-field games.Intuitively, mean-field games describes the dynamics of a system with an infinite population. It is useful in approximating systems with a large population, for which an exact result is computationally intractable. By adding in ambiguity aversion, the mean-field game reflects how players in the population should act if they wish to protect themselves from model misspecification.Two applications of mean-field games are considered through two distinct approaches.The broker execution problem is investigated in a multi-agent framework containing (i) a major agent who is liquidating a large number of shares, (ii) a number of minor agents (high-frequency traders (HFTs)) who search for statistical arbitrage strategies, and (iii) noise traders who buy and sell for exogenous reasons. All optimizing agents (the broker and HFTs) trade against noise traders as well as one another. We use a mean-field game approach to solve the problem and obtain a set of decentralized feedback trading strategies for the major and minor agents. Furthermore, the mean-field game strategies have an N -Nash equilibrium property where N → 0 as N → ∞.The second application focuses on interbank borrowing and lending, which may induce systemic risk into financial markets. A simple model of this is to assume that log-monetary reserves are coupled, and that banks can also borrow/lend from/to a central bank. When all banks optimize their cost of borrowing and lending, this leads to a stochastic game. We account for model uncertainty by recasting the problem as a robust stochastic game and succeed in providing a strategy which leads to a Nash equilibria for ii both the finite game and the mean-field game limit. We prove that an -Nash equilibrium exists and show that when firms are ambiguity-averse, default probabilities can be reduced relative to their ambiguity-neutral counterparts.Finally, this thesis develops a modified stochastic maximum principle for min-max problems, and derives new existence and uniqueness results for a mean-field game with ambiguity averse players. An -Nash equilibrium is shown to exist for this class of mean-field games, and the mean-field game equations are explicitly derived in the linearquadratic framework.iii

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Section: Impact Of Execution Quantity On Performancesupporting

confidence: 69%

Section: Optimal Executionmentioning

confidence: 99%

Mean-Field Games and Ambiguity Aversion

Huang

2017

SSRN Journal

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“…Among the many extensions and generalizations that explore the broad theory of MFGs as well as their applications, we highlight the following works: Huang () and Nourian and Caines (), who investigate MFGs with combination of major and minor agents; Carmona and Delarue (), who develop a probabilistic analysis of MFGs, as well as the works of Cirant (); Bensoussan, Huang, and Laurière (), who introduce MFGs with heterogeneous populations of agents. This theory has seen applications in various financial contexts, such as Guéant, Lasry, and Lions (), who explore various applications of MFGs in economics; Carmona, Fouque, and Sun () and Huang and Jaimungal (), who study systemic risk; Huang, Jaimungal, and Nourian (), who study algorithmic trading in the presence of a major agent and a population of minor agents; Cardaliaguet and Lehalle (), who investigate optimal execution; and Firoozi and Caines (, ), who look at MFGs with partial information on states and apply it to algorithmic trading.…”

Section: Introductionmentioning

confidence: 99%

Mean‐field games with differing beliefs for algorithmic trading

Casgrain

Jaimungal

2020

Mathematical Finance

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Even when confronted with the same data, agents often disagree on a model of the real-world. Here, we address the question of how interacting heterogenous agents, who disagree on what model the real-world follows, optimize their trading actions. The market has latent factors that drive prices, and agents account for the permanent impact they have on prices. This leads to a large stochastic game, where each agents' performance criteria is computed under a different probability measure. We analyse the mean-field game (MFG) limit of the stochastic game and show that the Nash equilibria is given by the solution to a non-standard vector-valued forward-backward stochastic differential equation. Under some mild assumptions, we construct the solution in terms of expectations of the filtered states. We prove the MFG strategy forms an -Nash equilibrium for the finite player game.Lastly, we present a least-squares Monte Carlo based algorithm for computing the optimal control and illustrate the results through simulation in market where agents disagree on the model. 1. Introduction. Financial markets are immensely complicated dynamic systems which incorporate the interactions of millions of individuals on a daily basis. Market participants vary immensely, both in terms of their trading objectives and in their beliefs on the assets they are trading. All of these participants compete with one another in an attempt to achieve their own personal objectives in the most efficient way possible. Traded assets may also be driven by latent factors, and agents must dynamically incorporate data into their trading decisions.In this paper, we propose a game theoretic model in which a large population of heterogeneous agents all trade the same asset. This model considers heterogeneity not only from the point of view of an individual's trading objectives and risk appetite, but also from the point of view of each agent's beliefs regarding the performance of the asset they are trading. We pay particular attention to the information each agent is privy to, in an attempt to render the framework as realistic as possible, while maintaining some semblance of tractability.We study the equilibrium of these markets by using the theory of mean-field games (MFGs), which serves to describe Game-Theoretic models as the number of participating agents becomes extremely large. The general theory of mean-field games already has a large * SJ would like to acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference numbers RGPIN-2018-05705 and RGPAS-2018-522715]

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“…An optimal execution problem in algorithmic trading with the linear models in [21] was formulated as for the nonlinear major-minor (MM) MFG model in [22]. The completely observed and partially observed major minor linear quadratic MFG theory was first applied to an optimal execution problem with linear models (see e.g., [21]) in [23], and subsequently in [24][25][26]. Optimal stopping and switching time problems are addressed for competitive market participants in [27].…”

Section: Introductionmentioning

confidence: 99%

Mean Field Game Systems with Common and Latent Processes

Firoozi

Jaimungal

Caines

2018

2018 IEEE Conference on Decision and Control (CDC)

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In many stochastic games stemming from financial models, the environment evolves with latent factors and there may be common noise across agents' states.Two classic examples are: (i) multi-agent trading on electronic exchanges, and (ii) systemic risk induced through inter-bank lending/borrowing. Moreover, agents' actions often affect the environment, and some agent's may be small while others large. Hence sub-population of agents may act as minor agents, while another class may act as major agents. To capture the essence of such problems, here, we introduce a general class of non-cooperative heterogeneous stochastic games with one major agent and a large population of minor agents where agents interact with an observed common process impacted by the mean field. A latent Markov chain and a latent Wiener process (common noise) modulate the common process, and agents cannot observe them. We use filtering techniques coupled with a convex analysis approach to (i) solve the mean field game limit of the problem, (ii) demonstrate that the best response strategies generate an ǫ-Nash equilibrium for finite populations, and (iii) obtain explicit characterisations of the best response strategies.

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Mean Field Game ε-Nash equilibria for partially observed optimal execution problems in finance

Cited by 24 publications

References 9 publications

Mean-Field Games and Ambiguity Aversion

Mean-Field Games and Ambiguity Aversion

Mean‐field games with differing beliefs for algorithmic trading

Mean Field Game Systems with Common and Latent Processes

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