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Pricing Convertible Bonds with the Penalty TF Model Using Finite Element Method

Author

Listed:
  • Rakhymzhan Kazbek

    (Nazarbayev University)

  • Yogi Erlangga

    (Zayed University)

  • Yerlan Amanbek

    (Nazarbayev University)

  • Dongming Wei

    (Nazarbayev University)

Abstract

In this paper, we discuss finite element methods (FEM) for solving numerically the so-called TF model, a PDE-based model for pricing convertible bonds. The model consists of two coupled Black-Scholes equations, whose solutions are constrained. The construction of the FEM is based on the P1 and P2 element, applied to the penalty-based reformulation of the TF model. The resultant nonlinear differential algebraic equations are solved using a modified Crank-Nicolson scheme, with non-linear part with non-smooth terms solved at each time step by Newton’s method. While P1-FEM demonstrates a comparable convergence rate to the standard finite difference method, a better convergence rate is achieved with P2-FEM. The fast convergence of P2-FEM leads to a significant reduction in CPU time, due to the reduction in the number of elements used to achieve the same accuracy as P1-FEM or FDM. As the Greeks are important numerical parameters in the bond pricing, we compute some Greeks using the computed solution and the corresponding FEM approximation functions.

Suggested Citation

  • Rakhymzhan Kazbek & Yogi Erlangga & Yerlan Amanbek & Dongming Wei, 2025. "Pricing Convertible Bonds with the Penalty TF Model Using Finite Element Method," Computational Economics, Springer;Society for Computational Economics, vol. 65(4), pages 1971-1998, April.
    • Handle: RePEc:kap:compec:v:65:y:2025:i:4:d:10.1007_s10614-024-10625-1
    DOI: 10.1007/s10614-024-10625-1
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    References listed on IDEAS

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    1. Ammann, Manuel & Kind, Axel & Wilde, Christian, 2008. "Simulation-based pricing of convertible bonds," Journal of Empirical Finance, Elsevier, vol. 15(2), pages 310-331, March.
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