A unified approach to informed trading via Monge-Kantorovich duality

TL;DR

This paper introduces a unified framework for informed trading models using Monge-Kantorovich duality, connecting optimal transport theory with financial market dynamics and deriving optimal strategies through stochastic PDEs.

Contribution

It develops a novel approach to model informed trading by integrating optimal transport theory with stochastic PDEs, providing explicit characterizations of optimal strategies.

Findings

Reformulation of the Kyle model as a terminal optimization problem with distributional constraints.

Derivation of market maker pricing rules using Kantorovich potentials.

Complete characterization of optimal informed trader strategies via filtering SPDEs.

Abstract

We solve a generalized Kyle model type problem using Monge-Kantorovich duality and backward stochastic partial differential equations. First, we show that the the generalized Kyle model with dynamic information can be recast into a terminal optimization problem with distributional constraints. Therefore, the theory of optimal transport between spaces of unequal dimension comes as a natural tool. Second, 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. Finally, we completely characterize the optimal strategies by analyzing the filtering problem from the market maker's point of view. In this context, the Kushner-Zakai filtering SPDE yields to an interesting backward stochastic partial differential equation whose measure-valued terminal condition comes from the optimal coupling of measures.