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

Paper: arXiv 2301.10734

Authors: Rakhymzhan Kazbek, Yogi Erlangga, Yerlan Amanbek, Dongming Wei

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.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous mathematical approach to convertible bond pricing using a two-coupled Black-Scholes system solved via P1/P2 Finite Element Methods and a penalty scheme. However, it lacks empirical validation with real market data, focusing primarily on numerical comparison against existing finite difference benchmarks.

Research Flowchart

  flowchart TD
    A[Goal: Value Convertible Bonds & Calculate Greeks via Penalty TF Model] --> B[Inputs: Black-Scholes Parameters, Inequality Constraints & Contract Terms]
    B --> C[Spatial Discretization: P1 & P2 Finite Element Methods]
    C --> D[Constraint Approximation: Penalty Method for Free Boundaries]
    D --> E[Time Integration: Modified Crank-Nicolson Scheme with Newton-Raphson]
    E --> F[Outcomes: Accurate CB Pricing, Derivatives/Greeks & Superiority over FDM]