Pricing a Defaultable Zero-Coupon Bond under Imperfect Information and Regime Switching

Abstract

We propose a pricing formula for a defaultable zero-coupon bond with imperfect information under a regime switching model using a structural form of credit risk modelling. This paper provides explicit representations of risky debt under regime switching with a constant interest rate and risky debt under regime switching with a regime switching interest rate. While the value of the firm’s equity is observed continuously, we assume that the total value of the firm is only observed at discrete times, such as the dates of the release of the firm’s annual reports, or quarterly reports. This uncertainty about the true value of the firm results in credit spreads that do not approach zero as the debt approaches maturity, which is a problem with many structural models. The firm’s value is typically decomposed into its equity and debt; however, we consider the asset–to–equity ratio, an accounting ratio used to examine a firm’s financial well-being. The parameters in our model are regime switching, where the regime can be thought of as the state of the economy. A Markov chain with a constant transition rate matrix produces the regime switching.

1. Introduction

The valuation of risky debt is fundamental to theoretical and empirical work in corporate finance [1]. Previous works such as [2] introduce the idea that the pricing of risky bonds is conditional on the firm value. The bond’s face value is paid in full if its firm’s value is above the default boundary. Otherwise, only a proportion of the claim is paid depending on the firm value and other liabilities of this firm. These are known as structural models of risky debt and are the criteria we use in this paper. Alternatively, many papers have found that this method and its extensions are not effective in pricing the credit spread as the bond’s maturity approaches zero; that is, typically, in these models, the credit spread approaches zero as the bond approaches maturity, which is not realistic. To solve this problem, [1,3,4] add a jump component to the firm value’s dynamic, which allows for a surprise default. Furthermore, many papers add different types of jumps; for example, see [5,6]. In addition, Merton’s credit risk modelling has been extended to a regime switching model with synchronous jumps in [7].

Structural bond pricing models commonly value risky debt as a derivative of the firm’s assets. This approach was introduced in the famous papers [8,9] and later [10] and the approach has since then drawn considerable attention in understanding credit risk by specifying a firm’s value process. One of the advantages of the structural bond pricing models is that it is possible to use price information from one class of securities, such as equity, to estimate the value of another, such as debt.

In the literature, it has been noted that one of the problems with structural models is that their credit spreads approach zero as the risky bond approaches its maturity (see [1]). Alternatively, other approaches have added jumps to the model of the value of the firm’s assets to add more uncertainty about whether or not there will be a default at maturity; see for example [4,5,11] and so on for different types of jump processes. As [12] states, “one can assume that at least one of the firm’s securities is traded and remains in a complete market setting. In such a framework, although a firm’s assets are not traded, their value can be replicated”.

In this paper we introduce our new methodology to price risky debt. Our approach is a simple way to keep the credit spreads from approaching zero at the bond’s maturity. Similar to [1], but simpler, our fundamental assumption is that the firm’s total value V is not continuously observable; i.e., it is only directly observable on specific dates, particularly when the firm’s annual or possibly quarterly reports are released. For this reason, we say that investors have imperfect information. On the other hand, the firm’s equity value, E, is continuously observed. Moreover, it turns out that working with multiplicative variables is more convenient mathematically. Therefore, we write $V=EH$, and thus define the ratio H as $H=V/E$. In industry, this is known as the asset–to–equity ratio which is an accounting ratio often used to examine a firm’s financial well-being. We then derive their stochastic differential equations, where the Brownian motion in E and H are correlated. Another assumption is that the default happens only at bond maturity if the firm’s value falls below a predetermined barrier.

Studying risky debt with imperfect information is one of our main contributions. Our method is similar to that employed in [1]. However, we assume the current released reports represent an accurate snapshot of the firm’s value rather than a noisy one. Because the firm’s reports are not given continuously, the lagged information is imperfect, particularly when the firm acquires different assets and liabilities over time, and their values vary. Thus, our key innovation is that, by assuming that the value of the firm is only observed (without noise) on fixed days, we can use this lagged information to derive explicit expressions for the price of risky debt, even with regime switching parameters.

In addition to modelling imperfect information, we also consider a structural model in which the parameters in the presented dynamics change due to regime switching, where the regime can be thought of as the state of the economy. A Markov chain with a constant transition rate matrix produces the regime switching. Since [13], many researchers have used the idea of regime switching (RS) in financial economics. One of their features is that the model dynamics can change over time according to the state of an underlying Markov chain, often interpreted as structural changes in economic conditions and different stages of business cycles. Default risk is influenced by business cycles and the macroeconomy, and it typically declines during economic expansion because this growth keeps the default rates low; the opposite is true in an economic recession. For this reason, it is reasonable to extend our idea to allow for regime switching parameters, as there is a need to develop some credit risk models that can take into account changes in market regimes. In particular, we provide an explicit representation of risky debt under regime switching with a constant interest rate (see Theorem 4), and risky debt under regime switching with a regime switching interest rate (see Theorem 5). Our approach can be extended nicely to price other derivatives, such as vulnerable options.

Although regime switching models have been widely used in credit risk modelling, they mainly focus on reduced-form models; see for example [14,15,16,17] among others. To our knowledge, some credit risk models with regime switching have recently been introduced to model default risk using the structural model, making it a hot topic to consider. For example, in pricing defaultable bonds under this framework, the authors of [18] price risky bonds using the Esscher transform, where a Markov-modulated generalized jump-diffusion model with a Markov-switching compensator governs the firm value. Alternatively, [19] models the firm value and the default boundary by two dependent regime switching jump diffusion processes, in which the Markov chain represents the states of an economy. Their numerical illustration suggests that the change of market regimes should be incorporated into the model for pricing credit derivatives.

Essentially, our model of imperfect information is a tractable approach that should give more realistic prices for short-term bonds, and the use of regime switching should give more realistic prices for long-term bonds.

This paper is arranged as follows. In Section 2, the model dynamics of the value of the firm and the Markov chain are introduced. Further, we study the moment-generating function of the logarithm of the value of the firm, $lnV\left(T\right)$, conditional on E as of date t, H as of date $t_k$, and the Markov chain as of date $t_k$, and give a justification as to why we used the methodology (see Section 3.1). In Section 3, we price the risky debt with two assumptions about the interest rate, which will define when the Markov chain is observed. Theorem 4 is a technical representation of risky debt under regime switching with a constant interest rate and Theorem 5 is the pricing for risky debt under regime switching with a regime switching interest rate. Section 4 concludes the paper.

2. The Model Dynamics

We consider a continuous-time financial market with a finite time horizon $T<\infty$, where we assume, for simplicity, that $(\mathrm{\Omega },\mathcal{F},\mathbb{Q})$ is a complete probability space and $\mathbb{Q}$ is an equivalent martingale measure. To be clear, a regime switching model is not a complete market (in the financial sense), and so there are many equivalent martingale measures. Thus, we are assuming that one particular equivalent martingale measure, under which all discounted price processes are $\mathbb{Q}$-martingales, is given. In general, it is not convenient to define the risk-neutral measure for the Markov chain, $\mathbf{X}$, defined in the following paragraph (see [7,20]), and so it is beyond the scope of this paper to discuss how this measure is selected.

Following [21], we suppose that the continuous-time, finite-state, observable Markov chain $\left\{{\mathbf{X}}_t\right\}$ on the probability space takes values in a set of standard unit vectors $\mathcal{E}=\left\{e_1,e_2,\cdots ,e_N\right\}\subset {\mathbb{R}}^N$, where for each $1\le j\le N$, $e_j$ is the vector with 1 in the jth entry and 0 elsewhere. The set $\mathcal{E}$ is called the canonical state space of $\mathbf{X}$.

Let $\mathbf{A}={\left(a_{ij}\right)}_{i,j=1,\cdots ,N}$ be the transition rate matrix of $\mathbf{X}$ under the measure $\mathbb{Q}$. Here for $i\ne j,a_{ij}$ is the (constant) transition intensity from state $e_i$ to state $e_j$ in a small interval of time, and satisfies $a_{ij}\ge 0$ for $i\ne j$ and ${\sum }_i^N{a}_{ij}=0$, for each $i,j=1,2,\dots ,N$. The semi-martingale representation of $\mathbf{X}$ is

$${\mathbf{X}}_t=X_0+{\int }_t^T\mathbf{A}{\mathbf{X}}_{s}ds+{\mathbf{M}}_t,t\in [0,T],$$

where $\mathbf{M}\left(t\right)$ is an $({\mathbb{F}}^{\mathbf{X}},\mathbb{Q})$ martingale in ${\mathbb{R}}^N$ and for each t, $\mathbf{X}\left(t\right)$ is bounded and its filtration is given by

$${\mathbb{F}}^{\mathbf{X}}:=\left\{{\mathcal{F}}_t^{\mathbf{X}},t\in \mathcal{T}\right\},\mathcal{T}=[0,T].$$

(1)

Note that ${\mathcal{F}}^{\mathbf{X}}\left(t\right)=\sigma \left\{\mathbf{X}\left(s\right):s\le t\right\}$, where and ${\mathcal{F}}^{\mathbf{X}}$ are assumed to be the right-continuous, $\mathbb{Q}$-complete, natural filtration generated by $\mathbf{X}$. (Note that $\mathbf{X}$ is càdlàg, and so is Riemann integrable, which implies that ${\int }_0^T{\mathbf{X}}_{s-}ds={\int }_0^T{\mathbf{X}}_{s+}ds={\int }_0^T{\mathbf{X}}_{s}ds$.) Let ${\mathbf{W}}_t=\left(W_1,W_2\right)$ be two standard Brownian motions which are mutually independent and also independent of ${\mathbf{X}}_t$ on the space $(\mathrm{\Omega },\mathcal{F},\mathbb{Q})$ and given a general column vector c, define a stochastic process $c_s$ as $c^{\top }{\mathbf{X}}_s$.

Recall that the fundamental assumption is that the firm’s total value V is not always observable. It can, however, be reflected in the decomposition of the equity value E and H, which is defined by the relation $V=EH$ or $H=V/E$. In our case, E is observable and tradable, and H is a ratio that could be explained in a stochastic differential equation, where the Brownian motion in E and H are correlated. In particular, the dynamics of E, H, and V under the risk-neutral measure $\mathbb{Q}$ evolve according to the following stochastic differential equations, where their parameters depend on the Markov chain $\mathbf{X}$:

$$\begin{array}{ccc}\hfill dE& =& r\left(t\right)Edt+{\sigma }_E\left(t\right)Ed{W}_1,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill dV& =& r\left(t\right)Vdt+{\sigma }_1\left(t\right)Vd{W}_1+{\sigma }_2\left(t\right)Vd{W}_2,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill dH& =& {\mu }_H\left(t\right)Hdt+{\sigma }_H\left(t\right)H\left({\rho }_{12}\left(t\right)d{W}_1+\sqrt{1-{\rho }_{12}^2\left(t\right)}d{W}_2\right),\hfill \end{array}$$

(4)

where H is derived by applying Itô’s lemma to the relation $H=V/E$. Here, ${\mu }_H\left(t\right)=-{\rho }_{12}\left(t\right){\sigma }_E\left(t\right){\sigma }_H\left(t\right)$, ${\sigma }_1\left(t\right)={\sigma }_E\left(t\right)+{\rho }_{12}\left(t\right){\sigma }_H\left(t\right)$ and ${\sigma }_2\left(t\right)={\sigma }_H\left(t\right)\sqrt{1-{\rho }_{12}^2\left(t\right)}.$

For ${\sigma }_E\left(t\right)$, which is the equity’s volatility, we assume that there is a constant $N\times 1$ vector, ${\sigma }_E={\left[{\sigma }_{E,1},{\sigma }_{E,2},\dots ,{\sigma }_{E,N}\right]}^{\top }$, such that ${\sigma }_E\left(t\right)={\sigma }_E^{\top }{\mathbf{X}}_t$, and similarly for the parameters ${\sigma }_H\left(t\right)$ and ${\rho }_{12}\left(t\right)$ and sometimes $r\left(t\right)$ (see Section 3.1 and Section 3.2). Further, we assume that ${\sigma }_E$ and ${\sigma }_H$ are strictly positive component-wise. Thus, the parameters change when the Markov chain jumps to a different state.

Note that Equations (2) and (3) assume risk neutrality, as their growth rates are equal to the risk-free rate. On the other hand, the drift for H in Equation (4) is determined by the Itô formula.

As discussed in the introduction, we assume that the value of the firm’s equity, E, is observed continuously over time, but the total value of the firm, V, can only be observed at dates $t_1,t_2,\dots ,t_N$. For the sake of generality, we also assume that the state of the Markov chain, $\mathbf{X}$, is not observed continuously. For convenience, we assume that $\mathbf{X}$ is observed at date $t_k$, where $t_k\le t<t_{k+1}$. The intuition behind this assumption is that $\mathbf{X}$ is hard to observe directly, and so it may take some time before the value of $\mathbf{X}$ can be determined, a technique known as smoothing. In Section 3.2, we will assume that $\mathbf{X}$ is observed at any date t, which is the most common assumption made in the literature.

Let us define the following $\sigma$-algebras:

$${\mathcal{F}}_1\left(t\right)=\sigma \left\{W_1\left(s\right):s\le t\right\}\mathrm{and}{\mathcal{F}}_2\left(t\right)=\sigma \left\{W_2\left(s\right):s\le t\right\}.$$

Recall that ${\mathcal{F}}^{\mathbf{X}}\left(t\right)=\sigma \left\{\mathbf{X}\left(s\right):s\le t\right\}$. Moreover, we define an enlarged filtration

$${\mathcal{F}}^{*}(t,t_k)={\mathcal{F}}_1\left(t\right)\vee {\mathcal{F}}_2\left(t_k\right),$$

(5)

where the notation “ ∨ ” represents the minimal $\sigma$-field containing ${\mathcal{F}}_1\left(t\right)$ and ${\mathcal{F}}_2\left(t_k\right)$, and also we know that ${\mathcal{F}}_1\left(t_k\right)\subseteq {\mathcal{F}}_1\left(t\right)$ as $t_k\le t\le t_{k+1}$. We assume that the filtration given above satisfies the usual conditions. In addition, we assume ${\mathcal{F}}_T:={\mathcal{F}}^{*}(T,T)\vee {\mathcal{F}}^{\mathbf{X}}\left(T\right)=\mathcal{F}$.

We derive some technical results regarding the moment-generating function of $lnV\left(T\right)$ conditional on ${\mathcal{F}}^{*}\left(t,t_k\right)$ and ${\mathcal{F}}^{\mathbf{X}}\left(t_k\right)$. Throughout this section, we assume that the risk-free rate is regime switching. Then, in the following section, we price risky debt and consider two cases. In the first case, we assume that the risk-free rate is constant, but the state of the Markov chain is only observed at date $t_k$. Then, in the following subsection, we assume the risk-free rate is regime switching, and the Markov chain is observed at any date t.

For the calculations, we rewrite the processes for $E,H$ in differential form using It$\hat{\mathrm{o}}$’s lemma as follows

$$dln\left(E_t\right)=\left(r\left(t\right)-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(t\right)\right)dt+{\sigma }_E\left(t\right)d{W}_1\left(t\right)$$

(6)

and

$$dln\left(H_t\right)=\left({\mu }_H\left(t\right)-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(t\right)\right)dt+{\sigma }_H\left(t\right)\left(\rho \left(t\right)d{W}_1\left(t\right)+\sqrt{1-{\rho }^2\left(t\right)}d{W}_2\left(t\right)\right).$$

(7)

Note that integration with respect to the processes $ln\left(E_t\right)$ and $ln\left(H_t\right)$ is well defined, as the integrands are bounded and adapted to the filtration. Before proceeding to the main result, we will state some frequently used lemmas. The proofs of these results will be provided in the Appendix A.

Lemma 1. 

We assume that $t_k\le t<t_{k+1}\le T$. Given constant $N\times 1$ vectors $\mathbf{a}={\left[a_1,\dots ,a_N\right]}^{\top }$, $\mathbf{b}={\left[b_1,\dots ,b_N\right]}^{\top }$, and $\mathbf{c}={\left[c_1,\dots ,c_N\right]}^{\top }$, and defining the processes $a\left(t\right)\equiv a_t={\mathbf{a}}^{\top }{\mathbf{X}}_t,b\left(t\right)\equiv b_t={\mathbf{b}}^{\top }{\mathbf{X}}_t$, and $c\left(t\right)\equiv c_t={\mathbf{c}}^{\top }{\mathbf{X}}_t$, we have,

$$\begin{array}{cc}& \mathbb{E}\left[e^{{\int }_{t_k}^T{c}_{s}ds+{\int }_{t_k}^T{a}_{s}d{W}_1\left(s\right)+{\int }_{t_k}^T{b}_{s}d{W}_2\left(s\right)}|{\mathcal{F}}^{\mathbf{X}}\left(T\right),{\mathcal{F}}^{*}\left(t,t_k\right)\right]\hfill \\ & =exp\left\{{\int }_{t_k}^T{c}_{s}ds+{\int }_{t_k}^t{a}_{s}d{W}_1\left(s\right)\right\}exp\left\{{\displaystyle \frac{1}{2}}{\int }_t^T{a}_s^{2}ds+{\displaystyle \frac{1}{2}}{\int }_{t_k}^T{b}_s^{2}ds\right\}.\hfill \end{array}$$

See Appendix A.1 for the proof of Lemma 1.

Remark 1. 

The main expectation we want to evaluate under regime switching is

$$\mathbb{E}\left[e^{uln\left(V_T\right)}|{\mathcal{F}}^{*}\left(t,t_k\right),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]=\mathbb{E}\left[\mathbb{E}\left[e^{uln\left(V_T\right)}|{\mathcal{F}}^{*}\left(t,t_k\right),{\mathcal{F}}^{\mathbf{X}}\left(T\right)\right]|{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right],$$

(8)

which we now derive. This conditional expectation represents the moment-generating function for $ln\left(V_T\right)$.

Lemma 2. 

The conditional moment-generating function of $ln\left(V_T\right)$ is given by

$$\begin{array}{cc}\hfill & \mathbb{E}\left[e^{uln\left(V_T\right)}|{\mathcal{F}}^{*}\left(t,t_k\right),{\mathcal{F}}^{\mathbf{X}}\left(T\right)\right]\hfill \\ \hfill & =exp\left\{uln\left(E_t\right)+uln\left(H_{t_k}\right)+uln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}\hfill \\ \hfill & \times exp\left\{u{\int }_t^T{c}_1\left(s\right)ds+u^2{\int }_t^T{c}_2\left(s\right)ds\right\}\hfill \\ \hfill & \times exp\left\{u{\int }_{t_k}^t{b}_1\left(s\right)ds+u^2{\int }_{t_k}^t{b}_2\left(s\right)ds+u{\int }_{t_k}^{t}ln\left({\displaystyle \frac{E_t}{E_s}}\right){\beta }_{EH}^{T}d{\mathbf{X}}_s\right\},\hfill \end{array}$$

(9)

where

$$\begin{array}{cc}\hfill & c_1\left(s\right)=r\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)+{\mu }_H\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right),\hfill \\ & c_2\left(s\right)={\displaystyle \frac{1}{2}}{\left({\sigma }_E\left(s\right)+{\sigma }_H\left(s\right)\rho \left(s\right)\right)}^2+{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\left(1-{\rho }^2\left(s\right)\right),\hfill \\ & b_1\left(s\right)=\left({\mu }_H\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\right)-{\beta }_{EH}\left(s\right)\left(r\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)\right),\hfill \\ & b_2\left(s\right)={\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\left(1-{\rho }^2\left(s\right)\right),\hfill \end{array}$$

${\mu }_H=-2{\sigma }_H{\sigma }_E$, and ${\beta }_{EH}\left(t\right):={\displaystyle \frac{{\sigma }_H\left(t\right)\rho \left(t\right)}{{\sigma }_E\left(t\right)}}$.

See Appendix A.1 for the proof of Lemma 2.

We now define processes $U_1$ from $t_k\le s\le t$ and $U_2$ from $t\le s\le T$. We find $U_2$ first, as it is easier to derive than $U_1$.

Theorem 1. 

For $t\le s\le T$, consider the process,

$$U_2(t,T)=exp\left\{{\int }_t^T\tilde{b}\left(s\right)ds\right\},$$

(10)

where $\tilde{b}\left(s\right)={\tilde{\mathbf{b}}}^{\top }{\mathbf{X}}_s$ with $\tilde{\mathbf{b}}$ equal to a constant $N\times 1$ vector. Then,

$$\mathbb{E}\left[U_2(t,T)|{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]=U_2(t,t){\mathbf{1}}^{\top }exp\left\{(diag\left(\tilde{\mathbf{b}}\right)+\mathbf{A})(T-t)\right\}{\mathbf{X}}_t.$$

(11)

See Appendix A.1 for the proof of Theorem 1.

The next Lemma will be used in proving Theorem 2.

Lemma 3. 

Let $H(t,T)$ be an $N\times 1$ vector-valued function such that

$$H(t,T)=exp\left\{\mathbf{C}(T-t)+{\int }_t^T\mathbf{B}\left(s\right)ds\right\}H(t,t),$$

(12)

where $\mathbf{B}$ is an $N\times N$ matrix-valued function and $\mathbf{C}$ is a constant $N\times N$ matrix. Also, for any $N\times N$ matrix, $\mathbf{M}$, $exp\left\{\mathbf{M}\right\}$ is an an exponential matrix; that is, $exp\left\{\mathbf{M}\right\}=\mathbf{I}+\mathbf{M}+{\displaystyle \frac{1}{2!}}{\mathbf{M}}^2+{\displaystyle \frac{1}{3!}}{\mathbf{M}}^3+\dots$ Then,

$$H(t,T)=H(t,t)+\mathbf{C}{\int }_t^{T}H(t,s)ds+{\int }_t^T\mathbf{B}\left(s\right)H(t,s)ds,$$

(13)

where the initial condition $H(t,t)$ is a given constant vector. Moreover, if Equation (13) holds, then Equation (12) follows. This converse holds because the integral equation has a unique solution.

See Appendix A.1 for the proof of Lemma 3.

Recall that $\mathbf{B}$ and $\mathbf{C}$ are two $N\times N$ matrices. The Hadamard product of $\mathbf{B}$ and $\mathbf{C}$, denoted by $\mathbf{B}\circ \mathbf{C}$, is the $N\times N$ matrix having a $(i,j)$th element equal to $B_{ij}{C}_{ij}$.

The next three results are needed in Section 3.1.

Theorem 2. 

For $t_k\le s\le t$, consider the process,

$$\begin{array}{cc}\hfill U_1\left(t_k,t\right)& =exp\left\{{\int }_{t_k}^t\tilde{c}\left(s\right)ds+{\int }_{t_k}^{t}uln\left({\displaystyle \frac{E_{t-}}{E_{s-}}}\right){\beta }_{EH}^{\top }d{\mathbf{X}}_s\right\}\hfill \\ \hfill & =exp\left\{{\int }_{t_k}^t\tilde{c}\left(s\right)ds+{\int }_{t_k}^{t}uln\left({\displaystyle \frac{E_{t-}}{E_{s-}}}\right){\mathbf{X}}_{s-}^{\top }\mathbf{B}d{\mathbf{X}}_s\right\},\hfill \end{array}$$

(14)

where $\tilde{c}\left(s\right)={\tilde{\mathbf{c}}}^{\top }{\mathbf{X}}_s$ with $\tilde{\mathbf{c}}$ equal to a constant $N\times 1$ vector and $\mathbf{B}$ is a constant $N\times N$ matrix

$$\mathbf{B}=\left[\begin{array}{c}{\beta }_{EH}^{\top }\\ ⋮\\ {\beta }_{EH}^{\top }\end{array}\right]=\left[\begin{array}{ccc}{\beta }^1& \cdots & {\beta }^K\\ ⋮& \ddots & ⋮\\ {\beta }^1& \cdots & {\beta }^K\end{array}\right].$$

Then,

$$\mathbb{E}\left[U_1\left(t_k,t\right){\mathbf{X}}_t|{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]=exp\left\{(diag\left(\tilde{\mathbf{c}}\right)+\mathbf{A})\left(t-t_k\right)\right\}{U}_1\left(t_k,t_k\right){\mathbf{X}}_{t_k},$$

(15)

where $U_1\left(t_k,t_k\right)=1$.

See Appendix A.1 for the proof of Theorem 2.

Then Equation (9) can be written as

$$\begin{array}{ccc}\hfill & & \mathbb{E}\left[e^{uln\left(V_T\right)}|{\mathcal{F}}^{*}\left(t,t_k\right),{\mathcal{F}}^{\mathbf{X}}\left(T\right)\right]\hfill \\ & =& exp\left\{uln\left(E_t\right)+uln\left(H_{t_k}\right)+uln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}{U}_1\left(t_k,t\right)U_2\left(t,T\right).\hfill \end{array}$$

(16)

Now our main expectation in Equation (8) is presented in the following theorem.

Theorem 3. 

Using the tower property, we can prove the following result

$$\begin{array}{ccc}\hfill & & \mathbb{E}\left[\mathbb{E}\left[e^{uln\left(V_T\right)}|{\mathcal{F}}^{*}\left(t,t_k\right),{\mathcal{F}}^{\mathbf{X}}\left(T\right)\right]|{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ & =& {\mathbf{1}}^{\top }exp\left\{uln\left(E_t\right)+uln\left(H_{t_k}\right)+uln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}\hfill \\ & & \times exp\left\{(diag\left(\tilde{\mathbf{b}}\right)+\mathbf{A})(T-t)\right\}\times exp\left\{(diag\left(\tilde{\mathbf{c}}\right)+\mathbf{A})\left(t-t_k\right)\right\}.\hfill \end{array}$$

(17)

Proof. 

We have,

$$\begin{array}{cc}\hfill & \mathbb{E}\left[\mathbb{E}\left[e^{uln\left(V_T\right)}|{\mathcal{F}}^{*}\left(t,t_k\right),{\mathcal{F}}^{\mathbf{X}}\left(T\right)\right]|{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ \hfill & =\mathbb{E}\left[exp\left\{uln\left(E_t\right)+uln\left(H_{t_k}\right)+uln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}{U}_1\left(t_k,t\right)U_2\left(t,T\right){\mathbf{X}}_T|{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ \hfill & =exp\left\{uln\left(E_t\right)+uln\left(H_{t_k}\right)+uln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}\mathbb{E}\left[U_1\left(t_k,t\right)U_2\left(t,T\right){\mathbf{X}}_T|{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ \hfill & =exp\left\{uln\left(E_t\right)+uln\left(H_{t_k}\right)+uln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}\hfill \\ & \times \mathbb{E}\left[\mathbb{E}\left[U_1\left(t_k,t\right)U_2(t,T){\mathbf{X}}_T|{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]|{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ \hfill & =exp\left\{uln\left(E_t\right)+uln\left(H_{t_k}\right)+uln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}\hfill \\ \hfill & \times \mathbb{E}\left[U_1\left(t_k,t\right)\mathbb{E}\left[U_2(t,T){\mathbf{X}}_T|{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]|{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ \hfill & =exp\left\{uln\left(E_t\right)+uln\left(H_{t_k}\right)+uln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}exp\left\{(diag\left(\tilde{\mathbf{b}}\right)+\mathbf{A})(T-t)\right\}\hfill \\ & \times \mathbb{E}\left[U_1\left(t_k,t\right)U_2(t,t){\mathbf{X}}_t|{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ \hfill & =exp\left\{uln\left(E_t\right)+uln\left(H_{t_k}\right)+uln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}\hfill \\ & \times exp\left\{(diag\left(\tilde{\mathbf{b}}\right)+\mathbf{A})(T-t)\right\}\mathbb{E}\left[U_1\left(t_k,t\right){\mathbf{X}}_t|{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ \hfill & ={\mathbf{1}}^{\top }exp\left\{uln\left(E_t\right)+uln\left(H_{t_k}\right)+uln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}\hfill \\ \hfill & \times exp\left\{(diag\left(\tilde{\mathbf{b}}\right)+\mathbf{A})(T-t)\right\}exp\left\{(diag\left(\tilde{\mathbf{c}}\right)+\mathbf{A})\left(t-t_k\right)\right\}.\hfill \end{array}$$

We use Theorems 1 and 2 as well as the tower property in the third equality, and in the sixth equality we use the fact that $U_2(t,t)=1$. □

Now, in the next section, we want to price the risky debt under regime switching with imperfect information.

3. Risky Debt under Regime Switching Model

In this section, we present the price of a defaultable zero-coupon bond with regime switching in the structural form of credit risk modelling. Although regime switching models have been widely used in credit risk modelling, previous work mainly focused on the reduced-form models, and little research has discussed the structural model with defaultable bonds.

The literature in this area is still maturing, and people continue to work on this problem. The paper [18] extends the model in [9] as their dynamic of the firm value consists of a Markov-modulated generalized jump-diffusion model where the jump component is described by a random measure, in which the jump sizes and jump times can be correlated. In addition, they allow for a Markov-switching compensator that switches over time as modelled by a continuous-time Markov chain according to the states of the economy.

For our calculations, recall the following $\sigma$-algebras

$${\mathcal{F}}_1\left(t\right)=\sigma \left\{W_1\left(s\right):s\le t\right\}\mathrm{and}{\mathcal{F}}_2\left(t\right)=\sigma \left\{W_2\left(s\right):s\le t\right\}\mathrm{and}{\mathcal{F}}^{\mathbf{X}}\left(t\right)=\sigma \left\{\mathbf{X}\left(s\right):s\le t\right\}.$$

As in Equation (5), we have ${\mathcal{F}}^{*}(t,t_k)={\mathcal{F}}_1\left(t\right)\vee {\mathcal{F}}_2\left(t_k\right)$, where the notation “ ∨ ” represents the minimal $\sigma$-field containing ${\mathcal{F}}_1\left(t\right)$ and ${\mathcal{F}}_2\left(t\right)$.

3.1. Constant Interest Rate

We now price risky debt in a regime switching framework; i.e., the parameters for the processes, V, E, and H are regime switching, according to the Markov chain, $\mathbf{X}$. Here, V represents the value of the firm, E is the market value of the firm’s equity, and $V=HE$, as usual.

In this sub-section, we assume that the state of the Markov chain is only observed at certain dates. Therefore, we assume that the risk-free rate is constant; otherwise, the interest rate would not be continuously observable, in general. The technical results that we will mainly use are Lemma 1 and the moment-generating function in Theorem 3.

Recall that state of the Markov chain, $\mathbf{X}$, is not observed continuously. For convenience, we assume that $\mathbf{X}$ is observed at date $t_k$, where $t_k\le t<t_{k+1}\le t_N$. So, the price of a defaultable zero-coupon with face value $1, but now conditional on $\mathbf{X}\left(t_k\right)$, is given by

$$\begin{array}{cc}\hfill P_t\left(E,H,V\right)& ={\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-r(T-t)}\left(1_{\left\{V\left(T\right)\ge D^{*}\right\}}+1_{\left\{V\left(T\right)<D^{*}\right\}}.{\displaystyle \frac{(1-\alpha )}{D}}V\left(T\right)\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ & ={\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-r(T-t)}{1}_{\left\{V\left(T\right)\ge D^{*}\right\}}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ & +{\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-r(T-t)}{1}_{\left\{V\left(T\right)<D^{*}\right\}}{\displaystyle \frac{(1-\alpha )}{D}}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ & ={\mathrm{e}}^{-r(T-t)}{\mathbb{E}}^{\mathbb{Q}}\left[1_{\left\{V\left(T\right)\ge D^{*}\right\}}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ & +{\displaystyle \frac{(1-\alpha )}{D}}{\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-r(T-t)}{1}_{\left\{V\left(T\right)<D^{*}\right\}}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right],\hfill \end{array}$$

(18)

where

• $D^{*}$ is a constant default boundary such that a credit loss occurs if the value of the firm’s assets satisfy $V\left(T\right)<D^{*}$.
• D is the value of total liabilities given by $D^{*}$ plus an additional liability as there is a possibility of a counter-party keeping operation even while $V\left(T\right)<D^{*}$, $D=D^{*}+\mathrm{additional}\mathrm{liability}.$
• $\alpha$ is the deadweight cost related to the bankruptcy of the firm, expressed as a percentage of $V\left(T\right)$.

The entire claim is paid out when $V\left(T\right)\ge D^{*}$. However, if the default occurs, only a fraction ${\displaystyle \frac{(1-\alpha )V\left(T\right)}{D}}$ of the claim is paid out, where ${\displaystyle \frac{V\left(T\right)}{D}}$ is the ratio representing the value of the firm which is available to pay the claim. Note that in most applications, $D=D^{*}$, but this notation is useful for extensions to modelling vulnerable options, which have counter-party risk.

For the expectation ${\mathbb{E}}^{\mathbb{Q}}\left[1_{\left\{V\left(T\right)\ge D^{*}\right\}}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]$ we need the characteristic function under the probability measure $\mathbb{Q}$ which is introduced in Equation (16).

Now, we focus on the conditional expectation

$${\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-r(T-t)}{1}_{\left\{V\left(T\right)<D^{*}\right\}}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right].$$

(19)

As mentioned in the introduction to this paper, recall that V is not a discrete-time stochastic process. The firm’s owners observe V continuously over time, but they do not allow outsiders to observe it, so outsiders only observe it on specific dates. Thus, V is defined for all t. It is simply the case that for pricing, we have to condition our expectations on information at date $t_k$.

Define the following equivalent probability measure at time t

$${\displaystyle \frac{d{\mathbb{Q}}^V}{d\mathbb{Q}}}={\displaystyle \frac{e^{-r\left(T-t\right)}V\left(T\right)}{V\left(t\right)}}$$

or at time zero

$${\displaystyle \frac{d{\mathbb{Q}}^V}{d\mathbb{Q}}}={\displaystyle \frac{e^{-r\left(T\right)}V\left(T\right)}{V\left(0\right)}}.$$

Here ${\mathbb{Q}}^V$ is the probability measure on $(\mathrm{\Omega },\mathcal{F})$. Note that this Radon–Nikodym derivative satisfies the necessary conditions to define a new probability measure (see e.g., [22] Theorem 1). Using abstract Bayes’ theorem to evaluate the conditional expectation in Equation (19), we have

$${\mathbb{E}}^V\left[1_{\left\{V\left(T\right)<D^{*}\right\}}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]={\displaystyle \frac{\mathbb{E}\left[{\mathrm{e}}^{-r\left(T-t\right)}{1}_{\left\{V\left(T\right)<D^{*}\right\}}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]}{\mathbb{E}\left[{\mathrm{e}}^{-r\left(T-t\right)}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]}},$$

(20)

and we see that

$$\begin{array}{ccc}\hfill & & \mathbb{E}\left[{\mathrm{e}}^{-r\left(T-t\right)}{1}_{\left\{V\left(T\right)<D^{*}\right\}}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ & =& \mathbb{E}\left[{\mathrm{e}}^{-r\left(T-t\right)}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ & & \times {\mathbb{Q}}^V\left\{V\left(T\right)<D^{*}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right\},\hfill \end{array}$$

(21)

where the $V\left(t\right)$ terms have cancelled. Then by Theorem 3, Equation (21) becomes

$$\begin{array}{cc}\hfill & \mathbb{E}\left[{\mathrm{e}}^{-r\left(T-t\right)}{1}_{\left\{V\left(T\right)<D^{*}\right\}}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ \hfill & ={\mathbf{1}}^{\top }{\displaystyle \frac{E_t}{E_{t_k}}}{V}_{t_k}exp\left\{ln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}exp\left\{(diag\left(\tilde{\mathbf{b}}\right)+\mathbf{A})(T-t)\right\}\hfill \\ \hfill & \times exp\left\{(diag\left(\tilde{\mathbf{c}}\right)+\mathbf{A})\left(t-t_k\right)\right\}{\mathbf{X}}_{t_k}\times {\mathbb{Q}}^V\left\{V\left(T\right)<D^{*}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right\},\hfill \end{array}$$

where

• $\tilde{b}\left(s\right)=-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)+{\mu }_H\left(s\right)+{\displaystyle \frac{1}{2}}{\left({\sigma }_E\left(s\right)+{\sigma }_H\left(s\right)\rho \left(s\right)\right)}^2-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right){\rho }^2\left(s\right)$,
• $\tilde{c}\left(s\right)=\left({\mu }_H\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\right)-{\beta }_{EH}\left(s\right)\left(r-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)\right)+{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\left(1-{\rho }^2\left(s\right)\right)$.

We have to evaluate ${\mathbb{Q}}^V\left(V\left(T\right)<D^{*}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right)$. We need the characteristic function of V under the probability measure ${\mathbb{Q}}^V$. The moment-generating function under this measure, which we denote by ${\varphi }^V\left(u\right)$, is as follows:

$$\begin{array}{cc}\hfill {\varphi }^V\left(u\right)& =E^{{\mathbb{Q}}^V}\left[{\mathrm{e}}^{ulnV\left(T\right)}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{E^{\mathbb{Q}}\left[{\mathrm{e}}^{ulnV\left(T\right)}{\mathrm{e}}^{-rT}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]}{E^{\mathbb{Q}}\left[{\mathrm{e}}^{-rT}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]}},\hfill \end{array}$$

(22)

where

$$\begin{array}{ccc}& & E^{\mathbb{Q}}\left[{\mathrm{e}}^{ulnV\left(T\right)}{\mathrm{e}}^{-rT}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]\hfill \\ & =& {\mathbf{1}}^{\top }exp\left\{-rt+\left(u+1\right)ln\left(E_t\right)+\left(u+1\right)ln\left(H_{t_k}\right)+\left(u+1\right)ln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}\hfill \\ & & \times exp\left\{(diag\left({\mathbf{b}}^{\prime }\right)+\mathbf{A})(T-t)\right\}exp\left\{(diag\left({\mathbf{c}}^{\prime }\right)+\mathbf{A})\left(t-t_k\right)\right\}{\mathbf{X}}_{t_k}.\hfill \end{array}$$

We can evaluate this by using Theorem 3 with the parameters defined in Equation (A11). Here, we have

• $b^{\prime }\left(s\right)=\left(u+1\right)\left(-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)+{\mu }_H\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\right)+{\left(u+1\right)}^2{c}_2\left(s\right),$
• $c^{\prime }\left(s\right)=\left(u+1\right)b_1\left(s\right)+{\left(u+1\right)}^2{b}_2\left(s\right),$

Next, we consider $E^{\mathbb{Q}}\left[{\mathrm{e}}^{-rT}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]$ following the same methodology with $u=1$, where

• $b^{\prime }\left(s\right)=\left(-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)+{\mu }_H\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\right)+c_2\left(s\right),$
• $c^{\prime }\left(s\right)=b_1\left(s\right)+b_2\left(s\right).$

Given the moment-generating function, the characteristic function is then just ${\varphi }^V\left(iu\right)$, and once we evaluate the characteristic function, we can find the probability by using the inverse Fourier transform technique as discussed, for example, in [23], i.e.,

$$P\{Y\ge a\}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }Re\left({\displaystyle \frac{e^{-iua}}{iu}}\varphi \left(u\right)\right)du.$$

(23)

We will use this in Theorem 4 below.

Theorem 4. 

The price of risky debt under regime switching with constant interest rate can be represented as the following

$$\begin{array}{cc}\hfill P_t(E,H,V)& ={\mathrm{e}}^{-r(T-t)}\mathbb{Q}\left(V\left(T\right)\ge D^{*}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right)\hfill \\ & +{\displaystyle \frac{E_t}{E_{t_k}}}{V}_{t_k}{\displaystyle \frac{(1-\alpha )}{D}}{\mathbf{1}}^{\top }exp\left\{-r\left(T-t\right)+ln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}\hfill \\ \hfill & \times exp\left\{(diag\left(\tilde{\mathbf{b}}\right)+\mathbf{A})(T-t)\right\}\times exp\left\{(diag\left(\tilde{\mathbf{c}}\right)+\mathbf{A})\left(t-t_k\right)\right\}{\mathbf{X}}_{t_k}\hfill \\ \hfill & \times {\mathbb{Q}}^V\left(V\left(T\right)<D^{*}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right),\hfill \end{array}$$

(24)

where $\tilde{\mathbf{b}}$, $\tilde{\mathbf{c}}$, and $\mathbf{J}$ are defined as in Equation (22) and the two probabilities in this expression are represented by

$$\mathbb{Q}\left(ln\left(V\left(T\right)\right)\ge ln\left(D^{*}\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }Re\left({\displaystyle \frac{e^{-ivln\left(D^{*}\right)}}{iv}}\varphi \left(iv\right)\right)dv$$

$${\mathbb{Q}}^V\left(ln\left(V\left(T\right)\right)<ln\left(D^{*}\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right)={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }Re\left({\displaystyle \frac{e^{-ivln\left(D^{*}\right)}}{iv}}{\varphi }^V\left(iv\right)\right)dv,$$

where $\varphi \left(iv\right)$ is defined as in Theorem 3 and ${\varphi }^V\left(iv\right)$ as in Equation (22).

3.2. Regime Switching Interest Rate

We now assume that the interest rate is regime switching, that is, $r\left(t\right)={\mathbf{r}}^{\top }\mathbf{X}\left(t\right)$ for a constant $N\times 1$ matrix, $\mathbf{r}={\left[r_1,\dots ,r_N\right]}^{\top }$. So, we assume that we can observe when there is a regime switch at any date. Loosely speaking, because E and r are observed continuously, we assume that $\mathbf{X}$ is, as well.

The price of a credit-risky bond is given by

$$\begin{array}{ccc}& & P_t\left(E_t,H_{t_k},V\right)\hfill \\ & & ={\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(s\right)ds}\left(1_{\left\{V\left(T\right)\ge D^{*}\right\}}+1_{\left\{V\left(T\right)<D^{*}\right\}}·{\displaystyle \frac{(1-\alpha )}{D}}V\left(T\right)\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]\hfill \\ & & ={\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(s\right)ds}{1}_{\left\{V\left(T\right)\ge D^{*}\right\}}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]\hfill \\ & & +{\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(s\right)ds}{1}_{\left\{V\left(T\right)<D^{*}\right\}}·{\displaystyle \frac{(1-\alpha )}{D}}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right].\hfill \end{array}$$

(25)

To solve ${\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(s\right)ds}{1}_{\left\{V\left(T\right)\ge D^{*}\right\}}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]$, we define the following measure:

$${\displaystyle \frac{d{\mathbb{Q}}^B}{d\mathbb{Q}}}={\displaystyle \frac{{\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(s\right)ds}|{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]}{{\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_0^{T}r\left(s\right)ds}\right]}}.$$

Here ${\mathbb{Q}}^B$ is a probability measure on $(\mathrm{\Omega },\mathcal{F})$. Note that this Radon–Nikodym derivative satisfies the necessary conditions to define a new probability measure (see e.g., [22] Theorem 1).

Let $B(t,T)={\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(s\right)ds}|{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]$ represent the price, as of date t, of a risk-free bond that pays $1 at date T. This implies that

$$\begin{array}{ccc}& & {\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(s\right)ds}{1}_{\left\{V\left(T\right)\ge D^{*}\right\}}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]\hfill \\ & & ={\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(s\right)ds}|{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]{\mathbb{E}}^{{\mathbb{Q}}^B}\left[1_{\left\{V\left(T\right)\ge D^{*}\right\}}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]\hfill \\ & & =B(t,T){\mathbb{E}}^{{\mathbb{Q}}^B}\left[1_{\left\{V\left(T\right)\ge D^{*}\right\}}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]\hfill \\ & & =B(t,T){\mathbb{Q}}^B\left(V\left(T\right)\ge D^{*}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right).\hfill \end{array}$$

(26)

Here, $B(t,T)$ can be represented as in [24] on page 284, which prices bonds in a model with a regime switching interest rate. We have

$$B(t,T)=\mathbb{E}\left[e^{-{\int }_t^{T}r\left(s\right)ds}|{\mathcal{F}}_X\left(t\right)\right]={\mathbf{1}}^{\mathrm{T}}exp\left\{(-diag\left[\mathbf{r}\right]+\mathbf{A})(T-t)\right\}{\mathbf{X}}_t.$$

(27)

To evaluate ${\mathbb{Q}}^B\left(V\left(T\right)\ge D^{*}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right)$, we need the characteristic function of V under the probability measure ${\mathbb{Q}}^B$, which is also called a T-forward measure. We have

$$\begin{array}{ccc}\hfill {\varphi }^{{\mathbb{Q}}^B}\left(iu\right)& =& {\mathbb{E}}^{{\mathbb{Q}}^B}\left[{\mathrm{e}}^{iulnV\left(T\right)}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]\hfill \\ & =& {\displaystyle \frac{{\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(t\right)dt}{\mathrm{e}}^{iulnV\left(T\right)}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]}{{\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(t\right)dt}|{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]}}\hfill \\ & =& {\displaystyle \frac{{\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(t\right)dt}{\mathrm{e}}^{iulnV\left(T\right)}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]}{B(t,T)}},\hfill \end{array}$$

(28)

where

$$\begin{array}{ccc}& & \mathbb{E}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(s\right)ds}{e}^{iuln\left(V_T\right)}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(T\right)\right]\hfill \\ & =& exp\left\{iuln\left(E_t\right)+iuln\left(H_{t_k}\right)+iuln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}\hfill \\ & & \times exp\left\{\left(iu-1\right){\int }_t^T{c}_0\left(s\right)ds+iu{\int }_t^T{c}_1\left(s\right)ds+{\left(iu\right)}^2{\int }_t^T{c}_2\left(s\right)ds\right\}\hfill \\ & & \times exp\left\{iu{\int }_{t_k}^t{b}_1\left(s\right)ds+{\left(iu\right)}^2{\int }_{t_k}^t{b}_2\left(s\right)ds+iu{\int }_{t_k}^{t}ln\left({\displaystyle \frac{E_t}{E_s}}\right){\beta }_{EH}^{T}d{\mathbf{X}}_s\right\},\hfill \end{array}$$

(29)

and

$$\begin{array}{cc}& c_0\left(s\right)=r\left(s\right)\hfill \\ & c_1\left(s\right)=-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)+{\mu }_H\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\hfill \\ & c_2\left(s\right)={\displaystyle \frac{1}{2}}{\left({\sigma }_E\left(s\right)+{\sigma }_H\left(s\right)\rho \left(s\right)\right)}^2+{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\left(1-{\rho }^2\left(s\right)\right)\hfill \\ & b_1\left(s\right)=\left({\mu }_H\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\right)-{\beta }_{EH}\left(s\right)\left(r\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)\right)\hfill \\ & b_2\left(s\right)={\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\left(1-{\rho }^2\left(s\right)\right).\hfill \end{array}$$

Using Theorem 1, we have,

$$\begin{array}{ccc}\hfill & & {\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(t\right)dt}{\mathrm{e}}^{iulnV\left(T\right)}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]\hfill \\ & =& exp\left\{iuln\left(E_t\right)+iuln\left(H_{t_k}\right)+iuln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^{\top }{\mathbf{X}}_{t_k}\right\}\hfill \\ & & \times exp\left\{iu{\int }_{t_k}^t{b}_1\left(s\right)ds+{\left(iu\right)}^2{\int }_{t_k}^t{b}_2\left(s\right)ds+iu{\int }_{t_k}^{t}ln\left({\displaystyle \frac{E_t}{E_s}}\right){\mathbf{X}}_{s-}^{\top }\mathbf{B}d{\mathbf{X}}_s\right\}\hfill \\ & & \times {\mathbf{1}}^{\mathrm{T}}exp\left\{\left(diag\left(b^{\prime }\right)+\mathbf{A}\right)(T-t)\right\}{\mathbf{X}}_t,\hfill \end{array}$$

(30)

where

• ${\beta }_{EH}^{\top }d{\mathbf{X}}_s={\mathbf{X}}_{s-}^{\top }\mathbf{B}d{\mathbf{X}}_s$,
• $b^{\prime }=(iu-1)c_0\left(s\right)+\left(iu\right)c_1\left(s\right)+{\left(iu\right)}^2{c}_2\left(s\right)$.

Combining Equations (30) and (27) gives Equation (28), the characteristic function under ${\mathbb{Q}}^B$:

$${\mathbb{Q}}^B\left(ln\left(V\left(T\right)\right)\ge ln\left(D^{*}\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }Re\left({\displaystyle \frac{e^{-iuln\left(D^{*}\right)}}{iu}}{\varphi }^{Q^B}\left(u\right)\right).$$

For the second term in Equation (25), we have

$$\begin{array}{ccc}& & {\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(t\right)dt}{1}_{\left\{V\left(T\right)<D^{*}\right\}}·{\displaystyle \frac{(1-\alpha )}{D}}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]\hfill \\ & & ={\displaystyle \frac{(1-\alpha )}{D}}{\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{-{\int }_t^{T}r\left(t\right)dt}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]\hfill \\ & & \times {\mathbb{E}}^{{\mathbb{Q}}^{V^r}}\left[1_{\left\{V\left(T\right)<D^{*}\right\}}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right],\hfill \end{array}$$

(31)

where we used abstract Bayes’ theorem and the change of measure

$${\displaystyle \frac{d{\mathbb{Q}}^{V^r}}{d\mathbb{Q}}}={\displaystyle \frac{e^{-{\int }_0^{T}rdt}V\left(T\right)}{V\left(0\right)}}.$$

Here ${\mathbb{Q}}^{V^r}$ is probability measure on $(\mathrm{\Omega },\mathcal{F})$. Note that this Radon–Nikodym derivative satisfies the necessary conditions to define a new probability measure (see e.g., [22] Theorem 1). Now, we have to evaluate ${\mathbb{Q}}^{V^r}\left(V\left(T\right)<D^{*}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right)$. We need the characteristic function of V under the probability measure ${\mathbb{Q}}^{V^r}$. The moment-generating function under this measure, which we denote by ${\varphi }^{V^r}\left(u\right)$, is as follows:

$$\begin{array}{cc}\hfill {\varphi }^{V^r}\left(u\right)& ={\mathbb{E}}^{{\mathbb{Q}}^{V^r}}\left[{\mathrm{e}}^{ulnV\left(T\right)}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{{\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{ulnV\left(T\right)}{e}^{-{\int }_0^{T}rdt}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]}{{\mathbb{E}}^{\mathbb{Q}}\left[e^{-{\int }_0^{T}rdt}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]}}.\hfill \end{array}$$

(32)

Conditional on the given filtration above, the $e^{-{\int }_0^{t}rdt}$ terms cancel out.

For ${\mathbb{E}}^{\mathbb{Q}}\left[{\mathrm{e}}^{ulnV\left(T\right)}{e}^{-{\int }_0^{T}rdt}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]$, from Equation (A10) and conditional on ${\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)$, we have

$$\begin{array}{cc}\hfill & {\varphi }^{V^r}\left(u\right)\hfill \\ \hfill & =exp\left\{\left(u+1\right)ln\left(E_t\right)+\left(u+1\right)ln\left(H_{t_k}\right)+\left(u+1\right)ln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}\hfill \\ \hfill & \times exp\left\{\left(u+1\right){\int }_{t_k}^t{b}_1\left(s\right)ds+{\left(\left(u+1\right)\right)}^2{\int }_{t_k}^t{b}_2\left(s\right)ds+\left(u+1\right){\int }_{t_k}^{t}ln\left({\displaystyle \frac{E_t}{E_s}}\right){\mathbf{X}}_{s-}^{\top }\mathbf{B}d{\mathbf{X}}_s\right\}\hfill \\ & \times {\mathbf{1}}^{\mathrm{T}}exp\left\{\left(diag\left(b^{\prime \prime }\right)+\mathbf{A}\right)(T-t)\right\}{\mathbf{X}}_t,\hfill \end{array}$$

where

• $b^{\prime \prime }=ur\left(s\right)+\left(u+1\right)\left(-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)+{\mu }_H\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right),\right)+{\left(u+1\right)}^2{c}_2\left(s\right)$
• $c_2$, $b_1$ and $b_2$ are defined by Equation (A10), and

${\mathbb{E}}^{\mathbb{Q}}\left[e^{-{\int }_0^{T}rdt}V\left(T\right)|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]$ is the same as above with $u+1=0$. Again, given the moment-generating function, the characteristic function is just ${\varphi }^{V^r}\left(iu\right)$. Once we evaluate the characteristic function, we can find the probability by using Equation (23), as we discuss in Theorem 5 below.

Theorem 5. 

The price of risky debt under regime switching with regime switching interest rate can be represented as the following

$$\begin{array}{ccc}& & P_t(E,H,V)\hfill \\ & =& B(t,T){\mathbb{Q}}^B\left(V\left(T\right)\ge D^{*}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right)\hfill \\ & & +{\displaystyle \frac{E_t}{E_{t_k}}}{V}_{t_k}{\displaystyle \frac{(1-\alpha )}{D}}exp\left\{ln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}\hfill \\ & & \times exp\left\{{\int }_{t_k}^t{b}_1\left(s\right)ds+{\int }_{t_k}^t{b}_2\left(s\right)ds+{\int }_{t_k}^{t}ln\left({\displaystyle \frac{E_t}{E_s}}\right){\mathbf{X}}_{s-}^{\top }\mathbf{B}d{\mathbf{X}}_s\right\}\hfill \\ & & \times {\mathbf{1}}^{\mathrm{T}}exp\left\{\left(diag\left(\hat{\mathbf{b}}\right)+\mathbf{A}\right)(T-t)\right\}{\mathbf{X}}_t\hfill \\ & & \times {\mathbb{Q}}^{V^r}\left(V\left(T\right)<D^{*}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right),\hfill \end{array}$$

(33)

where $\hat{\mathbf{b}}$ is the same as ${\mathbf{b}}^{″}$ with $u=1$. Also, $B(t,T)$, ${\mathbb{Q}}^B\left(V\left(T\right)\ge D^{*}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right)$, and ${\mathbb{Q}}^{V^r}\left(V\left(T\right)<D^{*}|{\mathcal{F}}^{*}(t,t_k),{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right)$ given by Equations (27), (28), and (32), respectively.

4. Conclusions

This paper considers the problem of pricing risky debt under the regime switching model with imperfect information. That is, since the firm’s reports are not given continuously, the lagged information is imperfect; particularly when the firm acquires different assets and liabilities over time, and their values vary. Thus, our key innovation is that, by assuming that the firm’s value is observed (without noise) on fixed days, we can use this lagged information to derive explicit expressions for the price of risky debt, even with regime switching parameters. Our assumption of imperfect information gives a more realistic approach to pricing risky debt for short maturities. The regime switching, in turn, gives reasonable outcomes for long maturities. From these assumptions, we develop a simple approach to valuing risky debt that incorporates default risk in their valuation, and this can be extended nicely to price other derivatives. In particular, we provide a technical representation of risky debt under regime switching with a constant interest rate (see Theorem 4), and risky debt under regime switching with a regime switching interest rate (see Theorem 5). It should be noted that regime switching is very adaptive, and this approach can be applied to different derivatives, possibly with stochastic volatility. Furthermore, an interesting extension of this paper would be to assume that some states of $\mathbf{X}$ can be observed continuously (e.g., states that affect E, or the stochastic interest rate, r), but other states can only be determined with a time delay (e.g., states affecting V). We leave these extensions for future research.

One limitation of this model is that, like the Merton model [9], it is hard to price bonds that may default before the maturity date, such as coupon bonds.

Appendix A.1. Proofs of Section 2

In this appendix, we provide the proofs of the results of this paper.

Proof of Lemma 1. 

We are given constant $N\times 1$ matrices $\mathbf{a}={\left[a_1,\dots ,a_N\right]}^{\top }$, $\mathbf{b}={\left[b_1,\dots ,b_N\right]}^{\top }$, and $\mathbf{c}={\left[c_1,\dots ,c_N\right]}^{\top }$, and we define the processes $a\left(t\right)\equiv a_t={\mathbf{a}}^{\top }{\mathbf{X}}_t,b\left(t\right)\equiv b_t={\mathbf{b}}^{\top }{\mathbf{X}}_t$, and $c\left(t\right)\equiv c_t={\mathbf{c}}^{\top }{\mathbf{X}}_t$. Note that

$$\begin{array}{cc}& \mathbb{E}\left[exp\left\{{\int }_{t_k}^T{c}_{s}ds+{\int }_{t_k}^T{a}_{s}d{W}_1\left(s\right)+{\int }_{t_k}^T{b}_{s}d{W}_2\left(s\right)\right\}|{\mathcal{F}}^{\mathbf{X}}\left(T\right),{\mathcal{F}}^{*}\left(t,t_k\right)\right]\hfill \\ & =exp\left\{{\int }_{t_k}^T{c}_{s}ds+{\int }_{t_k}^t{a}_{s}d{W}_1\left(s\right)\right\}\mathbb{E}\left[exp\left\{{\int }_t^T{a}_{s}d{W}_1\left(s\right)+{\int }_{t_k}^T{b}_{s}d{W}_2\left(s\right)\right\}|{\mathcal{F}}^{\mathbf{X}}\left(T\right),{\mathcal{F}}^{*}\left(t,t_k\right)\right]\hfill \\ & =exp\left\{{\int }_{t_k}^T{c}_{s}ds+{\int }_{t_k}^t{a}_{s}d{W}_1\left(s\right)\right\}exp\left\{{\displaystyle \frac{1}{2}}{\int }_t^T{a}_s^{2}ds+{\displaystyle \frac{1}{2}}{\int }_{t_k}^T{b}_s^{2}ds\right\}.\hfill \end{array}$$

The second equality follows from the fact that, conditional on ${\mathcal{F}}^{\mathbf{X}}\left(T\right)$, the integrals ${\int }_t^T{a}_{s}d{W}_1\left(s\right)$ and ${\int }_t^T{b}_{s}d{W}_2\left(s\right)$, are conditionally normally distributed with mean zero, and by the It$\hat{\mathrm{o}}$ isometry, their variances are given by ${\int }_t^T{a}_s^{2}ds\mathrm{and}{\int }_t^T{b}_s^{2}ds$, respectively. This proves the Lemma. □

Proof of Lemma 2. 

First, note that

$$\begin{array}{ccc}\hfill & & \mathbb{E}\left[e^{uln\left(V_T\right)}|{\mathcal{F}}^{*}\left(t,t_k\right),{\mathcal{F}}^{\mathbf{X}}\left(T\right)\right]\hfill \\ & & =\mathbb{E}\left[e^{uln\left(E_T\right)+uln\left(H_T\right)}|{\mathcal{F}}^{*}\left(t,t_k\right),{\mathcal{F}}^{\mathbf{X}}\left(T\right)\right]\hfill \\ & & =exp\left\{uln\left(E_t\right)+uln\left(H_{t_k}\right)+u{\int }_t^T\left(r\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)\right)ds\right\}\hfill \\ & & \times exp\left\{u{\int }_{t_k}^T\left({\mu }_H\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\right)ds+u{\int }_{t_k}^t{\sigma }_H\left(s\right)\rho \left(s\right)d{W}_1\left(s\right)\right\}\hfill \\ & & \times exp\left\{{\displaystyle \frac{1}{2}}{u}^2{\int }_t^T{\left({\sigma }_E\left(s\right)+{\sigma }_H\left(s\right)\rho \left(s\right)\right)}^{2}ds+{\displaystyle \frac{1}{2}}{u}^2{\int }_{t_k}^T{\sigma }_H^2\left(s\right)\left(1-{\rho }^2\left(s\right)\right)ds\right\}\hfill \\ & & =exp\left\{uln\left(E_t\right)+uln\left(H_{t_k}\right)\right\}\hfill \\ & & \times exp\left\{u{\int }_t^T\left(r\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)\right)ds+u{\int }_t^T\left({\mu }_H\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\right)ds\right\}\hfill \\ & & \times exp\left\{{\displaystyle \frac{1}{2}}{u}^2{\int }_t^T{\left({\sigma }_E\left(s\right)+{\sigma }_H\left(s\right)\rho \left(s\right)\right)}^{2}ds+{\displaystyle \frac{1}{2}}{u}^2{\int }_t^T{\sigma }_H^2\left(s\right)\left(1-{\rho }^2\left(s\right)\right)ds\right\}\hfill \\ & & \times exp\left\{u{\int }_{t_k}^t\left({\mu }_H\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\right)ds+{\displaystyle \frac{1}{2}}{u}^2{\int }_{t_k}^t{\sigma }_H^2\left(s\right)\left(1-{\rho }^2\left(s\right)\right)ds\right\}\hfill \\ & & \times exp\left\{u{\int }_{t_k}^t{\sigma }_H\left(s\right)\rho \left(s\right)d{W}_1\left(s\right)\right\}.\hfill \end{array}$$

(A1)

Here, the $u^2$ terms follow from Equations (6) and (7).

Next, we need to discuss some properties of the integral ${\int }_{t_k}^t{\sigma }_H\left(s\right)\rho \left(s\right)d{W}_1\left(s\right)$; in particular, we want to write this integral in terms of E. From Equation (6), note that

$$d{W}_1\left(t\right)={\displaystyle \frac{1}{{\sigma }_E\left(t\right)}}\left(dln\left(E_t\right)-\left(r\left(t\right)-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(t\right)\right)dt\right)$$

(A2)

which implies that

$$\begin{array}{cc}\hfill {\int }_{t_k}^t{\sigma }_H\left(s\right)\rho \left(s\right)d{W}_1\left(s\right)& ={\int }_{t_k}^t{\displaystyle \frac{{\sigma }_H\left(s\right)\rho \left(s\right)}{{\sigma }_E\left(s\right)}}\left(dln\left(E_s\right)-\left(r\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)\right)ds\right)\hfill \\ & ={\int }_{t_k}^t{\displaystyle \frac{{\sigma }_H\left(s\right)\rho \left(s\right)}{{\sigma }_E\left(s\right)}}dln\left(E_s\right)-{\int }_{t_k}^t{\displaystyle \frac{{\sigma }_H\left(s\right)\rho \left(s\right)}{{\sigma }_E\left(s\right)}}\left(r\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)\right)ds.\hfill \end{array}$$

(A3)

Let

$${\displaystyle \frac{{\sigma }_H\left(t\right)\rho \left(t\right)}{{\sigma }_E\left(t\right)}}:={\beta }_{EH}\left(t\right),$$

(A4)

which can be seen to be equal to ${\displaystyle \frac{{\sigma }_H^i{\rho }^i}{{\sigma }_E^i}}{1}_{\left\{{\mathbf{X}}_t={\mathbf{e}}_i\right\}}=:{\beta }^i{1}_{\left\{{\mathbf{X}}_i={\mathbf{e}}_i\right\}}={\beta }_{EH}^{\top }{\mathbf{X}}_t$, where

$${\beta }_{EH}^{\top }=\left({\beta }^1,\dots ,{\beta }^K\right),$$

(A5)

with

$${\beta }^i:={\displaystyle \frac{{\sigma }_H^i{\rho }^i}{{\sigma }_E^i}}.$$

(A6)

So, by Equations (A2)–(A6), we have

$$u{\int }_{t_k}^t{\sigma }_H\left(s\right)\rho \left(s\right)d{W}_1\left(s\right)=u{\int }_{t_k}^t{\beta }_{EH}\left(s\right)dln\left(E_s\right)-u{\int }_{t_k}^t{\beta }_{EH}\left(s\right)\left(r\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)\right)ds.$$

(A7)

To understand the distribution of the integral ${\int }_{t_k}^t{\beta }_{EH}\left(s\right)dln\left(E_s\right)$, we can use integration by parts:

$$\begin{array}{cc}\hfill & u{\int }_{t_k}^t{\beta }_{EH}\left(s\right)dln\left(E_s\right)\hfill \\ \hfill & =u{\beta }_{EH}\left(t\right)ln\left(E_t\right)-u{\beta }_{EH}\left(t_k\right)ln\left(E_{t_k}\right)-u{\int }_{t_k}^{t}ln\left(E_s\right)d{\beta }_{EH}\left(s\right)\hfill \\ \hfill & =uln\left(E_t\right){\beta }_{EH}^{\top }{\mathbf{X}}_t-uln\left(E_{t_k}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}-u{\int }_{t_k}^{t}ln\left(E_s\right)d{\beta }_{EH}\left(s\right)\hfill \\ \hfill & =uln\left(E_t\right){\beta }_{EH}^{\top }{\int }_{t_k}^{t}d{\mathbf{X}}_s+uln\left(E_t\right){\beta }_{EH}^{\top }{\mathbf{X}}_{t_k}-uln\left(E_{t_k}\right){\beta }_{EH}^{\top }{\mathbf{X}}_{t_k}\hfill \\ & -u{\int }_{t_k}^{t}ln\left(E_s\right)d{\beta }_{EH}\left(s\right)\hfill \\ \hfill & =uln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^{\top }{\mathbf{X}}_{t_k}+uln\left(E_t\right){\beta }_{EH}^{\top }{\int }_{t_k}^{t}d{\mathbf{X}}_s-u{\int }_{t_k}^{t}ln\left(E_s\right)d{\beta }_{EH}\left(s\right),\hfill \end{array}$$

(A8)

where we can define ${\int }_{t_k}^{t}ln\left(E_s\right)d{\beta }_{EH}\left(s\right)$ as

$$u{\int }_{t_k}^{t}ln\left(E_s\right)d{\beta }_{EH}\left(s\right)=u{\int }_{t_k}^{t}ln\left(E_s\right){\beta }_{EH}^{\top }d{\mathbf{X}}_s.$$

So, we have

$$\begin{array}{cc}\hfill u{\int }_{t_k}^t{\beta }_{EH}\left(s\right)dln\left(E_s\right)& =uln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^{\top }{\mathbf{X}}_{t_k}+uln\left(E_t\right){\beta }_{EH}^{\top }{\int }_{t_k}^{t}d{\mathbf{X}}_s-u{\int }_{t_k}^{t}ln\left(E_s\right){\beta }_{EH}^{\top }d{\mathbf{X}}_s\hfill \\ & =uln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^{\top }{\mathbf{X}}_{t_k}+u{\int }_{t_k}^{t}ln\left({\displaystyle \frac{E_t}{E_s}}\right){\beta }_{EH}^{\top }d{\mathbf{X}}_s.\hfill \end{array}$$

(A9)

Substituting Equation (A9) into Equation (A7) and then adding it to Equation (A1), we have shown that

$$\begin{array}{cc}\hfill & \mathbb{E}\left[e^{uln\left(V_T\right)}|{\mathcal{F}}^{*}\left(t,t_k\right),{\mathcal{F}}^{\mathbf{X}}\left(T\right)\right]\hfill \\ \hfill & =exp\left\{uln\left(E_t\right)+uln\left(H_{t_k}\right)+uln\left({\displaystyle \frac{E_t}{E_{t_k}}}\right){\beta }_{EH}^T{\mathbf{X}}_{t_k}\right\}\hfill \\ \hfill & \times exp\left\{u{\int }_t^T{c}_1\left(s\right)ds+u^2{\int }_t^T{c}_2\left(s\right)ds\right\}\hfill \\ \hfill & \times exp\left\{u{\int }_{t_k}^t{b}_1\left(s\right)ds+u^2{\int }_{t_k}^t{b}_2\left(s\right)ds+u{\int }_{t_k}^{t}ln\left({\displaystyle \frac{E_t}{E_s}}\right){\beta }_{EH}^{T}d{\mathbf{X}}_s\right\},\hfill \end{array}$$

(A10)

where

$$\begin{array}{cc}\hfill & c_1\left(s\right)=r\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)+{\mu }_H\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right),\hfill \\ \hfill & c_2\left(s\right)={\displaystyle \frac{1}{2}}{\left({\sigma }_E\left(s\right)+{\sigma }_H\left(s\right)\rho \left(s\right)\right)}^2+{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\left(1-{\rho }^2\left(s\right)\right),\hfill \\ \hfill & b_1\left(s\right)=\left({\mu }_H\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\right)-{\beta }_{EH}\left(s\right)\left(r\left(s\right)-{\displaystyle \frac{1}{2}}{\sigma }_E^2\left(s\right)\right),\hfill \\ \hfill & b_2\left(s\right)={\displaystyle \frac{1}{2}}{\sigma }_H^2\left(s\right)\left(1-{\rho }^2\left(s\right)\right),\hfill \end{array}$$

(A11)

and ${\mu }_H=-2{\sigma }_H{\sigma }_E$. □

Remark A1. 

For the distribution of the integral, ${\int }_{t_k}^t{\beta }_{EH}\left(s\right)dln\left(E_s\right)$, we need to stress our assumption regarding when the state of the Markov chain, $\mathbf{X}$, is observed or not observed. If we assume that $\mathbf{X}$ is observed continuously, then at date t, this integral is measurable with respect to σ-algebra defined on ${\mathcal{F}}^{*}(t,t_k)\vee {\mathcal{F}}^X\left(t\right)$. However, for the sake of generality, in this section, we assume that E is observed throughout the interval $[t_k,t]$, but $\mathbf{X}$ is only observable at time $t_k$; that is, we want to find conditional expectations given ${\mathcal{F}}^{*}(t,t_k)\vee {\mathcal{F}}^X\left(t_k\right)$.

Proof of Theorem 1. 

To prove this theorem, we start by applying the product rule to $exp\left\{{\int }_t^T\tilde{b}\left(s\right)ds\right\}{\mathbf{X}}_T$ which will allow us to find its expected value, which we denoted by $\mathbb{E}\left[U_2(t,T)|{\mathcal{F}}^{\mathbf{X}}\left(t\right)\right]$. By the product rule,

$$\begin{array}{cc}\hfill d\left(exp\left\{{\int }_t^T\tilde{b}\left(s\right)ds\right\}{\mathbf{X}}_T\right)& =exp\left\{{\int }_t^T\tilde{b}\left(s\right)ds\right\}d{\mathbf{X}}_t+exp\left\{{\int }_t^T\tilde{b}\left(s\right)ds\right\}\tilde{b}\left(t\right){\mathbf{X}}_{t_{-}}dt\hfill \\ & =U_2(t,T)\left(\mathbf{A}{\mathbf{X}}_{t}dt+M_t\right)+U_2(t,T)\tilde{b}\left(t\right){\mathbf{X}}_{t_{-}}dt.\hfill \end{array}$$

Note that $\tilde{b}\left(s\right)={\tilde{\mathbf{b}}}^{\top }{\mathbf{X}}_s$, with $\tilde{\mathbf{b}}$ equal to a constant $N\times 1$ vector. Also, $\left({\tilde{\mathbf{b}}}^{\top }{\mathbf{X}}_t\right){\mathbf{X}}_t=\mathrm{diag}\left(\tilde{\mathbf{b}}\right){\mathbf{X}}_t$ (to see this, it is not hard to confirm that $\left({\tilde{\mathbf{b}}}^{\top }{e}_i\right)e_i=\mathrm{diag}\left(\tilde{\mathbf{b}}\right)e_i$ for each unit vector, $e_i$; see, e.g., Remark 2.5 in Equation (11) from [25]). Integrating over t and then taking the expectation (note that M is martingale), we find that

$$\mathbb{E}\left[exp\left\{{\int }_t^T\tilde{b}\left(s\right)ds\right\}{\mathbf{X}}_T\right]=U_2(t,t){\mathbf{X}}_t+{\int }_t^T(diag\left(\tilde{\mathbf{b}}\right)+\mathbf{A})\mathbb{E}\left[exp\left\{{\int }_t^T\tilde{b}\left(s\right)ds\right\}{\mathbf{X}}_s\right]ds.$$

By Lemma 3, we obtain the result as desired. □

Proof of Lemma 3. 

By the definition of the exponential of a matrix, it is not hard to see that

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dT}}H(t,T)& ={\displaystyle \frac{d}{dT}}exp\left\{\mathbf{C}(T-t)+{\int }_t^T\mathbf{B}\left(s\right)ds\right\}H(t,t)\hfill \\ & =exp\left\{\mathbf{C}(T-t)+{\int }_t^T\mathbf{B}\left(s\right)ds\right\}(\mathbf{C}+\mathbf{B}\left(T\right))H(t,t)\hfill \\ & =(\mathbf{C}+\mathbf{B}\left(T\right))exp\left\{\mathbf{C}(T-t)+{\int }_t^T\mathbf{B}\left(s\right)ds\right\}H(t,t).\hfill \end{array}$$

Using the third equality and integration gives

$$\begin{array}{cc}\hfill H(t,T)& =H(t,t)+{\int }_t^T(\mathbf{C}+\mathbf{B}\left(s\right))exp\left\{\mathbf{C}(s-t)+{\int }_t^s\mathbf{B}\left(v\right)dv\right\}H(t,t)ds\hfill \\ & =H(t,t)+{\int }_t^T(\mathbf{C}+\mathbf{B}\left(s\right))H(t,s)ds,\hfill \end{array}$$

which proves the lemma. □

Proof of Theorem 2. 

With $t_k\le s\le t$, when ${\mathbf{X}}_{s-}={\mathbf{e}}_i$ and ${\mathbf{X}}_s={\mathbf{e}}_j$, it can be seen that, ${\mathbf{X}}_{s-}^{\top }\mathbf{B}d{\mathbf{X}}_s={\mathbf{e}}_i^{\top }\mathbf{B}\left({\mathbf{e}}_j-{\mathbf{e}}_i\right)={\mathbf{e}}_i^{\top }\left[\begin{array}{c}{\beta }^j\\ ⋮\\ {\beta }^j\end{array}\right]-{\mathbf{e}}_i^{\top }\left[\begin{array}{c}{\beta }^i\\ ⋮\\ {\beta }^i\end{array}\right]={\beta }^j-{\beta }^i$, where ${\beta }^i\ne 0$. Recall that

$$U_1\left(t_k,t\right)=exp\left\{{\int }_{t_k}^t\tilde{c}\left(s\right)ds+{\int }_{t_k}^{t}uln\left({\displaystyle \frac{E_{t-}}{E_{s-}}}\right){\mathbf{X}}_{s-}^{\top }\mathbf{B}d{\mathbf{X}}_s\right\},$$

where $\tilde{c}\left(s\right)={\tilde{\mathbf{c}}}^{\top }{\mathbf{X}}_s$ with $\tilde{\mathbf{c}}$ equal to a constant $N\times 1$ vector and $\mathbf{B}$ is the constant $N\times N$ matrix

$$\mathbf{B}=\left[\begin{array}{c}{\beta }_{EH}^{\top }\\ ⋮\\ {\beta }_{EH}^{\top }\end{array}\right]=\left[\begin{array}{ccc}{\beta }^1& \cdots & {\beta }^K\\ ⋮& \ddots & ⋮\\ {\beta }^1& \cdots & {\beta }^K\end{array}\right].$$

Note that

$${\int }_{t_k}^{t}uln\left({\displaystyle \frac{E_{t-}}{E_{s-}}}\right){\mathbf{X}}_{s-}^{\top }\mathbf{B}d{\mathbf{X}}_s=uln\left(E_{t-}\right){\int }_{t_k}^t{\mathbf{X}}_{s-}^{\top }\mathbf{B}d{\mathbf{X}}_s-{\int }_{t_k}^{t}uln\left(E_{s-}\right){\mathbf{X}}_{s-}^{\top }\mathbf{B}d{\mathbf{X}}_s.$$

We want to find $d\left(U_1(t_k,s){\mathbf{X}}_s\right)$, but first, we will find $d{U}_1\left(t_k,s\right)$ by Itô’s formula for semimartingales:

$$d{U}_1\left(t_k,s\right)=U_1\left(t_k,s-\right)\tilde{c}\left(s\right)ds+\Delta U_1\left(t_k,s\right).$$

(A12)

In this case,

$$\Delta U_1\left(t_k,s\right):=U_1\left(t_k,s\right)-U_1\left(t_k,s-\right)=U_1\left(t_k,s-\right)\left(e^{uln\left({\displaystyle \frac{E_s}{E_{s-}}}\right)\left({\beta }_j-{\beta }_i\right)}-1\right)=0,$$

(A13)

since E is continuous. When $t_k\le s\le t$, we have,

$$\begin{array}{c}\hfill d\left(U_1\left(t_k,s\right){\mathbf{X}}_s\right)=U_1\left(t_k,s-\right)d{\mathbf{X}}_s+{\mathbf{X}}_{s-}d{U}_1\left(t_k,s\right).\end{array}$$

(A14)

We can now write,

$$\begin{array}{ccc}& & d\left(U_1\left(t_k,s\right){\mathbf{X}}_s\right)\hfill \\ & =& U_1\left(t_k,s-\right)d{\mathbf{X}}_s+{\mathbf{X}}_{s-}d{U}_1\left(t_k,s\right)\hfill \\ & =& U_1\left(t_k,s-\right)d{\mathbf{X}}_s+{\mathbf{X}}_{s-}{U}_1\left(t_k,s-\right){\tilde{\mathbf{c}}}^{\top }{\mathbf{X}}_{s-}ds\hfill \\ & =& U_1\left(t_k,s-\right)d{\mathbf{X}}_s-U_1\left(t_k,s-\right)\mathbf{A}{\mathbf{X}}_{s}ds+U_1\left(t_k,s-\right)diag\left(\tilde{\mathbf{c}}\right){\mathbf{X}}_{s-}ds+U_1\left(t_k,s-\right)\mathbf{A}{\mathbf{X}}_{s}ds\hfill \\ & =& U_1\left(t_k,s-\right)\left(d{\mathbf{X}}_s-\mathbf{A}{\mathbf{X}}_{s}ds\right)+U_1\left(t_k,s-\right)diag\left(\tilde{\mathbf{c}}\right){\mathbf{X}}_{s-}ds+U_1\left(t_k,s-\right)\mathbf{A}{\mathbf{X}}_{s}ds\hfill \\ & =& U_1\left(t_k,s-\right)d{M}_s^1+U_1\left(t_k,s-\right)diag\left(\tilde{\mathbf{c}}\right){\mathbf{X}}_{s-}ds+U_1\left(t_k,s-\right)\mathbf{A}{\mathbf{X}}_{s}ds,\hfill \end{array}$$

where $M^1$ is martingale. In integral form, this can be written.

$$\begin{array}{ccc}\hfill U_1\left(t_k,t\right){\mathbf{X}}_t& =& U_1\left(t_k,t_k\right){\mathbf{X}}_{t_k}+{\int }_{t_k}^t{U}_1\left(t_k,s-\right)d{M}_s^1+{\int }_{t_k}^t{U}_1\left(t_k,s-\right)diag\left(\mathbf{c}\right){\mathbf{X}}_{s-}ds\hfill \\ & & +{\int }_{t_k}^t{U}_1\left(t_k,s-\right)\mathbf{A}{\mathbf{X}}_{s}ds.\hfill \end{array}$$

Taking the expected value, it follows that

$$\begin{array}{cc}\hfill \mathbb{E}\left[U_1\left(t_k,t\right){\mathbf{X}}_t|{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]=& U_1\left(t_k,t_k\right){\mathbf{X}}_{t_k}+diag\left(\mathbf{c}\right){\int }_{t_k}^t\mathbb{E}\left[U_1\left(t_k,s\right){\mathbf{X}}_s|{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]ds\hfill \\ & +\mathbf{A}{\int }_{t_k}^t\mathbb{E}\left[U_1\left(t_k,s\right){\mathbf{X}}_s|{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]ds.\hfill \end{array}$$

(A15)

Applying Lemma 3 with $H\left(t_k,t\right)=\mathbb{E}\left[U_1\left(t_k,t\right){\mathbf{X}}_t|{\mathcal{F}}^{\mathbf{X}}\left(t_k\right)\right]$ to Equation (A15) gives the desired result. □