Kyle-Back Models with risk aversion and non-Gaussian Beliefs

Abstract

We show that the problem of existence of equilibrium in Kyle's continuous time insider trading model ([31]) can be tackled by considering a system of quasilinear parabolic equation and a Fokker-Planck equation coupled via a transport type constraint. By obtaining a stochastic representation for the solution of such a system, we show the well-posedness of solutions and study the properties of the equilibrium obtained for small enough risk aversion parameter. In our model, the insider has exponential type utility and the belief of the market on the distribution of the price at final time can be non-Gaussian.

Multidimensional Kyle-Back model with a risk averse informed trader

“…Then, the algebraic condition identifying the price pressure was shown to be an optimal transport type condition between the law of ξ T in equilibrium and the belief of the market maker on the fundamental value. This condition yields, in the context of [8], to the existence of a fixed-point between a Fokker-Planck equation and a quasilinear parabolic partial differential equation. This fixed point allows the construction of an equilibrium.…”

“…Introduction. The objective of this paper is to continue the study of Kyle-Back models with a risk averse informed trader using the optimal transport theory based methodology initiated in [5,8]. In this paper, we study markets with multiple stocks and show the existence of equilibrium under the assumption of log-concavity of the distribution of the fundamental value of the assets.…”

“…In this paper, we aim to extend the results in [8] to a multi-dimensional case. A fundamental challenge in treating the case of the risk-averse insider is the fact that the total demand Y t is not a sufficient statistic for the pricing rule of the market maker and the strategy of the informed trader.…”

“…In [8], in order to relax the Gaussianity assumption of the fundamental value, the authors allow the price pressure to be a function of (ξ t ). Thus, (ξ t ) solves a stochastic differential equation driven by Y .…”

“…The multi-asset framework studied in this paper brings forward major challenges. First of all, although the quasilinear equation describing the pricing rule in [8] can be stated in this multi-asset framework, we are not able to construct other functions needed to carry out the existence proof. Thus, we derive novel tools to construct the price pressure function.…”