Consider a defaultable bond with maturity $T$. A defaultable bond means that the debtor may fail to pay the promised payments due to some credit events occurred before maturity. We consider the implied term structure of interest rates for this bond under two different assumptions.

Deterministic Assumption: Assume that under the risk-neutral measure, the default time $\tau$ of the bond is the first jump time of a non-homogeneous Poisson point process $N\left(t\right)$, where the intensity $\lambda \left(t\right)$ of $N\left(t\right)$ is a deterministic function of time $t$, and $\lambda \left(t\right)$ has at most a finite number of jump discontinuities in the interval $\left[0,T\right]$. The function $\lambda \left(t\right)$ is also referred to as the default intensity. In other words, there exists an exponential random variable $Y~\text{Exp}\left(1\right)$ such that

$\tau =\text{min}\left\{t>0;{\displaystyle {\int }_0^t\lambda \left(u\right)\text{d}u}\ge Y\right\}.$

In particular, when $\lambda \left(t\right)$ is constant and equal to $\lambda$, $\tau ~\text{Exp}\left(\lambda \right)$.

Let $P\left(t,T\right)$ denote the value of the bond at time $t$ given that no default has occurred ( $\tau >t$). Assume that the bond value after default follows the recovery of market value, i.e., there exists a constant $R\in \left[0,1\right)$ such that the value of the bond after default is $RP\left(\tau -,T\right)$, where $P\left(\tau -,T\right)$ is the left-sided limit of $P\left(t,T\right)$ as $t$ approaches $\tau$, that is $P\left(\tau -,T\right)=\underset{t\uparrow \tau }{\text{lim}}P\left(t,T\right)$. $R$ is called the recovery-rate. In the Deterministic Assumption, risk-free interest rate $r\left(t\right)$ is assumed to be a deterministic function of time $t$.

Stochastic Assumption: The risk-free interest rate $r\left(t\right)$ and default intensity $\lambda \left(t\right)$ follow some non-negative stochastic processes, and the recovery rate $R\in \left[0,1\right)$ is a random variable that follows some certain distribution. Further, $r\left(t\right)$, $\lambda \left(t\right)$ and $R$ are independent of the exponential random variable $Y$ introduced above.

Stochastic Assumption is more realistic. We consider the Deterministic Assumption only for simplicity. To some extent, the Deterministic Assumption serves as preparation for the Deterministic Assumption. In fact, given the filtration generated by stochastic $r\left(t\right)$, $\lambda \left(t\right)$ and $R$, the conditional expectation of the Stochastic Assumption can be transformed into a deterministic one. Please refer to Section 4 for details.

Consider the calculations of several probabilities under the Deterministic Assumption. For $0\le t\le T$, the probability that the bond has not defaulted by time $t$ is given by

$\text{Pr}\left(\tau >t\right)={\text{e}}^{-{\displaystyle {\int }_0^t\lambda \left(u\right)\text{d}u}}.$

Given $\tau >t$, the conditional probability that the bond has not defaulted by time $T$ is

$\text{Pr}\left(\tau >T|\tau >t\right)={\text{e}}^{-{\displaystyle {\int }_t^T\lambda \left(u\right)\text{d}u}}.$

For $t\le s<s+\text{Δ}s\le T$, assume that $\lambda \left(t\right)$ is continuous on the time interval $\left(s,s+\text{Δ}s\right]$. The conditional probability that the bond defaults in the interval $\left(s,s+\text{Δ}s\right]$, given the condition $\tau >t$ and $\text{Δ}s$ is sufficiently small, is given by

$\begin{array}{c}\text{Pr}\left(\tau \in \left(s,s+\text{Δ}s\right]|\tau >t\right)=\text{Pr}\left(\tau >s|\tau >t\right)-\text{Pr}\left(\tau >s+\text{Δ}s|\tau >t\right)\\ ={\text{e}}^{-{\displaystyle {\int }_t^s\lambda \left(u\right)\text{d}u}}-{\text{e}}^{-{\displaystyle {\int }_t^{s+\text{Δ}s}\lambda \left(u\right)\text{d}u}}\\ \approx \lambda \left(s\right){\text{e}}^{-{\displaystyle {\int }_t^s\lambda \left(u\right)\text{d}u}}\text{Δ}s.\end{array}$

Consider the price of a default-free bond under Deterministic Assumption. The price is derived in three different circumstances: zero-coupon bond, continuous coupon-paying bond and periodic coupon-paying bond.

Consider a default-free zero-coupon bond which pays a face value $1 only at maturity $T$. The price of the bond is given by

$P\left(t,T\right)={\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(u\right)\text{d}u}}.$(1)

Consider a default-free bond with continuous coupon payments and a face value $1. Assume the bond matures in $T$ years and pays a coupon $c\left(t\right)\text{d}t$ during the time interval $\left[t,t+\text{d}t\right]$, where $c\left(t\right)$ is referred to the coupon rate. The price of the bond is given by

$P\left(t,T\right)={\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(u\right)\text{d}u}}+{\displaystyle {\int }_t^{T}c\left(s\right){\text{e}}^{-{\displaystyle {\int }_t^{s}r\left(u\right)\text{d}u}}\text{d}s}.$(2)

Consider a default-free bond with annual coupon payments and a face value $1. Assume the bond matures in $T$ years, where $T$ is an integer, and the coupon payment in the $k$-th year is $c\left(k\right)$. The price of the bond is given by

$P\left(t,T\right)={\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(u\right)\text{d}u}}+{\displaystyle \underset{\left(t,T\right]}{\int }{\text{e}}^{-{\displaystyle {\int }_t^{v}r\left(u\right)\text{d}u}}\text{d}\mathcal{C}}\left(v\right),$(3)

where $\mathcal{C}\left(t\right)$ represents the cumulative coupon payments up to time $t$, i.e.,

$\mathcal{C}\left(t\right)=\underset{k\le t}{{\displaystyle \sum }}c\left(k\right).$

Clearly, $\mathcal{C}\left(t\right)$ is right-continuous with respect to $t$.

When $t$ is not an integer, $P\left(t,T\right)$ is continuous and differentiable with respect to $t$, and satisfies

$P\text{'}\left(t,T\right)=r\left(t\right)P\left(t,T\right).$(4)

When $t$ is an integer, $P\left(t,T\right)$ is right-continuous with respect to $t$, and satisfies

$\text{Δ}P\left(t,T\right)=P\left(t,T\right)-P\left(t-,T\right)=-c\left(t\right).$(5)

In other words, the expression for $P\left(t,T\right)$ given in Equation (3) is the solution that satisfies Equations (4)-(5) and terminal condition $P\left(T,T\right)=1$.

Consider a defaultable zero-coupon bond. The bond pays a face value $1 at maturity $T$ if no default occurs by $T$, and there is no payments made prior to that.

Theorem 1. The price of a defaultable zero-coupon bond at time $t$ is given by

$P\left(t,T\right)={\text{e}}^{-{\displaystyle {\int }_t^T\bar{r}\left(u\right)\text{d}u}},$(6)

where $\bar{r}\left(t\right)=r\left(t\right)+\left(1-R\right)\lambda \left(t\right)$.

Proof. By the risk-neutral pricing principle (see [3] for reference), the price of the bond under the risk-neutral probability, when $\tau >t$, is

$\begin{array}{c}P\left(t,T\right)=\mathbb{E}\left[{\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(u\right)\text{d}u}}{I}_{\left\{\tau >T\right\}}+{\text{e}}^{-{\displaystyle {\int }_t^{\tau }r\left(u\right)\text{d}u}}RP\left(\tau -,T\right)I_{\left\{\tau \le T\right\}}|\tau >t\right]\\ =A\left(t\right)+C\left(t\right),\end{array}$(7)

where

$A\left(t\right)={\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(u\right)+\lambda \left(u\right)\text{d}u}},$

$C\left(t\right)=R{\displaystyle {\int }_t^{T}P\left(s-,T\right)\lambda \left(s\right){\text{e}}^{-{\displaystyle {\int }_t^{s}r\left(u\right)+\lambda \left(u\right)\text{d}u}}\text{d}s}.$

Assume that $P\left(t,T\right)$ is continuously differentiable with respect to $t$. Note that

$A^{\prime }\left(t\right)=\left[r\left(t\right)+\lambda \left(t\right)\right]A\left(t\right),$

$C^{\prime }\left(t\right)=\left[r\left(t\right)+\lambda \left(t\right)\right]C\left(t\right)-R\lambda \left(t\right)P\left(t,T\right),$

differentiating both sides of Equation (7) with respect to $t$, we obtain

$\begin{array}{c}{P}^{\prime }\left(t,T\right)=A^{\prime }\left(t\right)+C^{\prime }\left(t\right)\\ =\left[r\left(t\right)+\lambda \left(t\right)\right]A\left(t\right)+\left[r\left(t\right)+\lambda \left(t\right)\right]C\left(t\right)-R\lambda \left(t\right)P\left(t,T\right)\\ =\left[r\left(t\right)+\lambda \left(t\right)\right]P\left(t,T\right)-R\lambda \left(t\right)P\left(t,T\right)\\ =\bar{r}\left(t\right)P\left(t,T\right).\end{array}$(8)

Equation (8) is a homogeneous first-order linear differential equation, the solution that satisfies Equation (8) and terminal condition $P\left(T,T\right)=1$ is given by

$P\left(t,T\right)={\text{e}}^{-{\displaystyle {\int }_t^T\bar{r}\left(u\right)\text{d}u}}.$(6)

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Remark 1. When $r\left(t\right)$, $\lambda \left(t\right)$ are constants, and equal to $r$, $\lambda$ respectively, we have

$P\left(t,T\right)={\text{e}}^{-\bar{r}\left(T-t\right)},$

where $\bar{r}=r+\left(1-R\right)\lambda$.

Consider a defaultable bond with continuous coupon payments, where the coupon rate $c\left(t\right)$ is a deterministic function of time $t$. The bond pays a face value $1 at maturity $T$ if no default occurs by $T$. In addition, the bond pays a coupon $c\left(t\right)\text{d}t$ during the time interval $\left[t,t+\text{d}t\right]$ if no default occurs by that time.

Theorem 2. The price at time $t$ of a defaultable bond with continuous coupon payments is given by

$P\left(t,T\right)={\text{e}}^{-{\displaystyle {\int }_t^T\bar{r}\left(u\right)\text{d}u}}+{\displaystyle {\int }_t^{T}c\left(s\right){\text{e}}^{-{\displaystyle {\int }_t^s\bar{r}\left(u\right)\text{d}u}}\text{d}s},$(9)

where $\bar{r}\left(t\right)=r\left(t\right)+\left(1-R\right)\lambda \left(t\right)$.

Proof. Under the risk-neutral probability, when $\tau >t$, we have

$\begin{array}{c}P\left(t,T\right)=\mathbb{E}[\left({\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(u\right)\text{d}u}}+{\displaystyle {\int }_t^{T}c\left(v\right){\text{e}}^{-{\displaystyle {\int }_t^{v}r\left(u\right)\text{d}u}}\text{d}v}\right)I_{\{\tau >T\}}\\ +\left(RP\left(\tau -,T\right){\text{e}}^{-{\displaystyle {\int }_t^{\tau }r\left(u\right)\text{d}u}}+{\displaystyle {\int }_t^{\tau }c\left(v\right){\text{e}}^{-{\displaystyle {\int }_t^{v}r\left(u\right)\text{d}u}}\text{d}v}\right)I_{\left\{\tau \le T\right\}}|\tau >t]\\ =A\left(t\right)+B\left(t\right)+C\left(t\right)+D\left(t\right),\end{array}$(10)

where

$A\left(t\right)={\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(u\right)+\lambda \left(u\right)\text{d}u}},$

$B\left(t\right)={\text{e}}^{-{\displaystyle {\int }_t^T\lambda \left(u\right)\text{d}u}}{\displaystyle {\int }_t^{T}c\left(v\right){\text{e}}^{-{\displaystyle {\int }_t^{v}r\left(u\right)\text{d}u}}\text{d}v},$

$C\left(t\right)=R{\displaystyle {\int }_t^{T}P\left(s-,T\right)\lambda \left(s\right){\text{e}}^{-{\displaystyle {\int }_t^{s}r\left(u\right)+\lambda \left(u\right)\text{d}u}}\text{d}s},$

$\begin{array}{c}D\left(t\right)={\displaystyle {\int }_t^T\lambda \left(s\right){\text{e}}^{-{\displaystyle {\int }_t^s\lambda \left(u\right)\text{d}u}}\left({\displaystyle {\int }_t^{s}c\left(v\right){\text{e}}^{-{\displaystyle {\int }_t^{v}r\left(u\right)\text{d}u}}\text{d}v}\right)\text{d}s}\\ ={\displaystyle {\int }_t^{T}c\left(v\right){\text{e}}^{-{\displaystyle {\int }_t^{v}r\left(u\right)\text{d}u}}\left({\displaystyle {\int }_v^T\lambda \left(s\right){\text{e}}^{-{\displaystyle {\int }_t^s\lambda \left(u\right)\text{d}u}}\text{d}s}\right)\text{d}v}.\end{array}$

Assume that $P\left(t,T\right)$ is continuously differentiable with respect to $t$. Note that

$A^{\prime }\left(t\right)=\left[r\left(t\right)+\lambda \left(t\right)\right]A\left(t\right),$

$B^{\prime }\left(t\right)=\left[r\left(t\right)+\lambda \left(t\right)\right]B\left(t\right)-c\left(t\right){\text{e}}^{-{\displaystyle {\int }_t^T\lambda \left(u\right)\text{d}u}},$

$C^{\prime }\left(t\right)=\left[r\left(t\right)+\lambda \left(t\right)\right]C\left(t\right)-R\lambda \left(t\right)P\left(t,T\right),$

$\begin{array}{c}{D}^{\prime }\left(t\right)=\left[r\left(t\right)+\lambda \left(t\right)\right]D\left(t\right)-c\left(t\right){\displaystyle {\int }_t^T\lambda \left(s\right){\text{e}}^{-{\displaystyle {\int }_t^s\lambda \left(u\right)\text{d}u}}\text{d}s}\\ =\left[r\left(t\right)+\lambda \left(t\right)\right]D\left(t\right)-c\left(t\right)\left[1-{\text{e}}^{-{\displaystyle {\int }_t^T\lambda \left(u\right)\text{d}u}}\right],\end{array}$

differentiating both sides of Equation (10) with respect to $t$, we obtain

$\begin{array}{c}{P}^{\prime }\left(t,T\right)=A^{\prime }\left(t\right)+B^{\prime }\left(t\right)+C^{\prime }\left(t\right)+D^{\prime }\left(t\right)\\ =\left[r\left(t\right)+\lambda \left(t\right)\right]A\left(t\right)\\ +\left[r\left(t\right)+\lambda \left(t\right)\right]B\left(t\right)-c\left(t\right){\text{e}}^{-{\displaystyle {\int }_t^T\lambda \left(u\right)\text{d}u}}\\ +\left[r\left(t\right)+\lambda \left(t\right)\right]C\left(t\right)-R\lambda \left(t\right)P\left(t,T\right)\\ +\left[r\left(t\right)+\lambda \left(t\right)\right]D\left(t\right)-c\left(t\right)\left[1-{\text{e}}^{-{\displaystyle {\int }_t^T\lambda \left(u\right)\text{d}u}}\right]\\ =\left[r\left(t\right)+\lambda \left(t\right)\right]P\left(t,T\right)-R\lambda \left(t\right)P\left(t,T\right)-c\left(t\right)\\ =\bar{r}\left(t\right)P\left(t,T\right)-c\left(t\right).\end{array}$(11)

Equation (11) is a first-order linear differential equation, and the solution satisfying Equation (11) and the terminal condition $P\left(T,T\right)=1$ is given by

$P\left(t,T\right)={\text{e}}^{-{\displaystyle {\int }_t^T\bar{r}\left(u\right)\text{d}u}}+{\displaystyle {\int }_t^{T}c\left(s\right){\text{e}}^{-{\displaystyle {\int }_t^s\bar{r}\left(u\right)\text{d}u}}\text{d}s}.$(9)

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Remark 2. When $r\left(t\right)$, $\lambda \left(t\right)$ and $c\left(t\right)$ are constants, and equal to $r$, $\lambda$ and $c$ respectively, the price of the bond is given by

$P\left(t,T\right)=\left(1-\frac{c}{\bar{r}}\right){\text{e}}^{-\bar{r}\left(T-t\right)}+\frac{c}{\bar{r}},$

where $\bar{r}=r+\left(1-R\right)\lambda$.

Consider a defaultable bond with annual coupon payments and a face value $1. Assume the bond matures in $T$ years, where $T$ is an integer. The coupon payment in the $k$-th year is constant $c\left(k\right)$.

Let $P\left(t,T\right)$ be the ex-coupon bond price, i.e., the bond price at time $t$, excluding the coupon payment made on that day, when $t$ is a coupon payment date. The bond value, given that default occurs on a coupon payment date $t$, is assumed to be $RP\left(t-,T\right)+c\left(t\right)$, meaning that the default does not affect the coupon payment itself.

Denote the cumulative coupons up to time $t$ by $\mathcal{C}\left(t\right)$, i.e.,

$\mathcal{C}\left(t\right)=\underset{k\le t}{{\displaystyle \sum }}c\left(k\right).$

The function $\mathcal{C}\left(t\right)$ is right-continuous with respect to $t$, and for any function $F\left(t\right)$ and $s>t$,

${\displaystyle \underset{k=\lfloor t\rfloor +1}{\overset{\lfloor s\rfloor }{\sum }}F\left(k\right)c\left(k\right)}={\displaystyle \underset{\left(t,s\right]}{\int }F}\left(v\right)\text{d}\mathcal{C}\left(v\right),$

where $\lfloor t\rfloor$ denotes the integer part of $t$. The integral on the right-hand side is a Stieltjes integral, which is right-continuous with respect to the upper limit of integration $s$.

Theorem 3. The price at time $t$ of a defaultable bond with periodic coupon payments is given by

$P\left(t,T\right)={\text{e}}^{-{\displaystyle {\int }_t^T\bar{r}\left(u\right)\text{d}u}}+{\displaystyle \underset{\left(t,T\right]}{\int }{\text{e}}^{-{\displaystyle {\int }_t^v\bar{r}\left(u\right)\text{d}u}}\text{d}\mathcal{C}}\left(v\right),$(12)

where $\bar{r}\left(t\right)=r\left(t\right)+\left(1-R\right)\lambda \left(t\right)$.

Proof. Under the risk-neutral probability, when $\tau >t$, the ex-coupon bond price is given by

$\begin{array}{c}P\left(t,T\right)=\mathbb{E}[\left({\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(u\right)\text{d}u}}+{\displaystyle \underset{\left(t,\tau \right]}{\int }{\text{e}}^{-{\displaystyle {\int }_t^{v}r\left(u\right)\text{d}u}}\text{d}\mathcal{C}\left(v\right)}\right)I_{\left\{\tau >T\right\}}\\ +\left(RP\left(\tau -,T\right){\text{e}}^{-{\displaystyle {\int }_t^{\tau }r\left(u\right)\text{d}u}}+{\displaystyle \underset{\left(t,\tau \right]}{\int }{\text{e}}^{-{\displaystyle {\int }_t^{v}r\left(u\right)\text{d}u}}\text{d}\mathcal{C}\left(v\right)}\right)I_{\left\{\tau \le T\right\}}|\tau >t]\\ =A\left(t\right)+B\left(t\right)+C\left(t\right)+D\left(t\right),\end{array}$(13)

where

$A\left(t\right)={\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(u\right)+\lambda \left(u\right)\text{d}u}},$

$B\left(t\right)={\text{e}}^{-{\displaystyle {\int }_t^T\lambda \left(u\right)\text{d}u}}{\displaystyle \underset{\left(t,T\right]}{\int }{\text{e}}^{-{\displaystyle {\int }_t^{v}r\left(u\right)\text{d}u}}\text{d}\mathcal{C}}\left(v\right),$

$C\left(t\right)=R{\displaystyle {\int }_t^{T}P\left(s-,T\right)\lambda \left(s\right){\text{e}}^{-{\displaystyle {\int }_t^{s}r\left(u\right)+\lambda \left(u\right)\text{d}u}}\text{d}s},$

$\begin{array}{c}D\left(t\right)={\displaystyle {\int }_t^T\lambda \left(s\right){\text{e}}^{-{\displaystyle {\int }_t^s\lambda \left(u\right)\text{d}u}}\left({\displaystyle \underset{\left(t,s\right]}{\int }{\text{e}}^{-{\displaystyle {\int }_t^{v}r\left(u\right)\text{d}u}}\text{d}\mathcal{C}\left(v\right)}\right)\text{d}s}\\ ={\displaystyle \underset{\left(t,T\right]}{\int }{\text{e}}^{-{\displaystyle {\int }_t^{v}r\left(u\right)\text{d}u}}}\left({\displaystyle {\int }_v^T\lambda \left(s\right){\text{e}}^{-{\displaystyle {\int }_t^s\lambda \left(u\right)\text{d}u}}\text{d}s}\right)\text{d}\mathcal{C}\left(v\right).\end{array}$

The calculation of $D\left(t\right)$ in the above equation involves the exchange of the order of double integration. Please refer to Appendix A for further details.

Assume that $P\left(t,T\right)$ is continuous and differentiable with respect to $t$ when $t$ is not an integer, and right-continuous with respect to $t$ when $t$ is an integer.

When $t$ is not an integer, since

$A^{\prime }\left(t\right)=\left[r\left(t\right)+\lambda \left(t\right)\right]A\left(t\right),$

$B^{\prime }\left(t\right)=\left[r\left(t\right)+\lambda \left(t\right)\right]B\left(t\right),$

$C^{\prime }\left(t\right)=\left[r\left(t\right)+\lambda \left(t\right)\right]C\left(t\right)-R\lambda \left(t\right)P\left(t,T\right),$

$D^{\prime }\left(t\right)=\left[r\left(t\right)+\lambda \left(t\right)\right]D\left(t\right),$

we have

$\begin{array}{c}{P}^{\prime }\left(t,T\right)=A^{\prime }\left(t\right)+B^{\prime }\left(t\right)+C^{\prime }\left(t\right)+D^{\prime }\left(t\right)\\ =\left[r\left(t\right)+\lambda \left(t\right)\right]\left[A\left(t\right)+B\left(t\right)+C\left(t\right)+D\left(t\right)\right]-R\lambda \left(t\right)P\left(t,T\right)\\ =\left[r\left(t\right)+\lambda \left(t\right)\right]P\left(t,T\right)-R\lambda \left(t\right)P\left(t,T\right)\\ =\bar{r}\left(t\right)P\left(t,T\right).\end{array}$(14)

When $t$ is an integer, since

$\text{Δ}A\left(t\right)=A\left(t\right)-A\left(t-\right)=0,$

$\text{Δ}B\left(t\right)=B\left(t\right)-B\left(t-\right)=-c\left(t\right){\text{e}}^{-{\displaystyle {\int }_t^T\lambda \left(u\right)\text{d}u}},$

$\text{Δ}C\left(t\right)=C\left(t\right)-C\left(t-\right)=0,$

$\begin{array}{c}\text{Δ}D\left(t\right)=D\left(t\right)-D\left(t-\right)=-c\left(t\right){\displaystyle {\int }_k^T\lambda \left(s\right){\text{e}}^{-{\displaystyle {\int }_t^s\lambda \left(u\right)\text{d}u}}\text{d}s}\\ =-c\left(t\right)\left[1-{\text{e}}^{-{\displaystyle {\int }_t^T\lambda \left(u\right)\text{d}u}}\right],\end{array}$

where the calculation of $\text{Δ}B\left(t\right)$ and $\text{Δ}D\left(t\right)$ can be further referred to Appendix B, it follows that

$\begin{array}{c}\text{Δ}P\left(t,T\right)=P\left(t,T\right)-P\left(t-,T\right)=\text{Δ}A\left(t\right)+\text{Δ}B\left(t\right)+\text{Δ}C\left(t\right)+\text{Δ}D\left(t\right)\\ =-c\left(t\right).\end{array}$(15)

The solution that satisfies Equations (14)-(15), and the terminal condition $P\left(T,T\right)=1$ is given by

$P\left(t,T\right)={\text{e}}^{-{\displaystyle {\int }_t^T\bar{r}\left(u\right)\text{d}u}}+{\displaystyle \underset{\left(t,T\right]}{\int }{\text{e}}^{-{\displaystyle {\int }_t^v\bar{r}\left(u\right)\text{d}u}}\text{d}\mathcal{C}}\left(v\right).$(12)

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