Portfolio Liquidation Games with Self-Exciting Order Flow

We analyze novel portfolio liquidation games with self-exciting order flow. Both the $N$-player game and the mean-field game are considered. We assume that players' trading activities have an impact on the dynamics of future market order arrivals thereby generating an additional transient price impact. Given the strategies of her competitors each player solves a mean-field control problem. We characterize open-loop Nash equilibria in both games in terms of a novel mean-field FBSDE system with unknown terminal condition. Under a weak interaction condition we prove that the FBSDE systems have unique solutions. Using a novel sufficient maximum principle that does not require convexity of the cost function we finally prove that the solution of the FBSDE systems do indeed provide open-loop Nash equilibria

The cavity method for minority games between arbitrageurs on financial markets

We use the cavity method from statistical physics for analyzing the transient and stationary dynamics of a minority game that is played by agents performing market arbitrage. On the level of linear response the method allows to include the reaction of the market to individual actions of the agents as well as the reaction of the agents to individual information items of the market. This way we derive a self-consistent solution to the minority game. In particular we analyze the impact of general nonlinear price functions on the amount of arbitrage if noise from external fluctuations is present. We identify the conditions under which arbitrage gets reduced due to the presence of noise. When the cavity method is extended to time dependent response of the market price to previous actions of the agents, the individual contributions of noise can be pursued over different time scales in the transient dynamics until a stationary state is reached and when the stationary state is reached. The contributions are from external fluctuations in price and information and from noise due to the choice of strategies. The dynamics explains the time evolution of scores of the agents' strategies: it changes from initially a random walk to non-Markovian dynamics and bounded excursions on an intermediate time scale to effectively random switching in the choice between strategies on long time scales. In contrast to the Curie-Weiss level of a mean-field approach, the market response included by the cavity method captures the realistic feature that the agents can have a preference for a certain choice of strategies without getting stuck to a single choice. The breakdown of the method in the phase transition region indicates possible market mechanisms leading to critical volatility and a possible regime shift.Comment: 36 pages, 7 figure

Models of Financial Markets with Extensive Participation Incentives

We consider models of financial markets in which all parties involved find incentives to participate. Strategies are evaluated directly by their virtual wealths. By tuning the price sensitivity and market impact, a phase diagram with several attractor behaviors resembling those of real markets emerge, reflecting the roles played by the arbitrageurs and trendsetters, and including a phase with irregular price trends and positive sums. The positive-sumness of the players' wealths provides participation incentives for them. Evolution and the bid-ask spread provide mechanisms for the gain in wealth of both the players and market-makers. New players survive in the market if the evolutionary rate is sufficiently slow. We test the applicability of the model on real Hang Seng Index data over 20 years. Comparisons with other models show that our model has a superior average performance when applied to real financial data.Comment: 17 pages, 16 figure

Dynamics of adaptive agents with asymmetric information

We apply path-integral techniques to study the dynamics of agent-based models with asymmetric information structures. In particular, we devise a batch version of a model proposed originally by Berg et al. [Quant. Fin. 1 (2001) 203], and convert the coupled multi-agent processes into an effective-agent problem from which the dynamical order parameters in ergodic regimes can be derived self-consistently together with the corresponding phase structure. Our dynamical study complements and extends the available static theory. Results are confirmed by numerical simulations.Comment: minor revision of text, accepted by JSTA