in a more general form, in Cui et al. (2018) Proposition 8 (i). The proof in
Appendix A differs from these previous proofs and provides a simple and
intuitive way of obtaining these results using basic probabilistic arguments.

###  B.2. Zero-Coupon Bond Option

The next result concerns the price of European call and put options on zero-
coupon bonds.

######  Corollary B.2.

Let Assumption 4.1 hold. Given that
$R^{(m)}_{t_{n_{1}}}=r_{j}\in\mathcal{S}_{R}^{(m)}$, the price at $t_{n1}\geq
0$ of a European call (resp. put) option with maturity $t_{n_{2}}>t_{n_{1}}$
on a zero-coupon bond maturing at time $T>t_{n_{2}}$ with a strike $K>0$ can
be approximated by

(38)
$\mathbb{E}\left[e^{-\int_{t_{n_{1}}}^{t_{n_{2}}}R_{s}^{(m)}\,\mathrm{d}s}h\left(P^{(m)}(t_{n_{2}},T)\right)\Big{|}R^{(m)}_{t_{n_{1}}}=r_{j}\right]=\mathbf{e}_{j}\left(e^{\left(\mathbf{Q}^{(m)}-\mathbf{D}_{m}\right)(t_{n_{2}}-t_{n_{1}})}\right)\mathbf{H},$

where $h(x)=\max(x-K,0)$ (resp. $h(x)=\max(K-x,0)$) denotes the payoff
function,
${P^{(m)}(t_{n_{2}},T):=\mathbb{E}\left[e^{-\int_{t_{n_{2}}}^{T}R_{s}^{(m)}\,\mathrm{d}s}\Big{|}R_{t_{n_{2}}}^{(m)}\right]}$
denotes the approximated zero-coupon bond price at $t_{n_{2}}$, and
$\mathbf{H}$ denotes a column vector of size $m\times 1$ whose $k$-th,
$h_{k}$, entry is given by

$h_{k}=h\left(P_{k}^{(m)}(t_{n_{2}},T)\right),$

with $P_{k}^{(m)}(t_{n_{2}},T)$ defined in (37).

##  Appendix C Extended Models of Brigo and Mercurio (2006)

In this section, we summarize the results of Section 4 under particular time-
inhomogeneous short-rate models, for which the short-rate process is obtained
by a time-deterministic shift of a time-homogeneous auxiliary process. More
precisely, we suppose that the short-rate process can be decomposed as

(39) $R_{t}=Y_{t}+\theta(t),$

for $t\geq 0$, where $\theta$ denotes a continuous deterministic function of
time and $Y$ denotes an auxiliary time-homogeneous diffusion process with the
following dynamics:

(40)
$\,\mathrm{d}Y_{t}=\mu_{Y}(Y_{t})\,\mathrm{d}t+\sigma_{Y}(Y_{t})\,\mathrm{d}W_{t},$

where $\mu_{Y},\sigma_{Y}:\mathcal{S}_{Y}\rightarrow\mathbb{R}$ are well-
behaved functions such that (40) has a unique in-law weak solution with
$\mathcal{S}_{Y}$ the state-space of $Y$. Examples of such diffusion processes
are listed in Table 3.

In this section, the auxiliary process $Y$ is approximated by a CTMC. We
denote by $Y^{(m)}$ the CTMC approximation of $Y$ taking values in a finite
state-space ${\mathcal{S}_{Y}^{(m)}=\\{y_{1},y_{2},\ldots,y_{m}\\}}$. The
time-independent generator of $Y^{(m)}$, denoted by $\mathbf{Q}^{(m)}_{Y}$, is
constructed as in (5), with functions $\mu_{R}$ and $\sigma_{R}$ replaced by
functions $\mu_{Y}$ and $\sigma_{Y}$ of (40), respectively. Moreover, since
the auxiliary process is time-homogenous, the generator of $Y^{(m)}$ does not
depend on time, such that
$\mathbf{Q}^{(m)}_{Y}:=\mathbf{Q}^{(m)}_{1}=\mathbf{Q}^{(m)}_{2}=\ldots=\mathbf{Q}^{(m)}_{N}.$
Using (39), the CTMC approximation of the short-rate process $R^{(m)}$ is
given by

$R^{(m)}_{t}=Y^{(m)}_{t}+\theta(t),$

for $t\geq 0$, with $\theta(0)=0$.

######  Remark C.1 (Weak convergence of the approximation).

The weak convergence of $Y^{(m)}$ to $Y$ follows from Theorem 5.1 of Mijatović
and Pistorius (2013). Then, since $R$ is a continuous transformation of $Y$,
we conclude that $R^{(m)}\Rightarrow R$ by the continuous mapping Theorem.

Recall that $\\{\mathbf{e}_{k}\\}_{k=1}^{m}$ denotes the standard basis in
$\mathbb{R}^{m}$, that is, $\mathbf{e}_{k}$ represents a row vector of size
$1\times m$ with a value of $1$ in the $k$-th entry and $0$ elsewhere,
$\mathbf{1}_{m\times 1}$ denotes an $m\times 1$ unit vector, and
$\mathbf{D}_{Y}:=\text{diag}(\bm{y})$ denotes an $m\times m$ diagonal matrix
with vector $\bm{y}=(y_{1},y_{2},\ldots,y_{m})$ on its diagonal, with
$y_{k}\in\mathcal{S}_{Y}^{(m)}$, $k=1,2,\ldots,m$.

###  C.1. Zero-Coupon Bond

The next corollary shows that the price of zero-coupon bonds inherits the
analytical tractability of the homogeneous auxiliary process $Y$.

######  Corollary C.1.

Suppose that there exists $y^{\star}\in\mathcal{S}_{Y}$ such that $Y_{t}\geq
y^{\star}$ for all $t\geq 0$. Then, the price at time $t\geq 0$ of a zero-
coupon bond with maturity $T\geq t$ can be approximated by

(41)
$P_{j}^{(m)}(t,T):=\mathbb{E}[e^{-\int_{t}^{T}R^{(m)}_{s}\,\mathrm{d}s}\big{|}R_{t}^{(m)}=y_{j}+\theta(t)]=e^{-\int_{t}^{T}\theta(s)\,\mathrm{d}s}\widetilde{P}_{j}^{(m)}(t,T),$

with

(42)
$\widetilde{P}_{j}^{(m)}(t,T):=\mathbb{E}\left[e^{-\int_{t}^{T}Y_{s}^{(m)}\,\mathrm{d}s}\big{|}Y_{t}^{(m)}=y_{j}\right]=\mathbf{e}_{j}e^{\left(\mathbf{Q}^{(m)}_{Y}-\mathbf{D}_{Y}\right)(T-t)}\mathbf{1}_{m\times
1},$

where $Y_{t}=Y_{t}^{(m)}=y_{j}\in\mathcal{S}_{Y}^{(m)}$.

The proof follows directly from Corollary B.1.

###  C.2. Zero-Coupon Bond Option

The next result concerns the price of European call and put options on zero-
coupon bonds. Using the homogeneous property of the auxiliary process, we can
obtain a simplified expression for (14).

######  Corollary C.2.

Suppose that there exists $y^{\star}\in\mathcal{S}_{Y}$ such that $Y_{t}\geq
y^{\star}$ for all $t\geq 0$. Given that
$R^{(m)}_{t_{n_{1}}}=y_{j}+\theta(t)$, with $y_{j}\in\mathcal{S}_{Y}^{(m)}$,
the price at $t_{n_{1}}\geq 0$ of a European call (resp. put) option with
maturity $t_{n_{2}}>t_{n_{1}}$ on a zero-coupon bond maturing at time
$T>t_{n_{2}}$ with a strike $K>0$ can be approximated by

(43)
$\mathbb{E}\left[e^{-\int_{t_{n_{1}}}^{t_{n_{2}}}R_{s}^{(m)}\,\mathrm{d}s}h\left(P^{(m)}(t_{n_{2}},T)\right)\Big{|}R^{(m)}_{t_{n_{1}}}=y_{j}+\theta(t)\right]\\\
\qquad\qquad\qquad=\mathbf{e}_{j}e^{-\int_{t_{n_{1}}}^{t_{n_{2}}}\theta(s)\,\mathrm{d}s}\left(e^{\left(\mathbf{Q}^{(m)}_{Y}-\mathbf{D}_{Y}\right)(t_{n_{2}}-t_{n_{1}})}\right)\mathbf{H},$

where $h(x)=\max(x-K,0)$ (resp. $h(x)=\max(k-x,0)$) denotes the payoff
function with
${P^{(m)}(t_{n_{2}},T):=\mathbb{E}\left[e^{-\int_{t_{n_{2}}}^{T}R_{s}^{(m)}\,\mathrm{d}s}\Big{|}R_{t_{n_{2}}}^{(m)}\right]}$
and $\mathbf{H}$ denotes a column vector of size $m\times 1$ whose $k$-th
entry, $h_{k}$, is given by

$h_{k}=h\left(P_{k}^{(m)}(t_{n_{2}},T)\right),$

with $P_{k}^{(m)}(t_{n_{2}},T)$ defined in (41).

###  C.3. Calibration to the Initial Term Structure of Interest Rates

When the short-rate process is of the form $R_{t}=Y_{t}+\theta(t)$, as in
Table 3, the fitting to the term structure of interest rates can be greatly
simplified, since the function $\theta$ now appears explicitly in the zero-
coupon bond formula (41).

Proposition C.1 provides an explicit expression for the function $\theta$ that
makes the model zero-coupon bond prices equal to the market prices when the
short-rate process is of the form (39). As in Section 4.4, we assume that the
time deterministic function $\theta$ is piecewise constant in time, such that

$\theta(t)=\sum_{n=1}^{N}\theta_{n}\mathbf{1}_{[t_{n-1},t_{n})}(t),$

for some
$\bm{\theta}=(\theta_{1},\theta_{2},\ldots,\theta_{N})\in\mathbb{R}^{N}$. The
objective is thus to find parameters $\bm{\theta}$ such that
$P^{(m)}_{j}(0,t_{n})=P^{\star}(0,t_{n})$ for $n=1,2,\ldots,N$, where
$P^{\star}$ represents the market zero-coupon bond prices. Those parameters
are called the calibrated parameters and are denoted by a star
$\bm{\theta}^{\star}$.

######  Proposition C.1.

Given $R_{0}=R^{(m)}_{0}=y_{j}\in\mathcal{S}_{Y}^{(m)}$, we have that

(44)
$\theta^{\star}_{n}:=\left\\{\begin{array}[]{ll}-\frac{1}{t_{n}}\ln\left(\frac{P^{\star}(0,t_{n})}{\widetilde{P}^{(m)}_{j}(0,t_{n})}\right)&\text{if
}n=1\\\
-\frac{1}{t_{n}-t_{n-1}}\ln\left(\frac{P^{\star}(0,t_{n})}{P^{\star}(0,t_{n-1})}\frac{\widetilde{P}_{j}^{(m)}(0,t_{n-1})}{\widetilde{P}_{j}^{(m)}(0,t_{n})}\right)&\text{if
}n=2,3,\ldots,N,\end{array}\right.$

where $\widetilde{P}_{j}^{(m)}$ is defined in (42).

The proof is straightforward from Corollary C.1.

Input: Initialize $\mathbf{Q}^{(m)}_{Y}$ and let $t\mapsto P^{\star}(0,t)$ be
the current market zero-bond curve

$N\in\mathbb{N}$, the number of time steps

$\Delta_{N}\leftarrow T/N$, the size of a time step

1 Set $t_{n}=n\Delta_{N}$, $n=1,2,\ldots,N$

2 Set $\mathbf{D}_{Y}\leftarrow\text{diag}(\bm{y})$ with
$\bm{y}=(y_{1},y_{2},\ldots,y_{m})$, $y_{k}\in\mathcal{S}^{(m)}_{Y}$,
$k=1,2,\ldots,m$

/* Adjusted transition probability matrix of $Y$ over a period of length
$\Delta_{N}$ */

3 $\mathbf{A}_{\Delta_{N}}\leftarrow
e^{(\mathbf{Q}^{(m)}_{Y}-\mathbf{D}_{Y})\Delta_{N}}$

/* Calibration to the current market zero-bond curve $t\mapsto P^{\star}(0,t)$
*/

4 Set
$\widetilde{\mathbf{P}}(t_{1})\leftarrow\mathbf{A}_{\Delta_{N}}\mathbf{1}_{m\times
1}$

5 Set
$\theta_{1}^{\star}=-\frac{1}{t_{1}}\ln\left(\frac{P^{\star}(0,t_{1})}{\mathbf{e}_{j}\widetilde{\mathbf{P}}(t_{1})}\right)$

6 for  _$n=2,\ldots,N$_ do

7
$\widetilde{\mathbf{P}}(t_{n})\leftarrow\mathbf{A}_{\Delta_{N}}\widetilde{\mathbf{P}}(t_{n-1})$

8
$\theta_{n}^{\star}=-\frac{1}{t_{n}-t_{n-1}}\ln\left(\frac{P^{\star}(0,t_{n})}{P^{\star}(0,t_{n-1})}\frac{\mathbf{e}_{j}\widetilde{\mathbf{P}}(t_{n-1})}{\mathbf{e}_{j}\widetilde{\mathbf{P}}(t_{n})}\right)$

9return $\\{\theta_{n}^{\star}\\}_{n=1}^{N}$

Algorithm 2 Calibration of $\bm{\theta}$ to the Current Market Term-Structure
- Extended Models of Brigo and Mercurio (2006)

The calibrated parameters in (44) can easily be obtained by calculating the
zero-coupon bond price at each time $\\{t_{1},t_{2},\ldots,t_{n}\\}$. This
procedure involves calculating matrix exponentials at each time step, which
can slow down the execution considerably. However, by taking advantage of the
homogeneous property of $Y$, the matrix exponentials can be calculated only
once at the beginning of the procedure, which makes it highly efficient. This
is illustrated in Algorithm 2. In Algorithm 2,
$\widetilde{\mathbf{P}}(t_{n}):=[\widetilde{P}_{j}^{(m)}(0,t_{n})]_{j=1}^{m}$
is a column vector of size $m\times 1$, representing the current zero-coupon
bond prices of the auxiliary process $Y^{(m)}$ for maturity $t_{n}$,
$n=1,2,\ldots,n$, with $\widetilde{P}_{j}^{(m)}(0,t_{n})$ defined in (42).

##  Appendix D Algorithms

This section presents the results of Section 5 into an algorithm format. More
precisely, Proposition 5.3 is provided in Algorithm 4. Using the results of
Mackay et al. (2023) Proposition 4.3, a fast version of Proposition 5.1 for
the pricing of European-style CBs is also provided in Algorithm 3, whereas the
fast-version of Proposition 5.3 for the pricing American-style CBs is provided
in Algorithm 5.

###  D.1. European-Style Convertible Bond

Using the tower property of conditional expectations and Proposition D.1
inspired from the work of Mackay et al. (2023), we present a new algorithm
(Algorithm 3) that speeds up the pricing of European-style CBs.

######  Proposition D.1 (Mackay et al. (2023), Proposition 4.3).

Let $h>0$ with $h\ll T$ and $0\leq t\leq T-h$. For any function $\phi$ for
which the expectation on the left-hand side of (45) is finite, we have that

(45)
$\mathbb{E}\left[\phi\left(t+h,X^{(m,M)}_{t+h},R^{(m)}_{t+h}\right)\big{|}X^{(m,M)}_{t}=x_{i},R^{(m)}_{t}=r_{j}\right]\\\
=\sum_{k=1}^{m}\mathbb{E}\left[\phi\left(t+h,X^{(m,M)}_{t+h},R^{(m)}_{t+h}\right)\big{|}R^{(m)}_{t}=R^{(m)}_{t+h}=r_{k},X^{(m,M)}_{t}=x_{i}\right]\\\
\times\mathbb{P}\left(R^{(m)}_{t+h}=r_{k}|R^{(m)}_{t}=r_{j}\right)+\widehat{c}(h),$

where $\widehat{c}(h)$ denotes a function satisfying $\lim_{h\rightarrow
0}\frac{\widehat{c}(h)}{h}=0$.

In their paper, Mackay et al. (2023) work with stochastic volatility models.
However, the reasoning behind the proof is the same for stochastic interest
rate models as in the present context.

The following notation is used in Algorithm 3.

  1. (1)

$\mathbf{B}=[b_{kl}]_{k,l=1}^{m,M}$ denotes a matrix of size $m\times M$,
containing the value of the CB.

  2. (2)

$\mathbf{B}_{*,l}=[b_{kl}]_{k=1}^{m}$ denotes the $l$-th column of
$\mathbf{B}$, $l=1,2,\ldots,M$,

  3. (3)

$\mathbf{B}_{k,*}=[b_{kl}]_{l=1}^{M}$ denotes the $k$-th row of $\mathbf{B}$,
$k=1,2,\ldots,m$.

  4. (4)

The symbol $\top$ indicates the matrix (vector) transpose operation.

Input: Initialize $\mathbf{Q}_{n}^{(m)}$ as in (5), and
$\mathbf{\Lambda}^{(n,M)}_{k}$ as in (8), for $k=1,2,\ldots,m$,
$n=1,2,\ldots,N$

$N\in\mathbb{N}$, the number of time steps,

$\Delta_{N}\leftarrow T/N$, the size of a time step

1 For each $k\in\\{1,2,\ldots,m\\}$, set $\mathbf{B}_{k,*}\leftarrow\left[\eta
e^{x_{l}+\rho f(r_{k})}\mathbf{1}_{\\{e^{x_{l}+\rho f(r_{k})}\geq
F/\eta\\}}+e^{-\int_{0}^{T}c_{u}\,\mathrm{d}u}F\mathbf{1}_{\\{e^{x_{l}+\rho
f(r_{k})}<F/\eta\\}}\right]_{l=1}^{M}$

2 for  _$n=N-1,\ldots,0$_ do

3 for  _$k=1,2,\ldots,m$_ do

4 $\mathbf{P}^{X}_{n,k}\leftarrow
e^{\mathbf{\Lambda}_{k}^{(n+1,M)}\Delta_{N}}e^{-r_{k}\Delta_{N}}$

5 $\mathbf{E}_{*,k}\leftarrow\mathbf{P}^{X}_{n,k}\mathbf{B}_{k,*}^{\top}$

6 $\mathbf{P}_{n}^{R}\leftarrow e^{\mathbf{Q}^{(m)}_{n+1}\Delta_{N}}$

7 for  _$l=1,2,\ldots,M$_ do

8 $\mathbf{B}_{*,l}\leftarrow\mathbf{P}_{n}^{R}\mathbf{E}_{l,*}^{\top}$

9

10return $b_{ji}$

Algorithm 3 European-style CB – Fast Algorithm

At each time step $n\in\\{1,2,\ldots,N\\}$, Proposition D.1 allows the
separation of the matrices $\mathbf{\Lambda}^{(n,M)}_{k}$ and
$\mathbf{Q}^{(m)}_{n}$. Hence, the matrix exponential of a large sparse matrix
$\mathbf{G}_{n}^{(mM)}$ of size $mM\times mM$ is replaced by $m$ calculations
of the exponential of an $M\times M$ matrix and one calculation of the
exponential of an $m\times m$ matrix. Numerical experiments in Section 6 show
the accuracy and efficiency of the fast algorithm empirically.

###  D.2. Convertible Bond (American-style)

Based on Proposition 5.3, Algorithm 4 provides the CTMC approximation for the
value of a CB given that $X_{0}^{(m,M)}=\ln(S_{0})-\rho
f(R_{0})=x_{i}\in\mathcal{S}^{(M)}_{X}$ and
$R_{0}^{(m)}=R_{0}=r_{j}\in\mathcal{S}^{(m)}_{R}$. The algorithm is set up for
zero-coupon CBs with no additional features, such as call and put options.
However, such extensions are straightforward and are discussed in Remark 5.4.

Similar to the European-style CB, the performance of Algorithm 4 can be
increased by assuming that the short-rate process is constant over small time
periods (Proposition D.1). Let ${\widetilde{\mathbf{H}}:=[\eta e^{x_{l}+\rho
f(r_{k})}]_{k,l=1}^{m,M}}$ be an be an $m\times M$ matrix representing the
conversion value. At each time step, matrices $\widetilde{\mathbf{B}}^{E}$,
$\widetilde{\mathbf{B}}^{CO}$, and $\widetilde{\mathbf{B}}$, of size $m\times
M$, contain the equity part, cash-only part, and the whole value of the CB,
respectively. Furthermore, we denote by $\widetilde{b}_{ij}$ (resp
$\widetilde{h}_{ij}$), the $(i,j)$-entry of $\widetilde{\mathbf{B}}$ (resp.
$\widetilde{\mathbf{H}}$), and define the matrix indicator
$\mathbf{1}_{\\{\widetilde{\mathbf{B}}=\widetilde{\mathbf{H}}\\}}$, where each
element $(i,j)$ of the matrix, denoted
$\mathbf{1}_{\\{\widetilde{\mathbf{B}}=\widetilde{\mathbf{H}}\\}}(i,j)$, is
given by
$\mathbf{1}_{\\{\widetilde{\mathbf{B}}=\widetilde{\mathbf{H}}\\}}(i,j)=\mathbf{1}_{\\{\widetilde{b}_{ij}=\widetilde{h}_{ij}\\}},$
for $1\leq i\leq m$, $1\leq j\leq M$. The fast Algorithm to value CBs is
provided in Algorithm 5.

Input: Initialize $\mathbf{G}_{n}^{(mM)}$ as in (10) for $n=1,2,\ldots,N$,
$\mathbf{H}^{CO}$ as in (29), and $\mathbf{H}^{E}_{n}$ as in (30), for
$n=0,1,\ldots,N$

$N\in\mathbb{N}$, the number of time steps,

$\Delta_{N}\leftarrow T/N$, the size of a time step

1 Set $\mathbf{D}_{mM}\leftarrow\text{diag}\left(\bm{d}\right)$ with
$\bm{d}=(d_{1},d_{2},\ldots,d_{mM})$, and
${d_{(k-1)M+l}=r_{k}\in\mathcal{S}_{R}^{(m)}}$, $k=1,2,,\ldots,m$,
$l=1,2,,\ldots,M$

2 Set $\mathbf{B}_{N}^{CO}\leftarrow\mathbf{H}^{CO}$,
$\mathbf{B}_{N}^{E}\leftarrow\mathbf{H}_{N}^{E}$,
$\mathbf{B}_{N}\leftarrow\mathbf{B}^{E}_{N}+\mathbf{B}^{CO}_{N}$

3 for  _$n=N-1,N-2,\ldots,0$_ do

4
$\mathbf{A}_{n+1}\leftarrow\exp\left\\{\Delta_{N}\left(\mathbf{G}^{(mM)}_{n+1}-\mathbf{D}_{mM}\right)\right\\}$,

5 $\mathbf{B}_{n}^{CO}\leftarrow
e^{-\int_{t_{n}}^{t_{n+1}}c_{u}\,\mathrm{d}u}\mathbf{A}_{n+1}\mathbf{B}_{n+1}^{CO}$,
$\mathbf{B}_{n}^{E}\leftarrow\mathbf{A}_{n+1}\mathbf{B}_{n+1}^{E}$,

6
$\mathbf{B}_{n}\leftarrow\max\left(\mathbf{H}_{n}^{E},\mathbf{B}_{n}^{E}+\mathbf{B}_{n}^{CO}\right)$

7 $\mathbf{B}_{n}^{CO}\leftarrow\mathbf{B}^{CO}_{n}\left(\mathbf{1}_{mM\times
1}-\mathbf{1}_{\\{\mathbf{B}_{n}=\mathbf{H}_{n}^{E}\\}}\right)$,
$\mathbf{B}_{n}^{E}=\mathbf{B}_{n}-\mathbf{B}_{n}^{CO}$

8

9

10$v^{(m,M)}(0,S_{0},R_{0})\leftarrow\mathbf{e}_{ji}\mathbf{B}_{0}$

return $v^{(m,M)}(0,S_{0},R_{0})$

Algorithm 4 American-style CB

Input: Initialize $\mathbf{Q}_{n}^{(m)}$ as in (5) and
$\mathbf{\Lambda}^{(n,M)}_{k}$ as in (8), for $k=1,2,\ldots,m$,
$n=1,2,\ldots,N$

$N\in\mathbb{N}$, the number of time steps,

$\Delta_{N}\leftarrow T/N$, the size of a time step

1 Set ${\widetilde{\mathbf{H}}:=[\eta e^{x_{l}+\rho f(r_{k})}]_{k,l=1}^{m,M}}$

2 Set $\widetilde{\mathbf{B}}_{k,*}^{E}\leftarrow\left[\eta e^{x_{l}+\rho
f(r_{k})}\mathbf{1}_{\\{e^{x_{l}+\rho f(r_{k})}\geq
F/\eta\\}}\right]_{l=1}^{M}$, $k=1,2,\ldots,m$

3 Set
$\widetilde{\mathbf{B}}_{k,*}^{CO}\leftarrow\left[F\mathbf{1}_{\\{e^{x_{l}+\rho
f(r_{k})}<F/\eta\\}}\right]_{l=1}^{M}$, $k=1,2,\ldots,m$

4
$\widetilde{\mathbf{B}}\leftarrow\widetilde{\mathbf{B}}^{E}+\widetilde{\mathbf{B}}^{CO}$

5 for  _$n=N-1,\ldots,0$_ do

6 for  _$k=1,2,\ldots,m$_ do

7 $\mathbf{P}^{X}_{n,k}\leftarrow
e^{\mathbf{\Lambda}_{k}^{(n+1,M)}\Delta_{N}}e^{-r_{k}\Delta_{N}}$

8 $\mathbf{E}^{CO}_{*,k}\leftarrow
e^{-\int_{t_{n}}^{t_{n+1}}c_{u}\,\mathrm{d}u}\mathbf{P}^{X}_{n,k}(\widetilde{\mathbf{B}}^{CO}_{k,*})^{\top}$,
$\mathbf{E}^{E}_{*,k}\leftarrow\mathbf{P}^{X}_{n,k}(\widetilde{\mathbf{B}}^{E}_{k,*})^{\top}$

9 $\mathbf{P}_{n}^{R}\leftarrow e^{\mathbf{Q}^{(m)}_{n+1}\Delta_{N}}$

10 for  _$l=1,2,\ldots,M$_ do

11
$\widetilde{\mathbf{B}}_{*,l}^{CO}\leftarrow\mathbf{P}_{n}^{R}(\mathbf{E}_{l,*}^{CO})^{\top}$,
$\widetilde{\mathbf{B}}_{*,l}^{E}\leftarrow\mathbf{P}_{n}^{R}(\mathbf{E}_{l,*}^{E})^{\top}$,
$\widetilde{\mathbf{B}}\leftarrow\max\left(\widetilde{\mathbf{H}},\widetilde{\mathbf{B}}^{E}+\widetilde{\mathbf{B}}^{CO}\right)$

12
$\widetilde{\mathbf{B}}^{CO}\leftarrow\widetilde{\mathbf{B}}^{CO}\left(\mathbf{1}_{m\times
M}-\mathbf{1}_{\\{\widetilde{\mathbf{B}}=\widetilde{\mathbf{H}}\\}}\right)$,
$\widetilde{\mathbf{B}}^{E}=\widetilde{\mathbf{B}}-\widetilde{\mathbf{B}}^{CO}$

13

14return $b_{ji}$

Algorithm 5 American-style CB – Fast Algorithm

##  Appendix E Supplemental Material

This document provides supplemental material to A Unifying Approach for the
Pricing of Debt Securities.

###  E.1. Closed-Form Expression for European-style Convertible Bonds under TF
approach

In the following, we derive a closed-form analytical formula for European-
style CBs (or when the conversion option can only be exercised at maturity).
We suppose that credit risk is incorporated into the model using the approach
of Tsiveriotis and Fernandes (1998).

Accordingly, we make some simplifying assumptions. We suppose that the
dynamics of the stock process are given by

(46) $\displaystyle\,\mathrm{d}S_{t}$
$\displaystyle=R_{t}S_{t}\,\mathrm{d}t+\widetilde{\sigma}_{S}S_{t}\left(\rho\,\mathrm{d}\widetilde{W}^{(2)}_{t}+\sqrt{1-\rho^{2}}\,\mathrm{d}\widetilde{W}^{(1)}_{t}\right),$
$\displaystyle\,\mathrm{d}R_{t}$ $\displaystyle=(\theta(t)-\kappa
R_{t})\,\mathrm{d}t+\widetilde{\sigma}_{R}\,\mathrm{d}\widetilde{W}_{t}^{(2)},$

with $\widetilde{\sigma}_{S},\widetilde{\sigma}_{R}>0$, $\rho\in[-1,1]$ and
$\widetilde{W}=\\{(\widetilde{W}^{(1)}_{t},\widetilde{W}^{(2)}_{t})\\}_{t\geq
0}$ is a standard bi-dimensional Brownian motion171717The formulation in (46)
in terms of independent Brownian motion is equivalent to that in (1) in terms
of correlated Brownian motion. Indeed, define
$Z_{t}=\rho\widetilde{W}^{(2)}_{t}+\sqrt{1-\rho^{2}}\widetilde{W}^{(1)}_{t}$,
$t\geq 0$. From Cholesky decomposition, the process $(Z,\widetilde{W}^{(2)})$
is a correlated Brownian motion with cross-variation
$[Z,\widetilde{W}^{(2)}]_{t}=\rho t$.. Under this assumption, the short-rate
process corresponds to the Hull–White model in Table 2. When function
$\theta(\cdot)$ is constant over time, then the short-rate dynamics collapsed
to the Vasicek model181818It suffices to set
$\theta(t)=\widetilde{\theta}\kappa$, for $0\leq t\leq T$, for some constant
$\widetilde{\theta}>0$ to obtain the Vasicek model with a long-term mean
parameter equal to $\widetilde{\theta}$., whereas when $\kappa=0$, the Ho–Lee
model is obtained. Under these three models, the short-rate process is
Gaussian, and the price of the zero-coupon bond can be obtained explicitly by

(47) $P(t,T)=e^{A(t,T)-B(t,T)R_{t}},$

for some time-deterministic functions $A$ and $B$ given in Table 12, see Björk
(2009), Section 24.4 for details. Finally, note that when
$\sigma_{R}(\cdot)=\kappa=\theta(\cdot)=0$, then the short-rate process is
constant to $R_{0}$ and (46) collapsed to the Black–Scholes model.

Model | $A(t,T)$ | $B(t,T)$
---|---|---
Vasicek | $\left(\theta-\frac{\widetilde{\sigma}_{R}^{2}}{2\kappa^{2}}\right)\left[B(t,T)-(T-t)\right]-\frac{\widetilde{\sigma}_{R}^{2}}{4\kappa}B^{2}(t,T)$ | $\frac{1}{\kappa}[1-e^{-\kappa(T-t)}]$
Ho–Lee | $\int_{t}^{T}\theta(s)(s-T)\,\mathrm{d}s+\frac{\widetilde{\sigma}_{R}^{2}}{2}\frac{(T-t)^{3}}{3}$ | $T-t$
Hull–White | $\int_{t}^{T}\frac{1}{2}\widetilde{\sigma}_{R}^{2}B(s,T)-\theta(s)B(s,T)\,\mathrm{d}s$ | $\frac{1}{\kappa}[1-e^{-\kappa(T-t)}]$
Table 12. Definition of $A(t,T)$ and $B(t,T)$ for different models of Tables 1
and 2.

The following proposition provides a closed-form pricing formula for European-
style CBs under general stochastic short-rate models (46). The pricing of
European call options under stochastic interest rate are discussed in Geman et
al. (1995) (Theorem 2 and Section 3.2), Björk (2009) (Section 26.5), and Brigo
and Mercurio (2006) (Appendix B), among others. The general proof uses the
change of numéraire techniques developed by Geman et al. (1995). An
alternative derivation is also provided in Abudy and Izhakian (2013) when the
short-rate dynamics is given by the Ornstein–Uhlenbeck process (or,
equivalently, the Vasicek model).

######  Proposition E.1.

Given $S_{t}=x>0$ and $R_{t}=r\in\mathbb{R}$, the price at time $t$ of a
European-style CB with maturity $T>0$, face value $F>0$, and conversion ratio
$\eta>0$ is given by

(48) $\displaystyle v_{e}(t,r,x)=\eta
x\Phi(d_{1})+e^{-\int_{t}^{T}c_{u}\,\mathrm{d}u}P(t,T)F\Phi(d_{2})$

where $\Phi(\cdot)$ denotes the cumulative distribution of a standard normal
distribution, ${d_{1}=\frac{\ln\frac{\eta
x}{FP(t,T)}+\frac{1}{2}V(t,T)}{\sqrt{V(t,T)}}}$, and
${d_{2}=\sqrt{V(t,T)}-d_{1}}$, with

$V(t,T)=\begin{cases}\widetilde{\sigma}_{S}^{2}(T-t)+\frac{2\rho\widetilde{\sigma}_{S}\widetilde{\sigma}_{R}}{\kappa}\left[(T-t)-B(t,T)\right]+\frac{\widetilde{\sigma}_{R}^{2}}{\kappa^{2}}\left[(T-t)-\frac{\kappa}{2}B^{2}(t,T)-B(t,T)\right]&\textrm{if
}\kappa\neq 0\\\
\widetilde{\sigma}_{S}^{2}(T-t)+\rho\widetilde{\sigma}_{S}\widetilde{\sigma}_{R}(T-t)^{2}+\frac{\widetilde{\sigma}_{R}^{2}}{3}(T-t)^{3}&\textrm{if
}\kappa=0,\end{cases}$

and ${P(t,T)=e^{A(t,T)+B(t,T)r}}$, where functions $A$ and $B$ are defined in
Table 12.

###### Proof.

Recall from (21) that

(49) $v_{e}(t,x,r)=\mathbb{E}\Big{[}e^{-\int_{t}^{T}R_{u}\,\mathrm{d}u}\eta
S_{T}\mathbf{1}_{\\{S_{T}\geq
F/\eta\\}}+e^{-\int_{t}^{T}R_{u}+c_{u}\,\mathrm{d}u}F\mathbf{1}_{\\{S_{T}<F/\eta\\}}\big{|}S_{t}=x,R_{t}=r\Big{]}.$

To solve this problem and avoid working with the joint density of
$(\int_{t}^{T}R_{u}\,\mathrm{d}u,S_{T})$, we use the change of numéraire
technique discussed in Geman et al. (1995), Theorem 2 and Section 3.2 (a).
Consequently, we introduce the $T$-forward measure, denoted by
$\mathbb{Q}^{T}$, which has the zero-coupon bond price process
$P=\\{P(t,T)\\}_{0\leq t\leq T}$ as numéraire. We also introduce the measure
$\mathbb{Q}_{S}$ under which the risky asset $S$ is the chosen numéraire.
Following Geman et al. (1995), Theorem 2, we have that

$\displaystyle v_{e}(t,x,r)$
$\displaystyle=P(t,T)\mathbb{E}^{\mathbb{Q}^{T}}\left[\eta
S_{T}\mathbf{1}_{\\{S_{T}\geq F/\eta\\}}\big{|}S_{t}=x,R_{t}=r\right]$
$\displaystyle\quad\quad+e^{-\int_{t}^{T}c_{u}\,\mathrm{d}u}P(t,T)F\mathbb{Q}^{T}\left(S_{T}<F/\eta\big{|}S_{t}=x,R_{t}=r\right)$
(50) $\displaystyle\begin{split}&=\eta x\mathbb{Q}_{S}\left(S_{T}\geq
F/\eta\big{|}S_{t}=x,R_{t}=r\right)\\\
&\quad\quad+e^{-\int_{t}^{T}c_{u}\,\mathrm{d}u}P(t,T)F\mathbb{Q}^{T}\left(S_{T}<F/\eta\big{|}S_{t}=x,R_{t}=r\right),\end{split}$

where the first term of the second equality follows from Corollary 2 of Geman
et al. (1995), and $\mathbb{E}^{\mathbb{Q}^{T}}[\cdot]$ denotes the
expectation under the $T$-forward measure. To solve (50), we need to find the
distribution of $S_{T}$ under measures $\mathbb{Q}^{T}$ and $\mathbb{Q}^{S}$,
respectively. This is what we do in the following.

The first step to obtain the dynamics of $S$ under $\mathbb{Q}^{T}$ consists
of finding the dynamics of the zero-coupon bond price process $P$ under
$\mathbb{Q}$. Accordingly, recall from (47) that

$P(t,T)=e^{A(t,T)-B(t,T)R_{t}},$

for some time-deterministic functions $A$ and $B$ defined in Table 12, and
note that

$A_{t}(t,T):=\frac{\partial A}{\partial
t}(t,T)=\theta(t)B(t,T)-\frac{1}{2}\widetilde{\sigma}_{R}^{2}B^{2}(t,T),\textrm{
and }B_{t}(t,T):=\frac{\partial B}{\partial t}(t,T)=\kappa B(t,T)-1,$

see Björk (2009), Section 24.4. Hence, applying Ito’s formula to $\ln P(t,T)$,
we obtain that

$\displaystyle\,\mathrm{d}\ln P(t,T)$
$\displaystyle=\left(A_{t}(t,T)-B_{t}(t,T)R_{t}\right)\,\mathrm{d}t-B(t,T)\,\mathrm{d}R_{t}$
$\displaystyle=\left[\theta(t)B(t,T)-\frac{1}{2}\widetilde{\sigma}_{R}^{2}B^{2}(t,T)-B_{t}(t,T)R_{t}-B(t,T)\left(\theta(t)-\kappa
R_{t}\right)\right]\,\mathrm{d}t-\widetilde{\sigma}_{R}B(t,T)\,\mathrm{d}\widetilde{W}^{(2)}_{t}$
(51)
$\displaystyle=\left[R_{t}-\frac{1}{2}\widetilde{\sigma}_{R}^{2}B^{2}(t,T)\right]\,\mathrm{d}t-\widetilde{\sigma}_{R}B(t,T)\,\mathrm{d}\widetilde{W}^{(2)}_{t},$

so that,

(52) $\displaystyle\,\mathrm{d}P(t,T)$
$\displaystyle=R_{t}P(t,T)\,\mathrm{d}t-\widetilde{\sigma}_{R}B(t,T)P(t,T)\,\mathrm{d}\widetilde{W}^{(2)}_{t}.$

Now, observe that

(53)
$\mathbb{Q}^{T}\left(S_{T}<F/\eta\big{|}S_{t}=x,R_{t}=r\right)=\mathbb{Q}^{T}\left(\frac{S_{T}}{P(T,T)}<F/\eta\big{|}S_{t}=x,R_{t}=r\right).$

From Theorem 1 (i) of Geman et al. (1995), we know that process
$f=\\{f_{t}:=\frac{S_{t}}{P(t,T)}\\}_{0\leq t\leq T}$ is a
$\mathbb{Q}^{T}$-local martingale (thus, it as no drift). Moreover, since a
change of measures only affects the drift of a process, we can deduce the
dynamics of process $f$ under $\mathbb{Q}^{T}$ from its dynamics under
$\mathbb{Q}$ by setting the drift term to nil, which we do next. Using (46) ,
(52) and Itô’s formula, we find that

$\displaystyle\,\mathrm{d}f_{t}$
$\displaystyle=\frac{1}{P(t,T)}\,\mathrm{d}S_{t}-\frac{S_{t}}{P^{2}(t,T)}\,\mathrm{d}P(t,T)-\frac{1}{P^{2}(t,T)}\,\mathrm{d}\langle
S_{t},P(t,T)\rangle+\frac{S_{t}}{P^{3}(t,T)}\,\mathrm{d}\langle P(t,T)\rangle$
(54)
$\displaystyle=\left(\rho\widetilde{\sigma}_{S}\widetilde{\sigma}_{R}B(t,T)+\widetilde{\sigma}_{R}^{2}B^{2}(t,T)\right)f_{t}\,\mathrm{d}t+\widetilde{\sigma}_{S}\sqrt{1-\rho^{2}}f_{t}\,\mathrm{d}\widetilde{W}^{(1)}_{t}+\left[\rho\widetilde{\sigma}_{s}+\widetilde{\sigma}_{R}B(t,T)\right]f_{t}\,\mathrm{d}\widetilde{W}^{(2)}_{t}.$

Because $f$ must be a local martingale under $\mathbb{Q}^{T}$, we deduce from
Girsanov theorem that process $(\widehat{W}^{(1)},\widehat{W}^{(2)})$, defined
by

(55)
$\begin{split}\,\mathrm{d}\widehat{W}^{(1)}_{t}&=\,\mathrm{d}\widetilde{W}^{(1)}_{t},\\\
\,\mathrm{d}\widehat{W}^{(2)}_{t}&=\,\mathrm{d}\widetilde{W}^{(2)}_{t}+\widetilde{\sigma}_{R}B(t,T)\,\mathrm{d}t,\end{split}$

for $0\leq t\leq T$, is a standard bi-dimensional Brownian motion under
$\mathbb{Q}^{T}$, which implies that

(56) $\displaystyle\,\mathrm{d}f_{t}$
$\displaystyle=\widetilde{\sigma}_{S}\sqrt{1-\rho^{2}}f_{t}\,\mathrm{d}\widehat{W}^{(1)}_{t}+\left[\rho\widetilde{\sigma}_{s}+\widetilde{\sigma}_{R}B(t,T)\right]f_{t}\,\mathrm{d}\widehat{W}^{(2)}_{t}.$

Hence, $\ln f_{T}$ given $\ln f_{t}$ is normally distributed with a mean
$\widehat{\mu}:=\ln f_{t}-\frac{1}{2}V(t,T)$ (see for instance, Björk (2009),
Lemma 4.15), and variance $\widehat{\sigma}^{2}:=V(t,T)$, where

$\displaystyle V(t,T)$
$\displaystyle=\int_{t}^{T}\widetilde{\sigma}_{S}^{2}(1-\rho^{2})+\left[\rho\widetilde{\sigma}_{S}+\widetilde{\sigma}_{R}B(s,T)\right]^{2}\,\mathrm{d}s$
(57)
$\displaystyle=\widetilde{\sigma}_{S}^{2}(T-t)+2\rho\widetilde{\sigma}_{S}\widetilde{\sigma}_{R}\int_{t}^{T}B(s,T)\,\mathrm{d}s+\widetilde{\sigma}_{R}^{2}\int_{t}^{T}B^{2}(s,T)\,\mathrm{d}s.$

The probability (53) can thus be calculated explicitly using the property of
the normal distribution, and we obtain that,

(58)
$\mathbb{Q}^{T}\left(f_{T}<F/\eta\big{|}f_{t}=x/P(t,T)\right)=\Phi\left(\frac{\ln\left(\frac{FP(t,T)}{\eta
x}\right)+\frac{1}{2}V(t,T)}{\sqrt{V(t,T)}}\right)=\Phi(d_{2}).$

This completes the proof for the second term of (50).

The challenge now consists of finding an expression for the first term of
(50). We thus need the distribution of $S_{T}$ under $\mathbb{Q}_{S}$. To this
end, we observe that

(59) $\begin{split}\mathbb{Q}_{S}\left(S_{T}\geq
F/\eta\big{|}S_{t}=x,R_{t}=r\right)&=\mathbb{Q}_{S}\left(1/S_{T}\leq\eta/F\big{|}S_{t}=x,R_{t}=r\right)\\\
&=\mathbb{Q}_{S}\left(P(T,T)/S_{T}\leq\eta/F\big{|}S_{t}=x,R_{t}=r\right)\\\
&=\mathbb{Q}_{S}\left(1/f_{T}\leq\eta/F\big{|}f_{t}=x/P(t,T)\right)\end{split}$

Similarly, as above, we use the results of Theorem 1 of Geman et al. (1995) to
conclude that process ${Y=\\{Y_{t}:=P(t,T)/S_{t}=1/f_{t}\\}_{0\leq t\leq T}}$
is a local martingale under $\mathbb{Q}_{S}$. Thus, it has no drift, and
because a change of measures only affects the drift of a process, we can
deduce the dynamics of process $Y$ under $\mathbb{Q}_{S}$ from its dynamics
under $\mathbb{Q}^{T}$, by setting the drift term to nil. This what we do in
the following. Using (56) and Itô’s formula, we find that

$\displaystyle\,\mathrm{d}Y_{t}$
$\displaystyle=-\frac{1}{f_{t}^{2}}\,\mathrm{d}f_{t}+\frac{1}{f_{t}^{3}}\,\mathrm{d}\langle
f_{t}\rangle$
$\displaystyle=\left[\widetilde{\sigma}_{S}^{2}(1-\rho^{2})+(\rho\widetilde{\sigma}_{S}+\widetilde{\sigma}_{R}B(t,T))^{2}\right]Y_{t}\,\mathrm{d}t$
$\displaystyle\quad\quad\quad-\widetilde{\sigma}_{S}\sqrt{1-\rho^{2}}Y_{t}\,\mathrm{d}\widehat{W}^{(1)}_{t}-\left[\rho\widetilde{\sigma}_{s}+\widetilde{\sigma}_{R}B(t,T)\right]Y_{t}\,\mathrm{d}\widehat{W}^{(2)}_{t}$

From the local martingale property of $Y$ under $\mathbb{Q}_{S}$, we deduce
from the Girsanov theorem that process $(\bar{W}^{(1)},\bar{W}^{(2)})$,
defined by

(60)
$\begin{split}\,\mathrm{d}\bar{W}^{(1)}_{t}&=\,\mathrm{d}\widehat{W}^{(1)}_{t}-\widetilde{\sigma}_{S}\sqrt{1-\rho^{2}}\,\mathrm{d}t,\\\
\,\mathrm{d}\bar{W}^{(2)}_{t}&=\,\mathrm{d}\widehat{W}^{(2)}_{t}-\rho\widetilde{\sigma}_{S}+\widetilde{\sigma}_{R}B(t,T)\,\mathrm{d}t,\end{split}$

for $0\leq t\leq T$, is a standard bi-dimensional Brownian motion under
$\mathbb{Q}_{S}$. Thus, the dynamics of $Y$ under the new measure are given by

$\displaystyle\,\mathrm{d}Y_{t}=-\widetilde{\sigma}_{S}\sqrt{1-\rho^{2}}Y_{t}\,\mathrm{d}\bar{W}^{(1)}_{t}-\left[\rho\widetilde{\sigma}_{s}+\widetilde{\sigma}_{R}B(t,T)\right]Y_{t}\,\mathrm{d}\bar{W}^{(2)}_{t}.$

From there, we conclude that $\ln Y_{T}$ given $\ln Y_{t}$ (or $\ln f_{t}$,
since $\ln Y_{t}=\ln 1/f_{t}$) is normally distributed with mean
${\bar{\mu}:=-\ln f_{t}-\frac{1}{2}V(t,T)}$ and variance
${\bar{\sigma}^{2}:=\widehat{\sigma}^{2}=V(t,T)}$, and the probability (59)
can be calculated explicitly as

(61)
$\mathbb{Q}_{S}\left(1/f_{T}\leq\eta/F\big{|}f_{t}=x/P(t,T)\right)=\Phi\left(\frac{\ln\left(\frac{\eta
x}{FP(t,T)}\right)+\frac{1}{2}V(t,T)}{\sqrt{V(t,T)}}\right)=\Phi(d_{1}).$

The final assertion then follows from (50),(57), (58), (61), and the
expression for function $B$ in Table 12.

∎

######  Remark E.1.

When $\sigma_{R}=\kappa=\theta(t)=0$ for all $t\in[0,T]$, the short-rate is
constant to $R_{0}$ over time and the model in (46) collapsed to the
Black–Scholes model. In that case, $A(t,T)=0$, $B(t,T)=T-t$, such that
$P(t,T)=e^{-(T-t)R_{0}}$, and $V(t,T)=\sigma_{S}^{2}(T-t)$. The general
formula (48) then becomes

(62) $\displaystyle v_{e}(t,r,x)=\eta
x\Phi(d_{1})+e^{-(T-t)R_{0}}e^{-\int_{t}^{T}c_{u}\,\mathrm{d}u}F\Phi(d_{2}),$

with $d_{1}=\frac{\ln\left(\frac{\eta
x}{F}\right)+\left(r+\frac{1}{2}\sigma_{S}^{2}\right)(T-t)}{\sigma_{S}\sqrt{T-t}}$,
and $d_{2}=\sigma_{S}\sqrt{T-t}-d_{1}$.

###  E.2. Additional Numerical Experiments

We now investigate the accuracy, convergence, and efficiency of the CTMC
method to approximate various debt securities in the Vasicek, CIR, and Dothan
models. Note that Assumption 4.1 is not respected under the Vasicek model.

Unless stated otherwise, we use the following model and CTMC parameters in all
numerical experiments, summarized in Table 13.

| ${r_{0}}$ | ${\kappa}$ | ${\theta}$ | ${\sigma}$ | ${m}$ | ${r_{1}}$ | ${r_{m}}$ | $\widetilde{\alpha}$ | $\Delta_{N}$
---|---|---|---|---|---|---|---|---|---
Vasicek | $0.04$ | $1$ | $0.04$ | $0.20$ | $160$ | $-30R_{0}$ | $25R_{0}$ | $0.5$ | 1/252
CIR | $0.04$ | $2$ | $0.035$ | $0.20$ | $160$ | $R_{0}/100$ | $7R_{0}$ | $0.5$ | 1/252
Dothan | 0.02 | 0 | N/A | 0.15 | $160$ | $R_{0}/100$ | $7R_{0}$ | $0.5$ | 1/252
Table 13. Model and CTMC parameters

To construct the state-space of the CTMC,
$\mathcal{S}^{(m)}_{R}=\\{r_{1},r_{2},\ldots,r_{m}\\}$ with $m\in\mathbb{N}$,
we use the non-uniform grid proposed by Tavella and Randall (Tavella and
Randall (2000), Chapter 5.3), as in Section 6.

####  E.2.1. Approximation of Zero-Coupon Bond Prices

We now examine the accuracy of the CTMC methods in calculating zero-coupon
bond prices, Corollary B.1. Under the Vasicek and CIR models, the price of
zero-coupon bonds has a closed-from expression, which can be found, for
instance, in Brigo and Mercurio (2006), Section 3.2. The analytical value of
the zero-coupon bond price can thus be used as a benchmark in our experiment.
We test the accuracy of the approximated prices across different values of
model parameters. The results are summarized in Table 14. Column “CTMC”
reports the CTMC approximated value using the results of Corollary B.1.

$\mathbf{\kappa}$ | CTMC | Benchmark | Abs. error
---|---|---|---
0.5 | 0.9625591 | 0.9625609 | 1.77E-06
1 | 0.8964870 | 0.8964877 | 7.12E-07
2 | 0.8661056 | 0.8661057 | 9.47E-08
3 | 0.8587974 | 0.8587974 | 2.27E-08
4 | 0.8560138 | 0.8560138 | 7.77E-09

$\mathbf{\kappa}$ | CTMC | Benchmark | Abs. error
---|---|---|---
0.5 | 0.8656670 | 0.8656663 | 7.09E-07
1 | 0.8676884 | 0.8676884 | 1.11E-08
2 | 0.8676884 | 0.8676884 | 1.11E-08
3 | 0.8681491 | 0.8681491 | 3.96E-10
4 | 0.8684110 | 0.8684110 | 1.98E-11

$\mathbf{R_{0}}$ | CTMC | Benchmark | Abs. error
---|---|---|---
0.02 | 0.9142555 | 0.9142630 | 7.43E-06
0.03 | 0.9053312 | 0.9053317 | 5.12E-07
0.04 | 0.8964870 | 0.8964877 | 7.12E-07
0.05 | 0.8877292 | 0.8877301 | 8.23E-07

$\mathbf{R_{0}}$ | CTMC | Benchmark | Abs. error
---|---|---|---
0.02 | 0.8763624 | 0.8763627 | 2.82E-07
0.03 | 0.8720147 | 0.8720147 | 2.21E-09
0.04 | 0.8676884 | 0.8676884 | 1.11E-08
0.05 | 0.8633835 | 0.8633835 | 2.90E-08

$\mathbf{\theta}$ | CTMC | Benchmark | Abs. error
---|---|---|---
0.01 | 0.9814527 | 0.9814529 | 1.93E-07
0.02 | 0.9522718 | 0.9522722 | 3.77E-07
0.03 | 0.9239585 | 0.9239590 | 5.50E-07
0.04 | 0.8964870 | 0.8964877 | 7.12E-07

$\mathbf{\theta}$ | CTMC | Benchmark | Abs. error
---|---|---|---
0.015 | 0.9303530 | 0.9303518 | 1.20E-06
0.025 | 0.8984741 | 0.8984740 | 1.82E-07
0.035 | 0.8676884 | 0.8676884 | 1.11E-08
0.045 | 0.8379576 | 0.8379576 | 5.63E-10

$\mathbf{\sigma}$ | CTMC | Benchmark | Abs. error
---|---|---|---
0.1 | 0.8630196 | 0.8630198 | 1.70E-07
0.2 | 0.8964870 | 0.8964877 | 7.12E-07
0.3 | 0.9551747 | 0.9551765 | 1.77E-06
0.4 | 1.0438353 | 1.0438513 | 1.60E-05

(a) Vasicek model

$\mathbf{\sigma}$ | CTMC | Benchmark | Abs. error
---|---|---|---
0.1 | 0.8673140 | 0.8673140 | 1.93E-10
0.2 | 0.8676884 | 0.8676884 | 1.11E-08
0.3 | 0.8683033 | 0.8683025 | 7.69E-07
0.4 | 0.8691384 | 0.8691428 | 4.36E-06

(b) CIR model

Table 14. Accuracy of the zero-coupon bond price approximation (37) under Vasicek and CIR models. Model and CTMC parameters are as listed in Table 13, and zero-coupon bond parameters are $t=0$ and $T=4$. Figure 6. Efficiency of the CTMC method in approximating zero-coupon bond prices under the Dothan model. Except for the number of grid points $m$, which range from $100$ to $1000$, model and CTMC parameters are as listed in Table 13. Zero-coupon bond price parameters are $t=0$ and $T=4$. |
---|---
(a) Vasicek | (b) CIR
Figure 7. Convergence pattern of the approximated zero-coupon bond prices.
Except for the number of grid points $m$, model and CTMC parameters are as
listed in Table 13. Zero-coupon bond price parameters are $t=0$ and $T=4$.

We observe that the approximation achieves a high level of precision across
all parameters, with an average calculation time of $0.009$ seconds,
illustrating the speed of the methodology.

Figures 6 demonstrate the efficiency of CTMC methodology in valuing zero-
coupon bond prices when the short-rate process follows a geometric Brownian
motion (the Dothan model). Thus, alternative methods such as binomial trees
(“Tree”) or Monte Carlo simulation (“Sim”) can be easily implemented. We
compare the performance of the CTMC approximation to the Cox–Ross–Rubinstein
binomial tree (Cox et al. (1979)) and to Monte Carlo simulation. For Monte
Carlo simulation, we use an exact scheme with a number of simulations ranging
from 10,000 to 200,000 with the same number of antithetic variables and
500-time steps per year. The benchmark is calculated using the CTMC method
with $m=5,000$. Figure 6 shows the high efficiency of the CTMC method compared
to these other numerical techniques. Indeed, CTMC approximation clearly
outperforms other methods in terms of both calculation time and precision.

Finally, Figure 7 shows the convergence pattern of the approximated zero-
coupon bond prices as the number of grid points $m$ increases. We observe that
the two models converge smoothly and rapidly to their analytical values.
Moreover, since Assumption 4.1 is not satisfied under the Vacisek model, the
results confirm that theoretical convergence is possible under less
restrictive conditions.

####  E.2.2. Approximation of the Zero-Coupon Bond Option Prices

In this section, we study the accuracy and the numerical convergence of the
zero-coupon bond option prices provided in (38), under the Vasicek and CIR
models. Under these two models, the price of zero-coupon bond options has a
closed-form expression, which can be found in Brigo and Mercurio (2006)
Section 3.2., and thus, can serve as a benchmark in our example. We test the
accuracy of the approximated option prices for different levels of moneyness
and volatilities. The results are summarized in Table 15. Column “price-to-
strike” shows the price-to-strike ratio, calculated as the actual zero-coupon
bond price over the option strike price $K>0$.

$\mathbf{\sigma}$ | price-to-strike | CTMC | Benchmark | Abs. error
---|---|---|---|---
$\mathbf{0.1}$ | 1.67 | 0.38319569 | 0.38319569 | 4.29E-09
1.25 | 0.22325433 | 0.22325433 | 4.25E-09
1.00 | 0.06581737 | 0.06581712 | 2.56E-07
0.83 | 0.01014165 | 0.01014179 | 1.40E-07
0.71 | 0.00000011 | 0.00000011 | 6.79E-10

$\mathbf{\sigma}$ | price-to-strike | CTMC | Benchmark | Abs. error
---|---|---|---|---
$\mathbf{0.1}$ | 1.67 | 0.38326833 | 0.38326833 | 5.81E-10
1.25 | 0.22191975 | 0.22191976 | 5.04E-10
1.00 | 0.06057118 | 0.06057118 | 4.27E-10
0.95 | 0.02020450 | 0.02020450 | 3.55E-09
0.92 | 0.00000000 | 0.00000000 | 1.22E-11

$\mathbf{\sigma}$ | price-to-strike | CTMC | Benchmark | Abs. error
---|---|---|---|---
$\mathbf{0.2}$ | 1.67 | 0.39232996 | 0.39232997 | 1.76E-08
1.25 | 0.22455160 | 0.22455166 | 5.84E-08
1.00 | 0.07590525 | 0.07590491 | 3.39E-07
0.83 | 0.01014165 | 0.01014179 | 1.40E-07
0.71 | 0.00053736 | 0.00053749 | 1.33E-07

$\mathbf{\sigma}$ | price-to-strike | CTMC | Benchmark | Abs. error
---|---|---|---|---
$\mathbf{0.2}$ | 1.67 | 0.38334989 | 0.38334989 | 3.29E-09
1.25 | 0.22190374 | 0.22190374 | 2.86E-09
1.00 | 0.06045761 | 0.06045761 | 2.43E-09
0.95 | 0.02020450 | 0.02020450 | 3.55E-09
0.92 | 0.00000690 | 0.00000690 | 5.39E-09

$\mathbf{\sigma}$ | price-to-strike | CTMC | Benchmark | Abs. error
---|---|---|---|---
$\mathbf{0.3}$ | 1.67 | 0.40772961 | 0.40772968 | 7.30E-08
1.25 | 0.22983971 | 0.22983969 | 1.39E-08
1.00 | 0.09108337 | 0.09108360 | 2.27E-07
0.83 | 0.01014165 | 0.01014179 | 1.40E-07
0.71 | 0.00465377 | 0.00465387 | 1.06E-07

$\mathbf{\sigma}$ | price-to-strike | CTMC | Benchmark | Abs. error
---|---|---|---|---
$\mathbf{0.3}$ | 1.67 | 0.38348301 | 0.38348311 | 1.01E-07
1.25 | 0.22187654 | 0.22187663 | 9.48E-08
1.00 | 0.06027987 | 0.06028002 | 1.53E-07
0.95 | 0.02020450 | 0.02020450 | 3.55E-09
0.92 | 0.00006003 | 0.00006008 | 5.13E-08

$\mathbf{\sigma}$ | price-to-strike | CTMC | Benchmark | Abs. error
---|---|---|---|---
$\mathbf{0.4}$ | 1.67 | 0.43035317 | 0.43036293 | 9.76E-06
1.25 | 0.24315058 | 0.24315678 | 6.20E-06
1.00 | 0.10988334 | 0.10988798 | 4.64E-06
0.83 | 0.01014165 | 0.01014179 | 1.40E-07
0.71 | 0.01314694 | 0.01315072 | 3.77E-06

(a) Vasicek model

$\mathbf{\sigma}$ | price-to-strike | CTMC | Benchmark | Abs. error
---|---|---|---|---
$\mathbf{0.4}$ | 1.67 | 0.38366164 | 0.38366831 | 6.68E-06
1.25 | 0.22183726 | 0.22184350 | 6.23E-06
1.00 | 0.06011562 | 0.06012477 | 9.15E-06
0.95 | 0.02020450 | 0.02020450 | 3.55E-09
0.92 | 0.00011258 | 0.00011395 | 1.37E-06

(b) CIR model

Table 15. Accuracy of the zero-coupon bond call option approximation (38)
under Vasicek and CIR models. Benchmark is calculated using closed-form
analytical formulas. Except for the number of grid points set to $m=1,000$,
model and CTMC parameters are as listed in Table 13. Zero-coupon bond call
option parameters using the notation of Corollary B.2: $t_{n_{1}}=0$,
$t_{n_{2}}=2$, and $T=4$.

We observe that the approximation achieves a high level of accuracy across all
volatilities and strikes. The convergence of the approximated call prices to
the analytical formulas is illustrated in Figure 8 for the two models.

|
---|---
(a) Vasicek | (b) CIR
Figure 8. Convergence pattern of the approximated zero-coupon bond call option
prices as the number of grid points $m$ increases. Benchmark is calculated
using closed-form analytical formulas. Except for the number of grid points
$m$, model and CTMC parameters are as listed in Table 13. Zero-coupon bond
call option parameters using the notation of Corollary B.2: $t_{n_{1}}=0$,
$t_{n_{2}}=2$, $T=4$, $K=0.9$.

We note that the approximated call prices converge rapidly to their analytical
values but exhibit a sawtooth pattern.

####  E.2.3. Approximation of Callable/Putable Bond Prices

We now examine the accuracy and the convergence of Proposition 4.3 in
approximating callable/putable bonds under the Vasicek model. Accordingly, we
consider a coupon-bearing bond with semi-annual coupons that mature in $4$
years $T=4$. The coupon rate, denoted below by $r_{c}$, is set to $4\%$ per
annum compounded semi-annually. The notional of the debt is set to $F=100$,
and we assume that it can be called at any time between the second and the
fourth year for no additional cost, that is, $K^{c}_{t}=100$ for $2\leq t\leq
T$ and we let $K_{t}^{c}\rightarrow\infty$ when $t<2$ (since exercise is not
allowed). Moreover, as there is no put feature, $K^{p}:=K^{p}_{t}=0$ for
$0\leq t\leq T$. Finally, we assume that accrued interest is paid to the
bondholder upon redemption. The contract specifications are summarized in
Table 20.

$F$ | $r_{c}$ | $T$ | $K_{t}^{c}$ | $K^{p}$
---|---|---|---|---
$100$ | $0.04$ | $4$ | $100$ | $0$
Table 16. Callable bond contract specifications202020$K_{t}^{c}$=100 for
$2\leq t\leq T$ and $K_{t}^{c}$ is set to a large constant when $t<2$.

Proposition 4.3 is also used to calculate the value of the straight bond
(i.e., the value of the coupon-bearing bond with no optionality, when
$K^{c}_{t}\rightarrow\infty$ and $K^{p}_{t}=0$ for all $t\in[0,T]$). The
results are summarized in Table 17(a), whereas those for the callable bond are
outlined in Table 17(b). The value of the optionality is obtained from the
difference between the value of the callable and the straight bonds. The call
option has a negative value because it is in favor of the issuer and, thus,
reduces the value of the bond. For the straight debt, a benchmark can be
obtained using a closed-form analytical formula since it can be decomposed in
a series of zero-coupon bonds (see Remark 4.5). For the callable debt, the
benchmark is calculated using CTMC approximation with $m=2,000$.

$\mathbf{\kappa}$ | CTMC | Benchmark | Rel. error
---|---|---|---
0.5 | 111.5779419 | 111.5781246 | 1.64E-06
1 | 104.6007798 | 104.6008544 | 7.13E-07
2 | 101.3606496 | 101.3606597 | 9.94E-08
3 | 100.5741608 | 100.5741632 | 2.42E-08
4 | 100.2732264 | 100.2732272 | 8.40E-09

$\mathbf{\kappa}$ | CTMC | Benchmark | Rel. error
---|---|---|---
0.5 | 94.4289102 | 94.4293068 | 4.20E-06
1 | 95.5617290 | 95.5616095 | 1.25E-06
2 | 96.9174873 | 96.9173130 | 1.80E-06
3 | 97.6890278 | 97.6890198 | 8.14E-08
4 | 98.1735503 | 98.1735203 | 3.06E-07

$\mathbf{R_{0}}$ | CTMC | Benchmark | Rel. error
---|---|---|---
0.02 | 106.6205602 | 106.6213440 | 7.35E-06
0.03 | 105.6061649 | 105.6062188 | 5.11E-07
0.04 | 104.6007798 | 104.6008544 | 7.13E-07
0.05 | 103.6050710 | 103.6051563 | 8.23E-07

$\mathbf{R_{0}}$ | CTMC | Benchmark | Rel. error
---|---|---|---
0.02 | 97.3030355 | 97.3030555 | 2.05E-07
0.03 | 96.4287924 | 96.4287773 | 1.57E-07
0.04 | 95.5617290 | 95.5616095 | 1.25E-06
0.05 | 94.7018444 | 94.7020397 | 2.06E-06

$\mathbf{\theta}$ | CTMC | Benchmark | Rel. error
---|---|---|---
0.01 | 113.7506391 | 113.7506574 | 1.61E-07
0.02 | 110.6102083 | 110.6102465 | 3.46E-07
0.03 | 107.5611515 | 107.5612085 | 5.30E-07
0.04 | 104.6007798 | 104.6008544 | 7.13E-07

$\mathbf{\theta}$ | CTMC | Benchmark | Rel. error
---|---|---|---
0.01 | 100.5832613 | 100.5844116 | 1.14E-05
0.02 | 98.9373817 | 98.9375213 | 1.41E-06
0.03 | 97.2623005 | 97.2631690 | 8.93E-06
0.04 | 95.5617290 | 95.5616095 | 1.25E-06

$\mathbf{\sigma}$ | CTMC | Benchmark | Rel. error
---|---|---|---
0.1 | 101.0176528 | 101.0176706 | 1.76E-07
0.2 | 104.6007798 | 104.6008544 | 7.13E-07
0.3 | 110.8775749 | 110.8777598 | 1.67E-06
0.4 | 120.3458095 | 120.3474922 | 1.40E-05

(a) Straight bond

$\mathbf{\sigma}$ | CTMC | Benchmark | Rel. error
---|---|---|---
0.1 | 97.0423712 | 97.0419581 | 4.26E-06
0.2 | 95.5617290 | 95.5616095 | 1.25E-06
0.3 | 95.3070805 | 95.3073647 | 2.98E-06
0.4 | 96.1965169 | 96.1965745 | 6.00E-07

(b) Callable bond

Table 17. Accuracy of Proposition 4.3 to approximate the values of straight and callable bonds under Vasicek model. Model and CTMC parameters are as listed in Table 13. Contract specifications are as listed in Table 20, with the call option exercise window starting from $t=2$ to maturity. |
---|---
(a) Straight bond | (b) Callable bond
Figure 9. Convergence pattern of the approximated prices of straight and
callable bonds as the number of grid points $m$ increases under the Vasicek
model. Model and CTMC parameters are as listed in Table 13. Contract
specifications are as listed in Table 20, with the call option exercise window
starting from $t=2$ to maturity.

Again, we note that the approximation achieves a high level of accuracy in a
fraction of a second across all model parameters. The average calculation time
is 0.02 seconds.

The convergence pattern of the straight and callable bonds is displayed in
Figure 9. We observe that the straight bond converges smoothly to the
analytical price. However, the callable debt exhibits a sawtooth pattern. For
the two securities, the absolute error decreases rapidly to $0$.

####  E.2.4. Approximation of Convertible Bond Prices

We now investigate the accuracy of Algorihtm 5 in approximating CB prices
under the Black–Scholes–Vasicek model, along with the numerical convergence of
the price estimates.

That is, we suppose that the stock price process follows a geometric Brownian
motion with stochastic interest rates satisfying

(63) $\displaystyle\,\mathrm{d}S_{t}$
$\displaystyle=R_{t}S_{t}\,\mathrm{d}t+\widetilde{\sigma}_{S}S_{t}\,\mathrm{d}W^{(1)}_{t},$
$\displaystyle\,\mathrm{d}R_{t}$ $\displaystyle=\kappa(\theta-
R_{t})\,\mathrm{d}t+\widetilde{\sigma}_{R}\,\mathrm{d}W^{(2)}_{t},$

with $\kappa$, $\theta$, $\widetilde{\sigma}_{S}$, $\widetilde{\sigma}_{R}$>0,
and $[W^{(1)},W^{(2)}]_{t}=\rho t$, $\rho\in[-1,1]$.

From Lemma 3.1, we find that
$f(r)=\frac{\widetilde{\sigma}_{S}}{\widetilde{\sigma}_{R}}r$. The dynamics of
the auxiliary process ${X_{t}=\ln(S_{t})-\rho f(R_{t})}$ can then be derived
as

(64)
$\begin{split}\,\mathrm{d}X_{t}&=\mu_{X}(t,R_{t})\,\mathrm{d}t+\sigma_{X}(R_{t})\,\mathrm{d}W^{\star}_{t}\\\
\,\mathrm{d}R_{t}&=\kappa(\theta-
R_{t})\,\mathrm{d}t+\widetilde{\sigma}_{R}\,\mathrm{d}W^{(2)}_{t},\end{split}$

with
$\mu_{X}(t,R_{t})=R_{t}-\frac{\widetilde{\sigma}_{S}^{2}}{2}-\rho\frac{\widetilde{\sigma}_{S}}{\widetilde{\sigma}_{R}}\kappa(\theta-
R_{t})$, $\sigma_{X}=\widetilde{\sigma}_{S}\sqrt{1-\rho^{2}}$, and
$X_{0}=\ln(S_{0})-\rho f(R_{0})$.

Unless stated otherwise, the model parameters for the short-rate process are
the same as those used in previous examples, reported in Table 13 under
Vasicek model. We suppose further that $\widetilde{\sigma}_{S}=0.2$ and
$\rho=-0.2$. The model parameters are summarized in Table 18.

Model | ${R_{0}}$ | ${\kappa}$ | ${\theta}$ | ${\widetilde{\sigma}_{R}}$ | $S_{0}$ | $\widetilde{\sigma}_{S}$ | $\rho$
---|---|---|---|---|---|---|---
Black–Scholes–Vasicek | $0.04$ | $1$ | $0.04$ | $0.20$ | $100$ | $0.2$ | $-0.2$
Table 18. Model parameters

The grid used to approximate the short-rate process,
$\mathcal{S}^{(m)}_{R}=\\{r_{1},r_{2},\ldots,r_{m}\\}$, and the auxiliary
process $\mathcal{S}^{(M)}_{X}=\\{x_{1},x_{2},\ldots,x_{M}\\}$, are
constructed using the methodology of Tavella and Randall (Tavella and Randall
(2000), Chapter 5), as explained in Section 6, with $\widetilde{\alpha}_{R}$
(resp. $\widetilde{\alpha}_{X}$) representing the non-uniformity parameter of
the grid for $R^{(m)}$ (resp. $X^{(m)}$). Unless otherwise indicated, all
numerical experiments are conducted using the CTMC parameters listed in Table
19.

| ${m}$ | $M$ | ${r_{1}}$ | ${r_{m}}$ | $\widetilde{\alpha}_{R}$ | ${x_{1}}$ | ${x_{M}}$ | $\widetilde{\alpha}_{X}$ | $\Delta_{N}$
---|---|---|---|---|---|---|---|---|---
Black–Scholes–Vasicek | $160$ | $100$ | $-30R_{0}$ | $25R_{0}$ | $0.5$ | $0.64X_{0}$ | $1.42X_{0}$ | $2$ | $1/100$
Table 19. CTMC parameters $F$ | $r_{c}$ | $T$ | $\eta$ | $c_{t}$
---|---|---|---|---
100 | 0 | 1 | 1 | 0.05
Table 20. CB contract specifications

The contract specifications are summarized in Table 20. We consider a zero-
coupon CB ($r_{c}=0$) with a notional $F=100$. We suppose that the bond can be
converted at any time from inception to maturity ($T=1$) at a conversion rate
$\eta=1$.

Under this set of parameters and when there is no credit risk ($c_{t}=0$ for
all $t\in[0,T]$), the valuation of American-style CBs is simplified to that of
European-style CBs, see Proposition 5.2. The results of Proposition E.1,
available online as supplemental material, can thus serve as a benchmark in
our analysis. When $c_{t}>0$ for some $t\in[0,T]$, the CTMC approximation is
compared with the least-squares Monte Carlo (“LSMC”) simulation method of
Longstaff and Schwartz (2001)212121For the LSMC method, we use $50,000$
simulation paths with $252$-time steps and the first three Laguerre
polynomials and a constant as basis functions.. The precision of the
estimation at a confidence level of 95% is provided under the LSMC approach.
The results are summarized in Tables 21 and 22.

$\mathbf{R_{0}}$ | CTMC | Benchmark | Rel. error
---|---|---|---
0.02 | 106.81214 | 106.82314 | 1.03E-04
0.03 | 106.51481 | 106.52372 | 8.37E-05
0.04 | 106.22685 | 106.23381 | 6.55E-05
0.05 | 105.94816 | 105.95328 | 4.83E-05

$\mathbf{R_{0}}$ | CTMC | LSMC | Precision
---|---|---|---
0.02 | 104.46808 | 104.47403 | 0.11544
0.03 | 104.24589 | 104.23711 | 0.11580
0.04 | 104.03224 | 104.01271 | 0.11659
0.05 | 103.82686 | 103.88733 | 0.11649

$\mathbf{\sigma_{R}}$ | CTMC | Benchmark | Rel. error
---|---|---|---
0.1 | 105.91095 | 105.91981 | 8.37E-05
0.2 | 104.51904 | 104.53120 | 1.16E-04
0.3 | 106.92857 | 106.93034 | 1.65E-05
0.4 | 107.97843 | 107.97101 | 6.87E-05

$\mathbf{\sigma_{R}}$ | CTMC | LSMC | Precision
---|---|---|---
0.1 | 103.76603 | 103.73569 | 0.11114
0.2 | 104.03224 | 104.01271 | 0.11659
0.3 | 104.65219 | 104.61190 | 0.12260
0.4 | 105.59650 | 105.52785 | 0.13238

$\mathbf{\theta}$ | CTMC | Benchmark | Rel. error
---|---|---|---
0.015 | 106.74428 | 106.74616 | 1.76E-05
0.025 | 106.56856 | 106.57216 | 3.37E-05
0.035 | 106.39609 | 106.40138 | 4.97E-05
0.045 | 107.28378 | 107.28108 | 2.52E-05

$\mathbf{\theta}$ | CTMC | LSCM | Precision
---|---|---|---
0.015 | 104.42155 | 104.40895 | 0.11553
0.025 | 104.28887 | 104.27291 | 0.11528
0.035 | 104.15911 | 104.14892 | 0.11577
0.045 | 104.03224 | 104.01271 | 0.11659

$\mathbf{\kappa}$ | CTMC | Benchmark | Rel. error
---|---|---|---
0.5 | 106.44436 | 106.44906 | 4.42E-05
1 | 106.22685 | 106.23381 | 6.55E-05
2 | 106.01009 | 106.01812 | 7.57E-05
3 | 105.92597 | 105.93366 | 7.26E-05
4 | 105.89402 | 105.90111 | 6.70E-05

(a) No credit risk, $c_{t}=0$ for all $t\in[0,T]$

$\mathbf{\kappa}$ | CTMC | LSMC | Precision
---|---|---|---
0.5 | 104.22325 | 104.23011 | 0.11946
1 | 104.03224 | 104.01271 | 0.11659
2 | 103.84523 | 103.81358 | 0.11260
3 | 103.77549 | 103.76427 | 0.11201
4 | 103.75119 | 103.72202 | 0.11033

(b) $c_{t}=0.05$ for all $t\in[0,T]$

Table 21. Accuracy of the CB price approximations, Algorithm 5, under
Black–Scholes–Vasicek model. Model, CTMC, and contract parameters are as
listed in Tables 18 and 19 and 20, respectively.

$\mathbf{S_{0}}$ | CTMC | Benchmark | Rel. error
---|---|---|---
90 | 101.12964 | 101.13025 | 6.06E-06
95 | 103.41896 | 103.42370 | 4.58E-05
100 | 106.22685 | 106.23381 | 6.55E-05
105 | 109.50964 | 109.51656 | 6.32E-05
110 | 113.20333 | 113.20847 | 4.54E-05

$\mathbf{S_{0}}$ | CTMC | LSMC | Precision
---|---|---|---
90 | 97.97114 | 97.89608 | 0.09261
95 | 100.73891 | 100.73928 | 0.10528
100 | 104.03224 | 104.01271 | 0.11659
105 | 107.75674 | 107.81978 | 0.12770
110 | 111.84378 | 111.92104 | 0.13691

$\mathbf{\sigma_{S}}$ | CTMC | Benchmark | Rel. error
---|---|---|---
0.1 | 102.99114 | 103.01936 | 2.74E-04
0.15 | 104.51904 | 104.53120 | 1.16E-04
0.2 | 106.22685 | 106.23381 | 6.55E-05
0.3 | 109.85751 | 109.86115 | 3.31E-05
0.4 | 113.58694 | 113.58916 | 1.96E-05

$\mathbf{\sigma_{S}}$ | CTMC | LSMC | Precision
---|---|---|---
0.1 | 101.13009 | 101.18147 | 0.06643
0.15 | 102.46320 | 102.48711 | 0.09210
0.2 | 104.03224 | 104.01271 | 0.11659
0.3 | 107.46459 | 107.44911 | 0.16793
0.4 | 111.04507 | 111.07825 | 0.22786

$\mathbf{\rho}$ | CTMC | Benchmark | Rel. error
---|---|---|---
-0.3 | 105.94165 | 105.94957 | 7.48E-05
-0.2 | 106.22685 | 106.23381 | 6.55E-05
0.2 | 107.28378 | 107.28108 | 2.52E-05
0.3 | 107.53082 | 107.52425 | 6.11E-05

(a) No credit risk, $c_{t}=0$ for all $t\in[0,T]$

$\mathbf{\rho}$ | CTMC | LSMC | Precision
---|---|---|---
-0.3 | 103.76729 | 103.75815 | 0.11911
-0.2 | 104.03224 | 104.01271 | 0.11659
0.2 | 105.02223 | 104.98302 | 0.10696
0.3 | 105.25520 | 105.19689 | 0.10441

(b) $c_{t}=0.05$ for all $t\in[0,T]$

Table 22. Accuracy of the CB price approximations, Algorithm 5, under
Black–Scholes–Vasicek model. Model, CTMC, and contract parameters are as
listed in Tables 18 and 19 and 20, respectively.

We note that the model achieves a high level of accuracy across all model
parameters. The average calculation time for the CTMC approximated prices is
less than 1.40 seconds.

|
---|---
(a) No credit risk, $c_{t}=0$ for all $t\in[0,T]$ | (b) $c_{t}=0.05$ for all $t\in[0,T]$
Figure 10. Convergence pattern of the CB price approximations, Algorithm 5,
under Black–Scholes–Vasicek model. Except for the number of grid points $M$ of
the auxiliary process, the model, CTMC, and contract parameters are as listed
in Tables 18, 19, and 20, respectively.

The convergence pattern of the approximation as $M$ increases is illustrated
in Figure 10. When there is no credit risk ($c_{t}=0$ for all $t\in[0,T]$),
the benchmark is calculated using the exact pricing formula of Proposition
E.1, available online as supplemental material. When credit risk is considered
($c_{t}>0$ for some $t\in[0,T]$), the benchmark is obtained using CTMC
approximation with $M=1,000$ and $\Delta_{N}=1/252$, all other CTMC parameters
from table 19. Figure 10 shows that the approximated prices converge rapidly
and smoothly to the benchmark prices.