Optimal Transport and Information Asymmetry Trading Models

We present two extensions of the Kyle-Back model and solve them with tools from optimal transport. In the first, the informed trader receives her private information over time as opposed to all at once from the start of the trading period. We show that this dynamic information model can be recast as a terminal optimization problem with distributional constraints. Therefore, the theory of optimal transport between spaces of unequal dimension comes as a natural tool. The pricing rule of the market maker and an optimality criterion for the problem of the informed trader are established using the Kantorovich potentials and transport maps. It turns out that the optimal strategy can be completely characterized by the market maker's filtering problem, and in particular the Kushner-Zakai SPDE. In the second extension, the signal is static, but there is stochastic noise volatility along with multiple traded assets. We start with the causal optimal coupling between the fundamental price of the assets and the Wiener process which drives the noise trades. From this, we derive a matrix-valued variational problem which characterizes the optimal rate by which the informed trader injects information into the market. This optimal rate completely characterizes the informed trader’s strategy and the market maker’s pricing rule. By considering dual formulation of this problem, we discover the equilibrium minimizes an average of the initial market depth and the noise traders’ slippage costs.