A Unifying Approach for the Pricing of Debt Securities

ArXiv ID: 2403.06303 “View on arXiv”

Authors: Unknown

Abstract

We propose a unifying framework for the pricing of debt securities under general time-inhomogeneous short-rate diffusion processes. The pricing of bonds, bond options, callable/putable bonds, and convertible bonds (CBs) is covered. Using continuous-time Markov chain (CTMC) approximations, we obtain closed-form matrix expressions to approximate the price of bonds and bond options under general one-dimensional short-rate processes. A simple and efficient algorithm is also developed to price callable/putable debt. The availability of a closed-form expression for the price of zero-coupon bonds allows for the perfect fit of the approximated model to the current market term structure of interest rates, regardless of the complexity of the underlying diffusion process selected. We further consider the pricing of CBs under general bi-dimensional time-inhomogeneous diffusion processes to model equity and short-rate dynamics. Credit risk is also incorporated into the model using the approach of Tsiveriotis and Fernandes (1998). Based on a two-layer CTMC method, an efficient algorithm is developed to approximate the price of convertible bonds. When conversion is only allowed at maturity, a closed-form matrix expression is obtained. Numerical experiments show the accuracy and efficiency of the method across a wide range of model parameters and short-rate models.

Keywords: Convertible Bonds, Short-rate Models, Continuous-Time Markov Chain (CTMC), Credit Risk, Derivatives Pricing, Fixed Income

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper presents advanced mathematical concepts including continuous-time Markov chain (CTMC) approximations, time-inhomogeneous diffusions, and closed-form matrix expressions, indicating high mathematical density. While it includes numerical experiments for accuracy and efficiency, the methodology is primarily theoretical with no indication of backtesting, real-world data implementation, or algorithmic trading readiness.

  flowchart TD
    A["Research Goal"] -->|Propose Unifying Framework| B(General Time-Inhomogeneous<br>Diffusion Processes)
    B --> C{"CTMC Approximation"}
    C -->|One-Dimensional| D["Closed-Form Matrix Expressions<br>Bonds & Bond Options"]
    C -->|Two-Dimensional| E["Efficient Algorithm<br>Convertible Bonds"]
    D --> F["Perfect Fit to Market<br>Term Structure"]
    E -->|Tsiveriotis-Fernandes| G["Integrated Credit Risk"]
    F --> H["Key Outcomes"]
    G --> H
    H -->|Numerical Experiments| I["High Accuracy &<br>Efficiency Verified"]