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

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.

Appendix A: Accuracy of the approximations

To demonstrate that our FEM approach to the TF problem described above may obtain first order accuracy, we consider the following linear model problem (Roache, 2001), which keeps the left hand side of the TF the system defined by (1) and (2), without constraints (and hence no penalty term), and coupon payment corresponding to the exact solution

$$\begin{aligned} U(x,\tau )&= S_{int}^2e^{2x}\sqrt{S_{int}e^x} - Fe^{-r\tau }\sqrt{S_{int}e^x}, \\ V(x,\tau )&= S_{int}^2e^{2x}\sqrt{S_{int}e^x} - Fe^{-r\tau }\sqrt{S_{int}e^x} + x^2\tau , \end{aligned}$$

where $\tau \in (0,1)$. This manufactured solution corresponds to the initial conditions

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle U(x,0) =S_{int}^2e^{2x}\sqrt{S_{int}e^x} - F\sqrt{S_{int}e^x}, \\ \displaystyle V(x,0) = S_{int}^2e^{2x}\sqrt{S_{int}e^x} - F\sqrt{S_{int}e^x}, \end{array}\right. } \end{aligned}$$

the boundary conditions

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle U(0,\tau ) = S_{int}^2\sqrt{S_{int}} - Fe^{-r\tau }\sqrt{S_{int}},\\ \displaystyle V(0,\tau )= S_{int}^2\sqrt{S_{int}} - Fe^{-r\tau }\sqrt{S_{int}}, \end{array}\right. }{\left\{ \begin{array}{ll} \displaystyle U(1,\tau ) = S_{int}^2e^2\sqrt{S_{int}e} - Fe^{-r\tau }\sqrt{S_{int}e}, \\ \displaystyle V(1,\tau ) = S_{int}^2e^2\sqrt{S_{int}e} - Fe^{-r\tau }\sqrt{S_{int}e} +\tau , \end{array}\right. } \end{aligned}$$

and the non-homogeneous BS equations

$$\begin{aligned} \displaystyle f_1&= U_{\tau } - \frac{\sigma ^2}{2}U_{xx}- \left( r- \frac{\sigma ^2}{2}\right) U_x +rU +r_cV, \\ \displaystyle f_2&= V_{\tau } - \frac{\sigma ^2}{2}V_{xx}- \left( r- \frac{\sigma ^2}{2}\right) V_x +(r +r_c)V, \end{aligned}$$

in the CB and COCB PDE, respectively. Applying FEM and the $\theta$-scheme leads to the numerical procedure

$$\begin{aligned} A_{11}\varvec{u}^{m+1}&= \widetilde{A}_{11}\varvec{u}^m - A_{12} \varvec{v}^{m+1} + \widetilde{A}_{12} \varvec{v}^m + \theta \Delta \tau \varvec{\beta }_1^{m+1} + (1-\theta ) \Delta \tau \varvec{\beta }_1^{m} + \hat{\varvec{b}}^m_{M,u} - \hat{\varvec{b}}^{m+1}_{M,u} \\&\quad +\theta \Delta \tau f_1 + (1-\theta )\Delta \tau f_1, \\ A_{22}\varvec{v}^{m+1}&= \widetilde{A}_{22}\varvec{v}^m + \theta \Delta \tau \varvec{\beta }_2^{m+1} + (1-\theta ) \Delta \tau \varvec{\beta }_2^{m} + \hat{\varvec{b}}^m_{M,v} - \hat{\varvec{b}}^{m+1}_{M,v} +\theta \Delta \tau f_2 + (1-\theta ) \Delta \tau f_2. \end{aligned}$$

Calculated errors are presented Figs. 7 and 8 using two measures (Amanbek & Wheeler, 2019; Amanbek et al., 2020):

$$\begin{aligned} \Vert Error\Vert _{L^2}&= \Vert U(x,1) - \varvec{u}^{n_t}\Vert _{L^2}, \\ \Vert Error\Vert _{L^\infty (L^2)}&= \max _{1\le m \le n_t}\left( \Vert U(x,m \Delta \tau ) - \varvec{u}^m \Vert _{L^2} \right) , \end{aligned}$$

where $\varvec{u}^m$ is the solution of the model problem at $\tau = m \Delta \tau$, computed by P1-FEM or P2-FEM. In Fig. 7, the errors are calculated for varying $\Delta \tau$ and a fixed value of h. The errors decrease as $\Delta \tau$ is reduced to 0, with a rate that is proportional to $\Delta \tau$ (first-order convergence).

Fig. 7

Error estimates of the model problem in $\tau \in [0,1]$, with $r = 0.05$,$r_c = 0.02$, $\sigma = 0.2$, $h_{P1} = 3\times 10^{-4}$, and $h_{P2} = 10^{-3}$

In Fig. 8, the errors are calculated for varying h and fixed $\Delta \tau$. The plots suggest convergence of P1-FEM and P2-FEM at the rate proportional to h and $h^2$, respectively (Theoretically, e.g. for P1-FEM the convergence rate is given by the relation $\Vert Error\Vert _{L^\infty (L^2)} \le C(h+\Delta \tau )$). This convergence rate is as expected for the linear model used but cannot, however, be expected when nonlinear (e.g., penalty) terms are added.

Fig. 8

Error estimates of the MMS model in $x \in [0,1]$, with $r = 0.05$,$r_c = 0.02$, $\sigma = 0.2$, and $\Delta \tau = 10^{-4}$