We study the high-frequency limits of strategies and costs in a Nash equilibrium for two agents that are competing to minimize liquidation costs in a discrete-time market impact model with exponentially decaying price impact and quadratic transaction costs of size \theta\ge0. We show that, for \theta=0, equilibrium strategies and costs will oscillate indefinitely between two accumulation points. For \theta>0, however, both strategiesand costs will converge towards limits that are independent of \theta. We then show that the limiting strategies form a Nash equilibrium for a continuous-time version of the model with \theta equal to a certain critical value \theta^*>0, and that the corresponding expected costs coincide with the high-frequency limits of the discrete-time equilibrium costs. For \theta\neq\theta^*, however, continuous-time Nash equilibria will typically not exist. Our results permit us to give mathematically rigorous proofs of numerical observations made in Schied and Zhang arXiv:1305.4013, 2013  . In particular, we provide a range of model parameters for which the limiting expected costs of both agents are decreasing functions of \theta. That is, for sufficiently high trading speed, raising additional transaction costs can reduce the expected costs of all agents.
We study the high-frequency limits of strategies and costs in a Nash equilibrium for two agents that are competing to minimize liquidation costs in a discrete-time market impact model with exponentially decaying price impact and quadratic transaction costs of size \theta\ge0. We show that, for \theta=0, equilibrium strategies and costs will oscillate indefinitely between two accumulation points. For \theta>0, however, strategies, costs, and total transaction costs will converge towards limits that are independent of \theta. We then show that the limiting strategies form a Nash equilibrium for a continuous-time version of the model with \theta equal to a certain critical value \theta^*>0, and that the corresponding expected costs coincide with the high-frequency limits of the discrete-time equilibrium costs. For \theta\neq\theta^*, however, continuous-time Nash equilibria will typically not exist. Our results permit us to give mathematically rigorous proofs of numerical observations made in Schied and Zhang ( 2013 ) . In particular, we provide a range of model parameters for which the limiting expected costs of both agents are decreasing functions of \theta. That is, for sufficiently high trading speed, raising additional transaction costs can reduce the expected costs of all agents.
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both strategiesand
strategies, costs, and total transaction
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arXiv:1305.4013,
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