Valuation of the Convertible Bonds under Penalty TF model using Finite Element Method

TL;DR

This paper develops a finite element method to numerically solve the Penalty TF model for convertible bond valuation, incorporating inequality constraints and calculating Greeks, showing favorable results compared to finite difference methods.

Contribution

It introduces a finite element approach with penalty methods and modified Crank-Nicolson scheme for pricing convertible bonds under the Penalty TF model, including Greeks computation.

Findings

Finite element solutions compare favorably with finite difference methods.

The method effectively handles inequality constraints in bond valuation.

Greeks are accurately computed using finite element approximation functions.

Abstract

In this paper, the TF system of two-coupled Black-Scholes equations for pricing the convertible bonds is solved numerically by using the P1 and P2 finite elements with the inequality constraints approximated by the penalty method. The corresponding finite element ODE system is numerically solved by using a modified Crank-Nicolson scheme, in which the non-linear system is solved at each time step by the Newton-Raphson method for non-smooth functions. Moreover, the corresponding Greeks are also calculated by taking advantage of the P1-P2 finite element approximation functions. Numerical solutions by the finite element method compare favorably with the solutions by the finite difference method in literature.