A two-factor structural model for valuing corporate securities

Abstract

We propose a general structural model for valuing risky corporate debt securities within a two-dimensional framework. The state variables in our model include the firm’s asset value, described as a geometric Brownian motion stochastic process, and the short-term interest rate, following a mean-reverting Ornstein–Uhlenbeck stochastic process. Our model accommodates flexible debt structure, multiple seniority classes, tax benefits, bankruptcy costs, and a stochastic endogenous default barrier. The proposed methodology relies on a two-dimensional dynamic program coupled with finite elements where key transition parameters are computed in closed form, and effective approximations using local interpolations are made during backward recursion. Our design incorporates space discretization without imposing time discretization, which is advantageous, particularly in the valuation of corporate bonds where exercise opportunities are often distant. Our methodology distinguishes itself by assuming a numerical error, setting it apart from statistical methods. Together, the above features establish dynamic programming coupled with finite elements as a competitive valuation approach as compared to its counterparts in the existing literature. We use parallel computing to enhance the efficiency of our methodology. We conduct a numerical and and an empirical investigation, both of which show consistency with several empirical evidence documented in the literature.

Appendices

Appendix A: Proof of Theorem 1

Proof

First, note that for any $\tau \in \mathcal {T}_k$, with $k<N$,

$$\begin{aligned} W_k^{(n)}= & {} (1-\alpha )V_n \quad \text { and } \mathcal {E}_k^{(k)} = 0;\\ TB_\tau ^{(k)}= & {} \left\{ b_k + e^{-r_{k+1}} TB_\tau ^{(k+1)} \right\} \textbf{1}(\tau>k);\\ D_\tau ^{(k)}= & {} (1-\alpha )V_k \textbf{1}(\tau =k) + \left\{ d_k + e^{-r_{k+1}} D_\tau ^{(k+1)} \right\} \textbf{1}(\tau >k). \end{aligned}$$

The proof is done by reverse induction on $k \le N$. We first prove the result for $k=N$. Then assuming the result is true for all $k\in \{n+1. \ldots , N\}$, we prove it is true for $k=n$.

Part (I): We prove the result for $k=N$, i.e., we prove (4), (6) and (7).

First, for any $\tau \in \mathcal {T}$ with $\tau \ge N$, one has

$$\begin{aligned} E\left( \mathcal {E}_\tau ^{(N)}|\mathcal {F}_N\right) = \mathcal {B}_N\textbf{1}(\tau >N), \end{aligned}$$

where $\mathcal {B}_N = V_N + b_N-d_N$. Hence

$$\begin{aligned} \mathcal {T}_N = \left\{ \tau \in \mathcal {T}; \tau \ge N\; \text { and } \{\tau>N\}\subset \{\mathcal {B}_N>0\} \right\} . \end{aligned}$$

Now, for any $\tau \in \mathcal {T}_N$,

$$\begin{aligned} J_\tau ^{(N)}= & {} -\alpha V_N \textbf{1}(\tau =N)+ b_N \textbf{1}(\tau =N+1) \\= & {} -\alpha V_N \textbf{1}(\mathcal {B}_N\le 0) -\alpha V_N \textbf{1}(\tau =N)\textbf{1}(\mathcal {B}_N>0) + b_N \textbf{1}(\tau =N+1) \\\le & {} -\alpha V_N \textbf{1}(\mathcal {B}_N \le 0) + b_N \textbf{1}(\mathcal {B}_N>0) \end{aligned}$$

As a result, if $\tau _N^\star = N\textbf{1}(\mathcal {B}_N \le 0)+ (N+1)\textbf{1}(\mathcal {B}_N>0)$, then

$$\begin{aligned} \mathcal {E}_N = E\left( \mathcal {E}_{\tau _N^\star }^{(N)}|\mathcal {F}_N \right) =\max (\mathcal {B}_N,0) \end{aligned}$$

and

$$\begin{aligned} {\bar{J}}_N =E\left( J_{\tau _N^\star }^{(N)}|\mathcal {F}_N \right) = -\alpha V_N \textbf{1}(\mathcal {E}_N= 0) + b_N\textbf{1}(\mathcal {E}_N>0). \end{aligned}$$

Hence (4), (6) and (7) hold for $k=N$.

Part (II): Next suppose the result is true for all $k>n$, i.e., (5)– (8) hold for all $k>n$, We need to show that they also hold for $k=n$.

For $\tau \in \mathcal {T}_n$,

$$\begin{aligned} E\left( J_\tau ^{(n)} |\mathcal {F}_n\right)= & {} -\alpha V_n \textbf{1}(\tau =n)+ b_n \textbf{1}(\tau>n) + E\left( e^{-r_{n+1}} J_{\tau \vee (n+1)}^{(n+1)} |\mathcal {F}_n\right) \textbf{1}(\tau>n)\\= & {} -\alpha V_n \textbf{1}(\tau =n)+ b_n \textbf{1}(\tau>n) \\{} & {} \qquad + E\left\{ e^{-r_{n+1}} E\left( J_{\tau \vee (n+1)}^{(n+1)}|\mathcal {F}_{n+1}\right) |\mathcal {F}_n\right\} \textbf{1}(\tau>n)\\\le & {} -\alpha V_n \textbf{1}(\tau =n)+ \left\{ b_n + E\left( e^{-r_{n+1}}{\bar{J}}_{n+1}|\mathcal {F}_n\right) \right\} \textbf{1}(\tau >n), \end{aligned}$$

where the last inequality comes from the fact that if $\tau \in \mathcal {T}_n$, then $\tau \vee (n+1) \in \mathcal {T}_{n+1}$. As a result, since $\tau>n\subset \left\{ \mathcal {B}_n^{(\tau )}>0\right\}$, it follows that

$$\begin{aligned} E\left( J_\tau ^{(n)} |\mathcal {F}_n\right) \le -\alpha V_n\textbf{1}\left( \mathcal {B}_n^{(\tau )} \le 0\right) + \left\{ b_n + E\left( e^{-r_{n+1}}{\bar{J}}_{n+1}|\mathcal {F}_n\right) \right\} \textbf{1}\left( \mathcal {B}_n^{(\tau )}>0\right) . \end{aligned}$$

By the induction hypothesis and (2), it follows that

$$\begin{aligned} \mathcal {B}_n^{(\tau )} \le V_n+b_n-d_n - E\left( e^{-r_{n+1}} V_{n+1}|\mathcal {F}_n\right) + E\left( e^{-r_{n+1}} \mathcal {E}_{n+1}|\mathcal {F}_n \right) , \end{aligned}$$

so $\left\{ \mathcal {B}_n^{(\tau )}> 0 \right\} \subset \{\mathcal {E}_n>0\}$. As a result, setting $c_n = b_n + E\left( e^{-r_{n+1}}\bar{J}_{n+1}|\mathcal {F}_n\right)$, one may conclude that

$$\begin{aligned} -\alpha V_n\textbf{1}\left( \mathcal {B}_n^{(\tau )} \le 0\right) + c_n \textbf{1}\left( \mathcal {B}_n^{(\tau )}>0\right) \le -\alpha V_n\textbf{1}\left( \mathcal {E}_n = 0\right) + c_n \textbf{1}\left( \mathcal {E}_n >0\right) , \end{aligned}$$

since

$$\begin{aligned} c_n \left\{ \textbf{1}\left( \mathcal {E}_n>0\right) -\textbf{1}\left( \mathcal {B}_n^{(\tau )}>0\right) \right\} \ge 0 \ge \alpha V_n \left\{ \textbf{1}\left( \mathcal {E}_n =0\right) -\textbf{1}\left( \mathcal {B}_n^{(\tau )}\le 0\right) \right\} . \end{aligned}$$

Therefore ${\bar{J}}_n \le -\alpha V_n\textbf{1}\left( \mathcal {E}_n = 0\right) + c_n \textbf{1}\left( \mathcal {E}_n >0\right)$. Next, from (6), it then follows that $\tau _n^\star \vee (n+1) = \tau _{n+1}^\star$, so

$$\begin{aligned} {\bar{J}}_n= & {} E\left( J_{\tau _n^\star }^{(n)} |\mathcal {F}_n\right) \nonumber \\= & {} -\alpha V_n\textbf{1}(\mathcal {E}_n = 0)+ \left\{ b_n + E\left( e^{-r_{n+1}}{\bar{J}}_{n+1}|\mathcal {F}_n\right) \right\} \textbf{1}(\mathcal {E}_n>0). \end{aligned}$$

Hence (5)–(8) also hold $k=n$, completing the proof. $\square$

Appendix B: Forward measure

The forward measure ${\mathbb {P}}^{T_F}$ for any date $T_F$ is the measure associated with taking the bond $B(t,T_F)$ as a numeraire asset. Under the forward measure, the ratio $B(t,T)/B (t, T_F)$ is a martingale for $T\le T_F$. From Girsanov’s Theorem, it follows that the process $W^{T_F}$ defined by

$$\begin{aligned} dW_t^{T_F} = dZ_t^1 +\frac{\sigma _r}{\alpha } (1 - e^{-\alpha (T_F-t)}), \end{aligned}$$

is standard Brownian motion under ${\mathbb {P}}_{T_F}$. Thus, the dynamic of the interest rate becomes

$$\begin{aligned} dr_t=\left( \theta -\alpha r_t - \frac{\sigma _r^2}{\alpha }(1- e^{-\alpha (T_F-t)})\right) dt+\sigma _r dW_t^{T_F}, \end{aligned}$$

with $\theta = \alpha \beta$ and the dynamic of $X_t = \ln (V_t)$ is

$$\begin{aligned} dX_t&= \left( r_t-\delta -\frac{\sigma _V^2}{2}- \frac{{\rho \sigma _V \sigma _r}}{\alpha }(1-e^{-\alpha (T_F-t)})\right) dt +\sigma _V\left( \rho dW_t^{T_F}+\sqrt{1-\rho ^2}dZ_t^2\right) . \end{aligned}$$

The solutions are given by

$$\begin{aligned} r_t \,\, =\,\,\,&r_ue^{-\alpha (t-u)}+ \left( \frac{\theta }{\alpha }-\frac{\sigma _r^2}{\alpha ^2}\right) \left( 1-e^{-\alpha (t-u)}\right) + \frac{\sigma ^2_r}{2\alpha ^2}\left( e^{-\alpha (T_F-t)}-e^{-\alpha (T_F+t-2u)}\right) +\\&\sigma _r\int _u^te^{-\alpha (t-s)}dW_s^{T_F},\\ X_t \,\, =\,\,\,&X_u+\beta (u,t)-\left( \frac{\sigma _V^2}{2}+\frac{\rho \sigma _V\sigma _r}{\alpha }\right) (t-u) + \frac{\rho \sigma _V\sigma _r}{\alpha ^2}\left( e^{-\alpha (T_F-u)}-e^{-\alpha (T_F-u)}\right) +\\&\int _u^t\left( \rho \sigma _V+ \frac{\sigma _r}{\alpha }\left( 1-e^{-\alpha (t-s)}\right) \right) dW^{T_F}_s, \quad \text {for } 0 \le u \le t, \end{aligned}$$

with

$$\begin{aligned} \beta (u,t)\,\,&=\,\,\,\frac{r_u}{\alpha }\left( 1-e^{-\alpha (t-u)}\right) +\left( \frac{\theta }{\alpha }-\frac{\sigma _r^2}{\alpha ^2}\right) \left( -\frac{1- e^{-\alpha (t-u)}}{\alpha }+t-u\right) \\&\quad +\frac{\sigma _r^2}{2\alpha ^3}\left( e^{-\alpha (T_F-t)}-2e^{-\alpha (T_F-u)}+e^{-\alpha (T_F+t-2u)}\right) . \end{aligned}$$

Under the forward measure, the pair $(X_t, r_t)$ follows a bivariate normal distribution with

$$\begin{aligned} E[X_t|X_u] \,\,&=\,\,\, X_u+\beta (u,t)-\left( \frac{\sigma _V^2}{2}+\frac{\rho \sigma _V\sigma _r}{\alpha }\right) (t-u)+\frac{\rho \sigma _V\sigma _r}{\alpha ^2}\left( e^{-\alpha (T_F-t)}-e^{-\alpha (T_F-u)}\right) ,\\ Var[X_t|X_u]\,\,&=\,\,\,\left( \sigma _V^2+\frac{2\rho \sigma _V\sigma _r}{\alpha }+\frac{\sigma _r^2}{\alpha ^2}\right) (t-u)-\frac{2\rho \sigma _V\sigma _r}{\alpha ^2} \times \left( 1-e^{-\alpha (t-u)}\right) - \\ {}&\frac{\sigma _r^2}{2\alpha ^3}\left( 3-4e^{-\alpha (t-u)}+e^{-2\alpha (t-u)}\right) , \\ E[r_t|r_u]\,\,&=\,\,\, r_ue^{-\alpha (t-u)}+\frac{\theta }{\alpha }\left( 1-e^{-\alpha (t-u)}\right) -\frac{\sigma _r^2}{\alpha ^2}\times \left( 1-e^{-\alpha (t-u)}\right) +\\ {}&\frac{\sigma _r^2}{2\alpha ^2}\left( e^{-\alpha (T_F-t)}-e^{-\alpha (T_F+t-2u)}\right) , \\ Var[r_t|r_u]\,\,&=\,\,\,\frac{\sigma _r^2}{2\alpha }\left( 1-e^{-2\alpha (t-u)}\right) , \end{aligned}$$

and

$$\begin{aligned} Cov[X_t, r_t|X_u, r_u] \,\, =\,\,\, \left( \frac{\rho \sigma _V\sigma _r}{\alpha }+\frac{\sigma _r^2}{\alpha ^2}\right) \left( 1-e^{-\alpha (t-u)}\right) -\frac{\sigma _r^2}{2\alpha ^2}\times \left( 1-e^{-2\alpha (t-u)}\right) . \end{aligned}$$

Appendix C: Transitions parameters

The transition parameters $T^{\nu \mu }_{klij}$ for $\nu$ and $\mu \in \{0,1\}$, $k\in \{1,\ldots ,p\}$, $l \in \{1,\ldots ,q\}$, $i \in \{0,\ldots ,p\}$, and $j \in \{0,\ldots ,q\}$ are calculated as follows:

$$\begin{aligned} T^{00}_{klij}\,\,=\,\,\,&{\mathbb {E}}^*\left[ {\mathbb {I}}\left( (V_{t_{n+1}}, r_{t_{n+1}}) \in R_{ij} \right) \mid (V_{t_n},r_{t_n})=(a_k,b_l)\right] \\ =\,\,\,&{\mathbb {Q}}^{*}\left[ (V_{t_{n+1}}, r_{t_{n+1}}) \in R_{ij} \mid (V_{t_n},r_{t_n})=(a_k,b_l) \right] \\ =\,\,\,&\int _{{x}_{k,i}}^{{x}_{k,i+1}} \int _{y_{l,j}}^{y_{l,j+1}} \phi (z_1,z_2,\rho ) dz_1dz_2\\ =\,\,\,&\Phi (x_{k,i+1},y_{l,j+1},\rho )- \Phi (x_{k,i},y_{l,j+1},\rho )-\Phi (x_{k,i+1},y_{l,j},\rho )+\Phi (x_{k,i},y_{l,j},\rho ), \end{aligned}$$

where

$$\begin{aligned} x_{k,i} \,\,=\,\,\,&\left( \log \left( a_{i}/a_k\right) -\eta _1\right) /\sqrt{\delta _1}\\ y_{l,j} \,\,=\,\,\,&\left( b_{j}-\eta _2\right) /\sqrt{\delta _2},\\ \eta _1\,\,=\,\,\,&\beta _l-\left( \frac{\sigma _V^2}{2}+\frac{\rho \sigma _V\sigma _r}{\alpha }\right) \Delta t+\frac{\rho \sigma _V\sigma _r}{\alpha ^2}\left( 1-e^{-\alpha \Delta t}\right) , \\ \delta _1 \,\,=\,\,\,&\left( \sigma _V^2+\frac{2\rho \sigma _V\sigma _r}{\alpha }+\frac{\sigma _r^2}{\alpha ^2}\right) \Delta t-\frac{2\rho \sigma _V\sigma _r}{\alpha ^2}\left( 1-e^{-\alpha \Delta t}\right) \\&\quad -\frac{\sigma _r^2}{2\alpha ^3}\left( 3-4e^{-\alpha \Delta t}+e^{-2\alpha \Delta t}\right) , \\ \eta _2 \,\,=\,\,\,&b_le^{-\alpha \Delta t}+\frac{\theta }{\alpha }\left( 1-e^{-\alpha \Delta t}\right) -\frac{\sigma _r^2}{\alpha ^2}\left( 1-e^{-\alpha \Delta t}\right) +\frac{\sigma _r^2}{2\alpha ^2}\left( 1-e^{-2\alpha \Delta t}\right) ,\\ \delta _2 \,\,=\,\,\,&\frac{\sigma _r^2}{2\alpha }\left( 1-e^{-2\alpha \Delta t}\right) ,\\ \beta _l\,\,=\,\,\,&\frac{r_l}{\alpha }\left( 1-e^{-\alpha \Delta t}\right) +\left( \frac{\theta }{\alpha }-\frac{\sigma _r^2}{\alpha ^2}\right) \left( -\frac{1- e^{-\alpha \Delta t}}{\alpha }+\Delta t\right) \\&\quad +\frac{\sigma _r^2}{2\alpha ^3}\left( 1-2e^{-\alpha \Delta t}+e^{-2\alpha \Delta t}\right) . \end{aligned}$$

${\mathbb {E}}^*$ is the expectation under the forward measure to the time $t_{n+1}$. The functions $\phi (\cdot ,\cdot ,\rho )$ and $\Phi (\cdot ,\cdot ,\rho )$ are the density and cumulative density functions, respectively, of the bivariate standard normal distribution with correlation coefficient $\rho$. The function $\Phi (\cdot ,\cdot ,\rho )$ is computed according to Genz (2004).

$$\begin{aligned} T^{10}_{klij}\,\,=\,\,\,&{\mathbb {E}}^{*}\left[ V_{t_{n+1}}{\mathbb {I}}\left( (V_{t_{n+1}}, r_{t_{n+1}}) \in R_{ij} \right) \mid (V_{t_n},r_{t_n})=(a_k,b_l)\right] \\ =\,\,\,&w^1_{k}\int _{x_{k,i}-\sqrt{\delta _1}}^{x_{k,i+1}-\sqrt{\delta _1}} \int _{y_{l,j}-\rho \sqrt{\delta _1}}^{y_{l,j+1}-\rho \sqrt{\delta _1}} \phi (u_1,u_2,\rho ) du_1du_2\\ =\,\,\,&w^1_{k}\Big [\Phi (x_{k,i+1}-\sqrt{\delta _1},y_{l,j+1}-\rho \sqrt{\delta _1},\rho ) -\Phi (x_{k,i}-\sqrt{\delta _1},y_{l,j+1}-\rho \sqrt{\delta _1},\rho )\\ \,\,\,&-\Phi (x_{k,i+1}-\sqrt{\delta _1},y_{l,j}-\rho \sqrt{\delta _1},\rho )+\Phi (x_{k,i}-\sqrt{\delta _1},y_{l,j}-\rho \sqrt{\delta _1},\rho )\Big ], \end{aligned}$$

where $w^1_{k}=a_k \exp \left( \eta _1+\delta _1/2\right)$.

$$\begin{aligned} T^{01}_{klij}\,\,=\,\,\,&{\mathbb {E}}^{*}\left[ e^{r_{t_{n+1}}}{\mathbb {I}}\left( (V_{t_{n+1}}, r_{t_{n+1}}) \in R_{ij} \right) \mid (V_{t_n},r_{t_n})=(a_k,b_l)\right] \\ =\,\,\,&w^2_{l}\int _{x_{k,i}-\rho \sigma _2\Delta t}^{x_{k,i+1}-\rho \sigma _2\Delta t} \int _{y_{l,j}- \sigma _2\Delta t}^{y_{l,j+1}- \sigma _2\Delta t} \phi (u_1,u_2,\rho ) du_1du_2\\ =\,\,\,&w^2_{l}\Big [\Phi (x_{k,i+1}-\rho \sqrt{\delta _2} ,y_{l,j+1}-\sqrt{\delta _2},\rho )\Phi (x_{k,i}-\rho \sqrt{\delta _2},y_{l,j+1}-\sqrt{\delta _2},\rho )\\ \,\,\,&-\Phi (x_{k,i+1}-\rho \sqrt{\delta _2},y_{l,j}-\sqrt{\delta _2},\rho )+\Phi (x_{k,i}-\rho \sqrt{\delta _2},y_{l,j}-\sqrt{\delta _2},\rho )\Big ], \end{aligned}$$

where $w^2_{l}=\exp \left( \eta _2+\delta _2/2\right)$.

$$\begin{aligned} T^{11}_{klij}\,\,=\,\,\,&{\mathbb {E}}^{*}\left[ V_{t_{n+1}}e^{r_{t_{n+1}}}{\mathbb {I}}\left( (V_{t_{n+1}}, r_{t_{n+1}}) \in R_{ij} \right) \mid (V_{t_n},r_{t_n})=(a_k,b_l)\right] \\ =\,\,\,&w_{1,k}w^2_{l}\exp \left( \rho \sqrt{\delta _1\delta _2}\right) \times \int _{x_{k,i}-\sqrt{\delta _1}-\rho \sqrt{\delta _2}}^{x_{k,i+1}-\sqrt{\delta _1}-\rho \sqrt{\delta _2}}\!\int _{y_{l,j}-\rho \sqrt{\delta _1} - \sqrt{\delta _2}}^{y_{l,j+1}- \rho \sqrt{\delta _1} -\sqrt{\delta _2}}\! \!\phi (u_1,u_2,\rho ) du_1du_2\\ =\,\,\,&w^1_{k}w^2_{l}\exp \left( \rho \sqrt{\delta _1\delta _2}\right) \times \Big [\Phi (x_{k,i+1}-\sqrt{\delta _1}-\rho \sqrt{\delta _2},y_{l,j+1}-\rho \sqrt{\delta _1} - \sqrt{\delta _2},\rho )\! \!\\ \,\,\,&-\Phi (x_{k,i}-\sqrt{\delta _1}-\rho \sqrt{\delta _2},y_{l,j+1}-\rho \sqrt{\delta _1} - \sqrt{\delta _2},\rho )\\ \,\,\,&-\Phi (x_{k,i+1}-\sqrt{\delta _1}-\rho \sqrt{\delta _2},y_{l,j}-\rho \sqrt{\delta _1} - \sqrt{\delta _2},\rho ) \\ \,\,\,&+\Phi (x_{k,i}-\sqrt{\delta _1}-\rho \sqrt{\delta _2},y_{l,j}--\rho \sqrt{\delta _1} - \sqrt{\delta _2},\rho ) \Big ]. \end{aligned}$$