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The two-body stochastic dealer model is revisited to provide an exactsolution to the average order-book profile using the kinetic approach. Thedealer model is a microscopic financial model where individual traders makedecisions on limit-order prices stochastically and then reach agreements ontransactions. In the literature, this model was solved for several cases: anexact solution for two-body traders $N=2$ and a mean-field solution for manytraders $N\gg 1$. Remarkably, while kinetic theory plays a significant role inthe mean-field analysis for $N\gg 1$, its role is still elusive for the case of$N=2$. In this paper, we revisit the two-body dealer model $N=2$ to clarify theutility of the kinetic theory. We first derive the exact master-Liouvilleequations for the two-body dealer model by several methods. We next illustratethe physical picture of the master-Liouville equation from the viewpoint of theprobability currents. The master-Liouville equations are then solved exactly toderive the order-book profile and the average transaction interval.Furthermore, we introduce a generalised two-body dealer model by incorporatinginteraction between traders via the market midprice and exactly solve the modelwithin the kinetic framework. We finally confirm our exact solution bynumerical simulations. This work provides a systematic mathematical basis forthe econophysics model by developing better mathematical intuition.

Exact solution to two-body financial dealer model: revisited from the viewpoint of kinetic theory