Numerical analysis for a unified 2 factor model of structural and reduced form types for corporate bonds with fixed discrete coupon

Technical Summary: Numerical Analysis for a Unified 2-Factor Model of Corporate Bonds with Fixed Discrete Coupons

Problem Statement
The paper addresses the pricing of defaultable corporate bonds that pay fixed discrete coupons. While significant theoretical work exists for zero-coupon bonds (Merton, 1977) and structural models for coupon bonds (Geske, 1977; Agliardi, 2011), there is a gap in analytical solutions for a unified model that combines structural and reduced-form approaches specifically for bonds with fixed discrete coupons.

The authors note that while a unified one-factor model for fixed discrete coupons has an analytical pricing formula, and a unified two-factor model for discounted discrete coupons has been solved, the unified two-factor model for fixed discrete coupons appears to lack an analytical solution. This is due to the complexity of the governing Partial Differential Equations (PDEs), which are parabolic equations involving three variables (firm value, short rate, and time) and mixed derivatives. Furthermore, the model must account for two types of default:

1. Expected Default: Occurs only at discrete coupon dates when the firm's equity is insufficient to cover debt and coupons.
2. Unexpected Default: Can occur at any time, governed by a constant default intensity (hazard rate).

Methodology
To solve this problem, the authors propose a unified two-factor model under the risk-neutral martingale measure, incorporating:

• Vasicek Model: For the stochastic short rate ($r$).
• Geometric Brownian Motion: For the firm value ($V$), with a constant dividend yield.
• Correlation: A correlation coefficient ($\rho$) between the Wiener processes driving the short rate and firm value.

The pricing problem is formulated as a set of initial value problems for PDEs (Equation 2.4). The solution involves a sequence of PDEs defined over sub-intervals between coupon dates, with boundary conditions that switch between the bond price plus the next coupon and the recovery value depending on whether the firm value covers the debt.

Since an analytical formula is deemed intractable for the general case ($i < N-1$), the authors employ an explicit finite difference scheme for numerical analysis. Key methodological steps include:

1. Variable Transformation: Transforming the firm value variable $V$ to $x = \ln V$ to handle the domain $V > 0$ as $x \in \mathbb{R}$.
2. Discretization: Using forward differences for time derivatives and central differences for spatial derivatives. For mixed derivatives ($\partial^2 B / \partial x \partial r$), the authors explore central, forward, and mixed forward-backward difference schemes depending on the sign of the correlation $\rho$.
3. Stability Analysis: A rigorous derivation of stability conditions for the explicit scheme. The authors demonstrate that the standard explicit scheme is generally unstable due to mixed derivatives. They derive specific constraints on the time step ($\Delta t$) and spatial steps ($\Delta x, \Delta r$) relative to the diffusion coefficients and correlation $\rho$ to ensure stability.

Key Contributions

1. Unified Model Formulation: The paper formalizes a unified two-factor model that integrates structural default (barrier crossing at discrete times) and reduced-form default (intensity-driven at any time) for fixed discrete coupon bonds.
2. Stability Conditions: The primary theoretical contribution is the derivation of stability conditions for the explicit finite difference scheme applied to this specific 3-variable parabolic PDE with mixed derivatives.
   • For $\rho = 0$, stability requires $\mu_x + \mu_r < 1$ and specific bounds on the drift terms.
   • For $0 < \rho < 1/2$, the authors show that stability requires $0 < \rho < 1/2$ and specific ratios between the spatial step sizes ($\sqrt{\mu_r/\mu_x}$).
   • For $-1 < \rho < 0$, similar constraints are derived using a mixed forward-backward difference for the mixed derivative.
3. Numerical Implementation: The authors provide a computational framework to calculate bond prices, credit spreads, and durations where analytical formulas are unavailable.

Results
Using the derived explicit scheme with $\rho = 0$, the authors present numerical results based on a specific set of parameters (e.g., $N=2$ coupons, Vasicek parameters, specific default intensities $\lambda_0, \lambda_1$). The analysis yields:

• Bond Prices:
  • Prices decrease as the short rate ($r$) increases.
  • Prices increase as the firm value ($V$) increases, exhibiting convexity with respect to $V$.
  • Prices exhibit jumps (discontinuities) on coupon dates, reflecting the payment of coupons and the reset of default risk conditions.
• Credit Spread (CS):
  • CS decreases as firm value increases.
  • CS increases rapidly as maturity approaches if the firm value is low.
  • CS decreases as the short rate increases (due to the discounting effect on the risk-free component).
  • CS increases with higher default intensity ($\lambda$).
  • CS exhibits jumps on coupon dates; the magnitude of the jump increases with the coupon size.
• Duration:
  • Duration increases with firm value and short rate.
  • Duration exhibits jumps after coupon payments, consistent with the definition of duration as a weighted average time to cash flows.

Significance and Claims
The paper modestly claims to provide a necessary numerical tool for a model that generalizes Agliardi's structural model into a unified framework for fixed discrete coupon bonds. The authors state that obtaining an analytical formula for this specific unified two-factor model is "difficult" and likely impossible in the general case. Therefore, the significance of the work lies in:

1. Establishing the stability conditions required to make the explicit finite difference scheme viable for this complex PDE system.
2. Demonstrating that the numerical results are compatible with real financial situations, as evidenced by the intuitive behavior of bond prices, credit spreads, and durations under varying market conditions (interest rates, firm value, default intensity).

The work serves as a bridge between theoretical structural/reduced-form unification and practical pricing for realistic bond structures (fixed discrete coupons) where closed-form solutions do not exist.