Double free boundary problem for defaultable corporate bond with credit rating migration risks and their asymptotic behaviors

Paper: arXiv 2301.10898

Authors: Yuchao Dong, Jin Liang, Claude-Michel Brauner

Abstract

In this work, a pricing model for a defaultable corporate bond with credit rating migration risk is established. The model turns out to be a free boundary problem with two free boundaries. The latter are the level sets of the solution but of different kinds. One is from the discontinuous second order term, the other from the obstacle. Existence, uniqueness, and regularity of the solution are obtained. We also prove that two free boundaries are $C^\infty$. The asymptotic behavior of the solution is also considered: we show that it converges to a traveling wave solution when time goes to infinity. Moreover, numerical results are presented.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using PDE theory to tackle a double free boundary problem arising from credit rating migrations and default risk. While the mathematical modeling, proofs of regularity, and asymptotic traveling wave analyses are sophisticated, the empirical aspect is minimal and relies primarily on basic numerical illustrations rather than real-world market data.

Research Flowchart

  flowchart TD
    A[Research Goal: Price Corporate Bond with<br>Credit Rating Migration Risk] --> B[Inputs: Discontinuous Variance,<br>Obstacle/Default Barrier, Bond Payoff]
    B --> C[Model Formulation: Partial Differential Equation<br>with Two Free Boundaries]
    C --> D[Mathematical & Asymptotic Analysis:<br>PDE Theory & Traveling Wave Analysis]
    C --> E[Numerical Simulations]
    D --> F[Key Outcomes:<br>1. Well-posedness: Existence, Uniqueness, Regularity<br>2. Smoothness: Both Free Boundaries are C^∞<br>3. Asymptotics: Convergence to Traveling Wave]
    E --> F