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A cross-border market model with limited transmission capacities

Cassandra Milbradt
Jul 2022
We develop a cross-border market model between two countries in which thetransmission capacities that enable transactions between market participants ofdifferent countries are limited. Starting from two so-called reduced-formrepresentations of national limit order book dynamics, we allow incoming marketorders to be matched with standing volumes of the foreign market, resulting incross-border trades. We introduce a microscopic model that consists of two bidand ask price processes, four queue length processes that describe the numberof unexecuted limit orders at the best bid and ask prices, and a capacityprocess. The latter counts the net number of executed cross-border trades overtime. Since the transmission capacities in our model are limited, our modelalternates between regimes in which cross-border trades are possible andregimes in which incoming market orders can only be matched against limitorders of the same origin. If the size of an individual order converges to zerowhile the order arrival rate tends to infinity, we derive a continuous-timelimit approximation of our microscopic market dynamics. In the limit the volumedynamics is approximated by a four-dimensional, non-negative semimartingalewith a non-trivial martingale part and reflections to the interior of$\mathbb{R}^4_+$ on hits of the (cumulative) queue length process of the axes.The approximation of the capacity dynamics turns out to be a bounded continuousprocess of finite variation. Since the tick size in our model is constant, theapproximation of the bid (resp. ask) price dynamics is a two-dimensional purejump process with jump times equal to those of the volume approximation. Theusefulness of the ability to transact across borders is illustrated through asimulation study of price evolution.