We investigate finite population games of optimal execution, taking place at a market with friction. The models over which we develop our results are akin to the standard Almgren-Chriss model with linear price impacts. On the one hand, at a temporary level, our perspective is rather similar to that of the aforementioned model. On the other hand, all players in the model will impact the asset's public price, yielding an aggregate permanent price impact. We propose to analyze two different settings . The first one comprises the case where there is no hierarchy among players, and there is a symmetry of information. In this setting, we obtain closed-form formulas to the Nash equilibrium in the most general setting, i.e., when players' preferences are completely heterogeneous. Particularizing to the case of homogeneous parameters, we show that the average optimal inventory of the finite population converges to its mean-field counterpart, uniformly over a fixed trading horizon , as the population size grows to infinity . In the second framework, we consider a major player, also called a leader, with the first move advantage, and a population of minor players, also known as followers, thought of as high-frequency traders, which trade on informational advantage against the leader. This leads to a model of McKean-Vlasov type for the dynamics of the asset's midprice. We prove the existence and uniqueness of the Stackelberg-Nash equilibrium for a reasonable set of model parameters. We also characterize it as the solution of an abstract vector forward-backward stochastic differential equation system .
We investigate stochastic differential games of optimal trading comprising a finite population. There are market frictions in the present framework, which take the form of stochastic permanent and temporary price impacts. Moreover, information is asymmetric among the traders, with mild assumptions. For constant market parameters, we provide specialized results. Each player selects her parameters based not only on her informational level but also on her particular preferences . The first part of the work is where we examine the unconstrained problem, in which traders do not necessarily have to reach the end of the horizon with vanishing inventory . In the sequel, we proceed to analyze the constrained situation as an asymptotic limit of the previous one. We prove the existence and uniqueness of a Nash equilibrium in both frameworks, alongside a characterization, under suitable weak interaction assumptions. We conclude the paper by presenting an extension of the basic model to a hierarchical market, for which we establish the existence, uniqueness, and characterization of a Stackelberg-Nash equilibrium .
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finite population
stochastic differential
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execution, taking place at a market with friction. The models over which we develop our results are akin to the standard Almgren-Chriss model with linear
trading comprising a finite population. There are market frictions in the present framework, which take the form of stochastic permanent and temporary
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On the one hand, at a temporary level, our perspective is rather similar to that of the aforementioned model. On the other hand, all players in the model will impact the asset's public price, yielding an aggregate permanent price impact. We propose to analyze two different settings
Moreover, information is asymmetric among the traders, with mild assumptions. For constant market parameters, we provide specialized results. Each player selects her parameters based not only on her informational level but also on her particular preferences
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one comprises the case where there is no hierarchy among players, and there is a symmetry of information. In this setting, we obtain closed-form formulas to the Nash equilibrium in the most general setting, i.e., when players' preferences are completely heterogeneous. Particularizing to
part of the work is where we examine the unconstrained problem, in which traders do not necessarily have to reach the end of
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case of homogeneous parameters, we show that the average optimal inventory of the finite population converges to its mean-field counterpart, uniformly over a fixed trading horizon , as the population size grows to infinity
horizon with vanishing inventory
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second framework, we consider a major player, also called a leader, with the first move advantage, and a population of minor players, also known as followers, thought of as high-frequency traders, which trade on informational advantage against the leader. This leads to a model of McKean-Vlasov type for the dynamics of the asset's midprice.
sequel, we proceed to analyze the constrained situation as an asymptotic limit of the previous one.
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the Stackelberg-Nash equilibrium for a reasonable set of model parameters. We also characterize it as the solution of an abstract vector forward-backward stochastic differential equation system
a Nash equilibrium in both frameworks, alongside a characterization, under suitable weak interaction assumptions. We conclude the paper by presenting an extension of the basic model to a hierarchical market, for which we establish the existence, uniqueness, and characterization of a Stackelberg-Nash equilibrium