[Paper Review] Over-the-counter market models with several assets

Alain Bélanger, Gaston Giroux+1 more|arXiv (Cornell University)|Aug 13, 2013

Stochastic processes and financial applications6 references3 citations

TL;DR

This paper extends over-the-counter (OTC) market models to multiple assets using systems of nonlinear ordinary differential equations (ODEs), analyzing steady-state equilibria and market stability. It derives explicit equilibrium prices for non-segmented and partially segmented markets with two or more assets, proving asymptotic stability via the Routh-Hurwitz criterion, offering a foundational framework for multi-asset OTC price discovery under search frictions.

ABSTRACT

We study two classes of over-the-counter markets specified by systems of ODE's, in the spirit of Duffie-Garleanu-Pedersen, Econometrica, 2005. We first compute the steady states for many of these ODE's. Then we obtain the prices at which investors trade with each other at these steady states. Finally, we study the stability of the solutions of these ODE's.

Motivation & Objective

• To model multi-asset over-the-counter (OTC) markets with search and bargaining frictions, extending single-asset models by Duffie, Gârleanu, and Pedersen.
• To analyze steady-state equilibria in OTC markets with multiple assets, particularly in non-segmented and partially segmented market structures.
• To derive explicit equilibrium prices for investors trading under these ODE-based market dynamics.
• To establish asymptotic stability of the ODE systems governing investor behavior and price formation.
• To provide a theoretical foundation for understanding illiquidity and price formation in opaque, decentralized financial markets with multiple assets.

Proposed method

• Models OTC markets with K ≥ 1 assets using systems of quadratic ODEs that describe investor transitions between states (e.g., holding, searching, trading).
• Introduces two market structures: non-segmented (no distinction in target asset during search) and partially segmented (searchers track intended asset to buy).
• Derives intrinsic value functions V(t,z) for each investor state z, solving them via integral equations and ODEs with time-dependent coefficients.
• Uses the Routh-Hurwitz criterion to assess asymptotic stability of the ODE systems, proving stability for non-segmented markets and for partially segmented markets with one or two assets.
• Computes equilibrium prices by solving for time-invariant solutions of the ODE systems under steady-state conditions.
• Employs matrix formulations (M matrices) to represent the dynamics of state transitions and solve for equilibrium distributions across investor types.

Experimental results

Research questions

• RQ1 What are the steady-state distributions of ownership and trading behavior in multi-asset OTC markets with search frictions?
• RQ2 How do equilibrium prices emerge in OTC markets with multiple assets under non-segmented and partially segmented search behavior?
• RQ3 What conditions ensure the asymptotic stability of the ODE systems modeling investor dynamics in multi-asset OTC markets?
• RQ4 How do the intrinsic values of assets evolve over time under different market segmentation assumptions?
• RQ5 What is the impact of asymmetric investor eagerness (buying vs. selling) on equilibrium price formation in multi-asset OTC markets?

Key findings

• For non-segmented markets with any number of assets, the paper explicitly computes the steady-state distribution of investor holdings and trading frequencies.
• In partially segmented markets with two assets, the steady-state distribution is derived analytically, showing how search targeting affects equilibrium outcomes.
• Equilibrium prices are derived as solutions to time-invariant ODE systems, with explicit expressions for intrinsic values V(t,z) in both market structures.
• The ODE systems for non-segmented markets are proven asymptotically stable for any number of assets using the Routh-Hurwitz criterion.
• For partially segmented markets, asymptotic stability is established for one and two assets, though the complexity of the criterion increases rapidly with more assets.
• Numerical examples confirm the convergence of the systems to equilibrium and validate the derived pricing formulas under various parameter settings.