Existence and uniqueness of a steady state for an OTC market with several assets

Abstract

We introduce and study a class of over-the-counter market models specified by systems of Ordinary Differential Equations (ODE's), in the spirit of Duffie-Gârleanu-Pedersen Duffie et al. (Econometrica 73(1):1815-1847, 2005). The key innovation is allowing for multiple assets. We show the existence and uniqueness of a steady state for these ODE's.

Dark Markets with Multiple Assets: Segmentation, Asymptotic Stability, and Equilibrium Prices

“…It was shown in [2] that every partially segmented OTC market model with several assets has a steady state which is unique. For the proof of asymptotic stability, we denote by x * the unique equilibrium state, that is F (x * ) = 0.…”

“…Following [2], the above dynamical system can be reduced to the following equivalent system of 2K equations:…”

“…The steady state equilibrium proportions of investors for a segmented OTC model with multiple assets is determined in [2] and its asymptotic stability is established above. In this section, we will use the equilibrium masses above, in order to compute the equilibrium bargaining prices P 1 , P 2 ,...,P K for the respective assets 1,2,...,K. The authors in [1] propose a method to show how to obtain the traded prices of the assets at the unique steady state (equilibrium) for non-segmented market models.…”

“…Also each investor either owns one unit of an asset i or does not own it. A transaction occurs when a high type investor who does not own a specific asset meets a seller of this asset, that is, a low liquidity type investor owning the asset (see [2]). We assume that the assets are differentiated by their liquidity and intrinsic value, so that we can compare the preferences of investors in relation to each asset and liquidity.…”

Some new results on Dufffie-type OTC markets

“…In many applications, it is more convenient to work with more than one kernel to describe the dynamics of the system. It is the case for instance in the models of Duffie-Gârleanu-Pedersen [6] and their extensions in Bélanger-Giroux-Moisan [2] and in Bélanger-Giroux-Ndouné [3].…”