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Probability and Computational Finance Seminar
Yan Xu
Carnegie Mellon University
Title: A martingale optimal transport problem motivated by Kyle's equilibrium.

Abstract: Given a random variable $X$ with value in $\mathbb{R}^2$, we consider the problem of maximizing $\mathbb{E}(c(X,Y))$ over all random variable $Y$ such that $(Y,X)$ forms a 1-step martingale, where $c(x,y)=-(x_1-y_1)(x_2-y_2)$ is the negative covariance. This problem can be considered as an optimal transport problem of Monge type. Its Kantorovich relaxation is to maximize $\int_{\mathbb{R}^4} c(x,y) d\gamma$ over all probability measure $\gamma$, whose $x$-marginal is fixed by $\mu=\mathcal{L}(X)$, and whose $y$-marginal is less than or equal to $\mu$ in convex order. We show that a measure $\gamma$ is optimal if and only if there is a maximal monotone set $G$ such that

(1) $G$ supports the $y$-marginal of $\gamma$,

(2) $c(x,y)=\sup_{z\in G} c(x,z)$ for every $(x,y)$ in the support of $\gamma$.

The motivation of our study comes from the classic Kyle model(1985) of insider trading. The geometric description of our optimal transport plan allows us to explicitly construct equilibrium to the Kyle model. In addition, we provide simple example where Kyle equilibrium does not exist, in which case the value function of the Monge-type problem is strictly less than that of the Kantorovich-type problem. To the best of our knowledge, this is the first study which links martingale optimal transportation and Kyle model.

Date: Monday, October 30, 2017
Time: 4:30 pm
Location: Wean Hall 8220
Submitted by:  Dmitry Kramkov