# PDE models for the valuation of a non callable defaultable coupon bond under an extended JDCEV model

## TL;DR

This paper develops PDE-based numerical methods for valuing non-callable, defaultable bonds under an extended JDCEV model that unifies credit and equity risks, with validation against Monte Carlo and asymptotic methods.

## Contribution

It introduces a PDE framework with specific numerical schemes for bond valuation under the JDCEV model, including existence proofs and real-market numerical validation.

## Key findings

- Numerical schemes accurately price bonds consistent with market data.
- The PDE approach aligns well with Monte Carlo and asymptotic methods.
- The model captures complex credit and equity risk interactions.

## Abstract

We consider a two-factor model for the valuation of a non callable defaultable bond which pays coupons at certain given dates. The model under consideration is the Jump to Default Constant Elasticity of Variance (JDCEV) model. The JDCEV model is an improvement of the reduced form approach, which unifies credit and equity models into a single framework allowing for stochastic and possible negative interest rates. From the mathematical point of view, the valuation involves two partial differential equation (PDE) problems for each coupon. First, we obtain the existence of solution for these PDE problems. In order to solve them, we propose appropriate numerical schemes based on a Crank-Nicolson semi-Lagrangian method for time discretization combined with bi-quadratic Lagrange finite elements for space discretization. Once the numerical solutions of the PDEs are obtained, a post-processing procedure is carried out in order to achieve the value of the bond. This post-processing includes the computation of an integral term which is approximated by using the composite trapezoidal rule. Finally, we present some numerical results for real market bonds issued by different firms in order to illustrate the proper behaviour of the numerical schemes. Moreover, we obtain an agreement between the numerical results from the PDE approach and those ones obtained by applying a Monte Carlo technique and an asymptotic approximation method.