Markovian Nash equilibrium in financial markets with asymmetric information and related forward-backward systems

TL;DR

This paper introduces a novel methodology for analyzing continuous-time Nash equilibria in financial markets with asymmetric information, accounting for risk-averse market makers and solving complex forward-backward stochastic differential equations.

Contribution

It develops a new approach that removes the risk neutrality restriction, proving the existence of Markovian solutions to forward-backward systems in such markets.

Findings

Existence of Markovian solutions to forward-backward systems

Equilibrium models explain stylized facts not captured by previous models

Market prices follow a quadratic backward stochastic differential equation

Abstract

This paper develops a new methodology for studying continuous-time Nash equilibrium in a financial market with asymmetrically informed agents. This approach allows us to lift the restriction of risk neutrality imposed on market makers by the current literature. It turns out that, when the market makers are risk averse, the optimal strategies of the agents are solutions of a forward-backward system of partial and stochastic differential equations. In particular, the price set by the market makers solves a nonstandard "quadratic" backward stochastic differential equation. The main result of the paper is the existence of a Markovian solution to this forward-backward system on an arbitrary time interval, which is obtained via a fixed-point argument on the space of absolutely continuous distribution functions. Moreover, the equilibrium obtained in this paper is able to explain several stylized facts which are not captured by the current asymmetric information models.