Valuation of Euro-Convertible Bonds in a Markov-Modulated, Cox–Ingersoll–Ross Economy

Abstract

This study investigates the valuation of Euro-convertible bonds (ECBs) using a novel Markov-modulated cojump-diffusion (MMCJD) model, which effectively captures the dynamics of stochastic volatility and simultaneous jumps (cojumps) in both the underlying stock prices and foreign exchange (FX) rates. Furthermore, we introduce a Markov-modulated Cox–Ingersoll–Ross (MMCIR) framework to accurately model domestic and foreign instantaneous interest rates within a regime-switching environment. To manage computational complexity, the least-squares Monte Carlo (LSMC) approach is employed for estimating ECB values. Numerical analyses demonstrate that explicitly incorporating stochastic volatilities and cojumps significantly enhances the realism of ECB pricing, underscoring the novelty and contribution of our integrated modeling approach.

1. Introduction

Euro-convertible bonds (ECBs) are hybrid securities issued by domestic companies abroad; they are denominated in foreign currencies and have the dual nature of bonds and stocks. Domestic enterprises issuing ECBs denominate the face value and interest payments in foreign currencies. However, the issuing company’s stock price will affect conversion, redemption, sale back, conversion price resetting, and other embedded aspects influencing ECBs. The stock price of the issuing company is based on the national currency. Therefore, for foreign investors holding ECBs, changes in the price of the underlying stocks and fluctuations in the foreign exchange (FX) rate, as well as the fluctuating interest rates in the two countries, will affect the value of the bonds held by ECB investors. In this complicated situation, the factors affecting the changes in the value of ECBs—namely, interest rates, FX rates, and underlying stock prices—will also act as related linkages. Companies and investors that issue ECBs must have a comprehensive understanding of the inextricable links among interest rates, FX rates, and the underlying stock price.

Issuing companies must analyze changes in these three markets and design ECBs’ issuance terms to reduce related costs. In addition, to reallocate their asset portfolios, investors must also understand how the correlation of these three factors affects the value of ECBs.

Previous studies have demonstrated that the positive and negative relationship between the stock market and the interest rate market may change with the business cycle (Rigobon and Sack [1]; Yang et al. [2]). In addition, Bianconi et al. [3] found that the degree of dependence between the BRIC stock markets and bond markets has increased significantly after the U.S. subprime mortgage financial crisis of 2008. In terms of the relationship between interest rates and FX markets, Bautista [4] used dynamic conditional correlation to analyze the interaction between interest rates and FX rate markets in the Philippines; they found a significant positive relationship between the two during periods of economic turbulence. Sanchez [5] found that the interest rate and the FX rate market have a negative relationship in terms of expansion and devaluation; contrastingly, there is a positive relationship between contraction and devaluation. Lothian and Wu [6] found that in the short term, the relationship between FX rate and interest rate markets deviates from the assumption of uncovered interest parity (UIP); however, in the long run, it conforms to the assumption of UIP. In other words, there is an inextricable relationship between interest rates and the FX market. Regarding the relationship between FX and stock markets, Soenen and Hennigar [7] found that it was not always positive; a negative relationship between the two variables was discovered during dire economic situations. Lee et al. [8] studied the relationship between FX and stock markets in Taiwan, Thailand, Malaysia, South Korea, and Indonesia. They found that when stock market volatility increases, the correlation between the two markets increases consequently. Yang et al. [9] studied Asia-Pacific countries and found a negative relationship between FX and stock markets. In addition, when a significant event occurs, the relationship between assets intensifies, resulting in cojumps (Andersen et al. [10]; Dungey et al. [11]; Lahaye et al. [12]; Lian and Chen [13]). At the same time, with the business cycle, the correlation between assets typically changes over time (Lian and Chen [14]) because the values of ECBs are affected by interest rates, FX rates, and underlying stocks. Therefore, constructing these three markets’ characteristics and analyzing their linkage has become an important aspect of ECB pricing.

Ingersoll [15,16] derived a closed-form solution for the convertible bond (CB) price. Brennan and Schwartz [17] focused more on market facts and, assuming that the issuing company paid dividends, developed a numerical algorithm for the CB price. Under the relaxation of the fixed interest rate, Brennan and Schwartz [18] improved upon the existing model to include stochastic interest rates and considered the correlation coefficient between the company value and the dynamics of interest rates to derive the theoretical price of CB. Carayannopoulos [19] modified the Vasicek interest-rate model used by Brennan and Schwartz [18] (where the interest rate may be negative), switched to the interest rate model by Cox et al. [20], and used the explicit finite difference method to evaluate the CB price. Furthermore, by extending the asset pricing formula of Duffie and Singleton [21], Carayannopoulos and Kalimipalli [22] obtained the CB pricing formula. According to Carayannopoulos and Kalimipalli’s [22] empirical results, when the conversion value of the CB option is in the money, the theoretical price is lower than the actual price. When the conversion value of the CB option is out of the money, the theoretical price is higher than the actual price. In addition, through the information on the ordinary corporate bonds of the issuing company, the authors also proved that the market undervalues the out-of-the-money CB in the US. Ammann et al. [23] studied the French CB market, showing that the discounted CB price is underestimated. Ammann et al. [24] used the Monte Carlo simulation and the GARCH (1,1) model to estimate the volatility and substituted the estimated volatility into the geometric Brownian motion stock price model to evaluate CBs issued by 32 companies in the US. For the valuation of ECBs, we consider the dynamic linkage among underlying stock prices, FX rates, and domestic/foreign instantaneous interest rates in a flexible framework.

Most exchange-traded option contracts are American-style, meaning that an investor has the right to exercise such options at any time before maturity. Many numerical methods have been presented to address the early-exercise feature in American derivative pricing, including the lattice (Cox et al. [25]), the finite difference (Brennan and Schwartz [26]; Hull and White [27]), and Monte Carlo simulation methods (Boyle et al. [28]; Gaß et al. [29]; Matsakos and Nield [30]). Longstaff and Schwartz [31] proposed an algorithm called the least-squares Monte Carlo (LSMC) method for pricing American options. This technique simulates forward paths using the Monte Carlo simulation and performs backward iterations by applying the least-squares approximation of the continuation function over a collection of basic functions (Malyscheff and Trafalis [32]; Stepniak and Palmowski [33]). This algorithm is simple to implement within existing Monte Carlo frameworks. It has the additional advantage that the continuation functions are constructed explicitly and can easily calibrate to existing market prices. Based on the advantages mentioned above, we use the LSMC approach to approximate the ECB prices.

This research contributes to the growing body of literature on financial modeling under complex market conditions, particularly in the context of derivatives pricing. By integrating cojumps and regime switching into a unified framework, the proposed Markov-modulated cojump-diffusion (MMCJD) model offers a more realistic representation of financial markets compared to traditional models that often assume continuous price movements or static parameters. This approach builds upon and extends prior work that has individually explored the implications of jumps (e.g., Merton [34]; Kou [35]) and regime switching (e.g., Hamilton [36]; Hardy [37]) in asset pricing. The incorporation of stochastic interest rates through the Markov-modulated Cox–Ingersoll–Ross (MMCIR) model further enhances the model’s realism, moving beyond simplified constant rate assumptions often found in basic pricing models. This is particularly relevant in volatile economic environments where interest rate fluctuations significantly impact derivative valuations. Our use of the LSMC algorithm for pricing ECBs, a common and computationally efficient method for American-style options, connects this work to the practical application of numerical methods in financial engineering.

The findings of this study, especially regarding the impact of cojumps and stochastic volatilities on ECB pricing, can inform risk management strategies, portfolio optimization, and the design of more sophisticated financial products. This work could be extended by exploring the hedging implications of such complex dynamics, investigating the impact of different jump distributions, or applying the model to other types of exotic derivatives or asset classes. Our methodology provides a flexible framework that can be adapted to analyze various financial instruments in environments characterized by sudden market shifts and interconnected asset behaviors, bridging theoretical advances with practical market applications.

In this study, to incorporate both cojumps and switching regimes between stock and FX rate markets simultaneously and parsimoniously, we model the underlying asset price dynamics by a Markov-modulated cojump-diffusion (MMCJD) model. The MMCJD model describes the cojumps as a compound Poisson process with log-normal jump amplitude, and the switching regime is captured by letting a continuous-time finite-state Markov chain model the market parameters. Furthermore, to avoid the disadvantage of possible negative rates in the Vasicek model, we develop a Markov-modulated Cox–Ingersoll–Ross (MMCIR) model to depict the dynamics of domestic/foreign instantaneous interest rates through time. After determining the martingale dynamics of such underlying asset prices, we employ the LSMC algorithm to approximate the values of ECBs. Numerical examples are provided to demonstrate the impact of stochastic volatilities and cojumps on ECB pricing.

The remainder of this study is organized as follows. The following section illustrates the model framework and the numerical method. Section 3 demonstrates the risk-neutral dynamics of asset prices and the valuation of ECBs, and Section 4 presents the numerical results. The study is concluded in Section 5.

2. Model Framework and Pricing Method

2.1. Euro-Convertible Bond Price Modeling

Let (Ω,  F,  P) be a complete probability space, where P is the physical measure. For all t ∈ [0,T], we define a continuous-time, finite-state Markov chain ξ(t) on (Ω,  F,  P) with state space Á = {1, 2, …,  I}. The states ξ(t) can be interpreted as an economy’s hidden states. Without any loss of generality, we can take state space Á for ξ(t) to be a finite set of unit vectors (e₁, e₂,…, e_I) with e_j = (0, …,1, …, 0) ∈ R^I. A semi-martingale representation decomposes a process into a predictable part (driven by dt) and a martingale part (representing unpredictable random fluctuations). (1) Π ξ(t)dt: This term represents the drift or predictable component of the change in ξ(t). The matrix Π governs how the process tends to move between states over infinitesimal time. (2) dM (t): This term represents the martingale difference, which captures the unexpected, random shocks or innovations in the system. The semi-martingale representation for ξ(t) is provided by Elliott et al. [38] as follows:

$$d\xi (t)=\mathsf{\Pi }\xi (t)dt+dM(t),$$

(1)

where M (t) is an R^I -valued martingale regarding the natural filtration generated by ξ(t) under P. Let Π= (π (i, j))_{i,j=1,2,…,I} denote the generator, or rate matrix, of ξ(t) under P, where π (i, j) is the transition intensity of ξ(t) from state e_i to state e_j.

Let S (t) represent the stock price at time t and X (t) be the domestic price at time t of one unit of a foreign currency. In addition, r_d (t) and r_f (t) are, respectively, used to denote the domestic and foreign risk-free instantaneous interest rates at time t. For pricing the ECBs, we further establish an MMCJD-MMCIR model to depict the dynamic behavior of stock prices and spot FX rates, as well as the stochastic processes of domestic and foreign risk-free instantaneous interest rates, which are given as follows:

$${\displaystyle \frac{dS(t)}{S(t-)}}=\left({\mu }_S(t)-{\lambda }_S(t){\beta }_S-{\lambda }_C(t){\beta }_S\right)dt+{\sigma }_S(t)d{W}_S(t)+\left(\exp \left(Y_S(t-)\right)-1\right)d{N}_S(t),$$

(2)

$${\displaystyle \frac{dX(t)}{X(t-)}}=\left({\mu }_X(t)-{\lambda }_X(t){\beta }_X-{\lambda }_C(t){\beta }_X\right)dt+{\sigma }_X(t)d{W}_X(t)+\left(\exp \left(Y_X(t-)\right)-1\right)d{N}_X(t),$$

(3)

$$d{r}_d(t)={\kappa }_d(t)(a_d(t)-r_d(t))dt+{\sigma }_d(t)\sqrt{r_d(t)}d{W}_d(t),$$

(4)

$$d{r}_f(t)={\kappa }_f(t)(a_f(t)-r_f(t))dt+{\sigma }_f(t)\sqrt{r_f(t)}d{W}_f(t),$$

(5)

for all t ∈ [0, T].

In a cross-currency economy, let $B_l(t)=\exp \left({\displaystyle {\int }_0^t{r}_l(s)ds}\right)$ denote the l^th country’s money market account at time t with an initial value Bl (0) = 1 for l ∈{d, f }. The domestic and foreign risk-free instantaneous interest rates are, respectively, defined by rd (t) and rf (t) using the following equation:

$$r_l(t)=〈r_l,\xi (t)〉, r_l=\left(r_l(1),r_l(2),\dots ,r_l(I)\right)\in {\left(0,\infty \right)}^I,$$

(6)

where $〈\cdot ,\cdot 〉$ denotes the inner product in R^I for all t ∈ [0,T], ξ(t) modulates (i) the appreciation rates—μS (t) and μX (t)—of the stock price and the spot FX rate; (ii) the stochastic volatilities, σS (t), σX (t), σd (t), and σf (t), of the stock price, the spot FX rate, and the domestic and foreign risk-free instantaneous interest rates; (iii) the stochastic arrival intensities, λS (t) and λX (t), of the Poisson processes, NS (t) and NX (t), and the speeds of mean reversion, kd (t) and kf (t); and (iv) the mean-reverting levels of the interest rate, ad (t) and af (t), of the drift terms on the domestic and foreign risk-free instantaneous interest rates, as demonstrated:

$${\mu }_k(t)=〈{\mu }_k,\xi (t)〉, {\mu }_k=\left({\mu }_k(1),{\mu }_k(2),\dots ,{\mu }_k(I)\right)\in {ℝ}^I, k\in \left\{S,X\right\},$$

(7)

$${\sigma }_v(t)=〈{\sigma }_v,\xi (t)〉, {\sigma }_v=\left({\sigma }_v(1),{\sigma }_v(2),\dots ,{\sigma }_v(I)\right)\in {(0,\infty )}^I, v\in \left\{S,X,d,f\right\},$$

(8)

$${\lambda }_z(t)=〈{\lambda }_z,\xi (t)〉, {\lambda }_z=\left({\lambda }_z(1),{\lambda }_z(2),\dots ,{\lambda }_z(I)\right)\in {\left(0,\infty \right)}^I, z\in \left\{S,X,C\right\},$$

(9)

$${\kappa }_l(t)=〈{\kappa }_l,\xi (t)〉, {\kappa }_l=\left({\kappa }_l(1),{\kappa }_l(2),\dots ,{\kappa }_l(I)\right)\in {ℝ}^I, l\in \left\{d,f\right\},$$

(10)

and

$$a_l(t)=〈a_l,\xi (t)〉, a_l=\left(a_l(1),a_l(2),\dots ,a_l(I)\right)\in {(0,\infty )}^I, l\in \left\{d,f\right\}.$$

(11)

In Equations (2)–(5), WS (t), WX (t), Wd (t), and Wf (t), are Wiener processes. The covariance matrix of such Weiner processes is given by the following:

$$\left[\begin{array}{cccc}1& {\rho }_{SX}& {\rho }_{Sd}& {\rho }_{Sf}\\ {\rho }_{SX}& 1& {\rho }_{Xd}& {\rho }_{Xf}\\ {\rho }_{Sd}& {\rho }_{Xd}& 1& {\rho }_{df}\\ {\rho }_{Sf}& {\rho }_{Xf}& {\rho }_{df}& 1\end{array}\right]t.$$

(12)

Under such dynamic processes, the two jump terms NS (t) and NX (t) are partially correlated, meaning that the FX rate jumps with positive probability p with an increase in stock prices. These two jump terms are constructed in this manner using three independent Poisson processes denoted by nS (t), nX (t), and nC (t). For k ∈{S, X}, the independent Poisson process nk (t) has a Markov-modulated intensity denoted by λk (t) with the following discrete probability density function:

$$P\left(n_k(t)=g\left|\xi (t)\right.\right)={\displaystyle \frac{{\left({\displaystyle {\int }_0^t{\lambda }_{ k}(s)ds}\right)}^g}{g!}}\exp \left(-{\displaystyle {\int }_0^t{\lambda }_{ k}(s)ds}\right), P-\mathrm{a}.\mathrm{s}.$$

(13)

The Poisson distributed random variable Nk (t) = nk (t) + nC (t) with Nk (t) ~ Poisson(λk (t) + λC (t)) is then defined. More specifically, two types of jumps exist: idiosyncratic jumps only for the kth asset with stochastic jump intensity λk (t) and simultaneous jumps with the arrival rate λC (t) for both assets. The NS (t) and NX (t) Poisson processes are capable of producing idiosyncratic jumps through nS (t) and nX (t) and simultaneous jumps through nC (t). A change in variables and integrating out nC (t) results in the joint probability density function for NS (t) and NX (t), provided by the following equation:

$$P\left(N_S(t)=m,N_X(t)=n\left|\xi (t)\right.\right)$$

$$={\displaystyle \sum _{w=0}^{\min \left(m,n\right)}\exp \left(-{\displaystyle {\int }_0^t\left({\lambda }_S(s)+{\lambda }_X(s)+{\lambda }_C(s)\right)ds}\right)}$$

$$\cdot {\displaystyle \frac{{\left({\displaystyle {\int }_0^t{\lambda }_S(s)ds}\right)}^{m-w}{\left({\displaystyle {\int }_0^t{\lambda }_X(s)ds}\right)}^{n-w}{\left({\displaystyle {\int }_0^t{\lambda }_C(s)ds}\right)}^w}{\left(m-w\right)!\left(n-w\right)!w!}}, P-\mathrm{a}.\mathrm{s}.$$

(14)

In addition, if a jump event occurs at time t, the random variable Yk(t), which controls the jump amplitude, is assumed to be normally distributed with mean uk and variance ${\delta }_k^2$, such as a compound Poisson process with a log-normal jump amplitude setting used by Merton [34]. Thus, the mean percentage jump amplitude of the kth asset price is ${\beta }_k=E\left[\exp \left(Y_k(t)\right)-1\right]=\exp \left(u_k+{\displaystyle \frac{1}{2}}{\delta }_k^2\right)-1={\varphi }_{Y_k}(1)-1$. In Equations (2) and (3), also assumed is that Yk (t) and Yk (s) are independent for t ≠ s, and in Equations (2)–(5), all of the random shocks Wv (t), Nk (t), ξ(t), and Yk (t) are mutually independent.

2.2. Least-Squares Monte Carlo Approach

Longstaff and Schwartz [31] demonstrated that an American-style option could be priced using the LSMC technique to achieve the desired accuracy level. This advantage supports use of the LSMC approach for pricing ECBs under the MMCJD-MMCIR model, in contrast with the shortcomings of the lattice pricing methods. Let w denote a sample path of the underlying asset price generated by the Monte Carlo simulation over a discrete set of τ exercise times 0 < t1 ≤ t2 ≤ … ≤ tt = T. The continuous exercise property of an ECB is approximated by taking sufficiently large τ. Let ECB (w, s;t, T) denote the path of cash flows generated, assuming that the ECB is not exercised at or before time t, and the ECB holder follows the optimal exercise policy for all subsequent s ∈ (t, T]. At maturity, the investor exercises the ECB if it is in the money. At time tk prior to expiration, the ECB holder must decide whether to exercise or continue and revisit the decision at the next time point. Although the ECB holder knows the immediate exercise payoff, they have no exact idea of the expected cash flows from continuation. According to the no-arbitrage pricing theory, H (w;tk), the continuation value at time tk for path w, is formally given as follows:

$$H\left(\omega ;t_k\right)=E^Q\left[{\displaystyle \sum _{j=k+1}^{\tau }\exp \left(-r\left(\omega ,t_j;t_k\right)\right)ECB\left(\omega ,t_j;t_k,T\right)}\left|F_{t_k}\right.\right],$$

(15)

where r (w, tj; tk) is the risk-free discount rate, Q is a martingale pricing measure, and the expected cash flows are taken conditional on the information set F_tk at time t_k. Assuming that the continuation value is estimated, we can decide whether to exercise at time tk or compare the immediate exercise value with the estimate of the continuation value. The procedure is repeated until exercise decisions have been determined for each exercise point on every path. Thus, by estimating the conditional expectation function for each exercise date, a complete specification of the optimal exercise strategy can be obtained along each path. Once the exercise strategy has been estimated, the approximate valuation of an ECB is achieved.

3. Fair Valuation of Euro-Convertible Bonds

3.1. Risk-Neutral MMCJD-MMCIR Dynamics

Here, let {${\left\{F_t^S\right\}}_{t\in \left[0,T\right]}$, ${\left\{F_t^X\right\}}_{t\in \left[0,T\right]}$, ${\left\{F_t^{r_d}\right\}}_{t\in \left[0,T\right]}$, ${\left\{F_t^{r_f}\right\}}_{t\in \left[0,T\right]}$, and ${\left\{F_t^{\xi }\right\}}_{t\in \left[0,T\right]}$ be the right-continuous, P -complete natural filtrations generated by stochastic processes ${\left\{S(t)\right\}}_{t\in \left[0,T\right]}$, ${\left\{X(t)\right\}}_{t\in \left[0,T\right]}$, ${\left\{r_d(t)\right\}}_{t\in \left[0,T\right]}$, ${\left\{r_f(t)\right\}}_{t\in \left[0,T\right]}$, and ${\left\{\xi (t)\right\}}_{t\in \left[0,T\right]}$, respectively. Furthermore, $F_t=F_t^S\vee F_t^X\vee F_t^{r_d}\vee F_t^{r_f}\vee F_t^{\xi }$ is defined as the enlarged σ -algebra for each t ∈ [0,T]. Trading is assumed to occur continuously over time in the [0,T] interval, and a filtered probability space $\left(\mathrm{\Omega },F,P\hspace{0.17em},{\left\{F_t\right\}}_{t\in \left[0,T\right]}\right)$ describes the uncertainty. In particular, we follow the assumption of Merton’s [34] diversifiable jump risk, which implies that no premium is paid for such a risk. We then use Merton’s approach for the MMCJD model to identify a risk-neutral pricing measure by changing the drift of the Brownian motion but leaving the other ingredients unchanged (Lian et al. [39]). Therefore, the jump risks are unchanged by the measured change; that is, the risk-neutral properties of the jump components of the underlying asset prices are assumed to be the same as their statistical properties.

Assume that we are interested in pricing ECBs, the correlation among the stock price, the spot FX rate, and the domestic and foreign risk-free instantaneous interest rates should be identified before the implementation of the Monte Carlo simulation. Consequently, in the risk-neutral circumstance, the correlation among the underlying Wiener processes can be calculated through the Cholesky decomposition. Then, the dynamic linkages among such underlying assets are expressed as follows:

$$\begin{array}{c}{\displaystyle \frac{dS(t)}{S(t-)}}=\left(r_d(t)-{\lambda }_S(t){\beta }_S-{\lambda }_C(t){\beta }_S\right)dt+\left(\exp \left(Y_S(t-)\right)-1\right)d{N}_S(t)\hfill \\ +{\sigma }_S(t)\left({\rho }_{Sf}d{W}_f(t)+{\displaystyle \frac{{\rho }_{Sd}-{\rho }_{Sf}{\rho }_{df}}{\sqrt{1-{\rho }_{df}^2}}}d{W}_d(t)+\sqrt{1-{\rho }_{Sf}^2-a_{Sd}^2}d{W}_S^1(t)\right)\hfill \end{array},$$

(16)

$$\begin{array}{l}{\displaystyle \frac{dX(t)}{X(t-)}}=\left(r_d(t)-r_f(t)-{\lambda }_X(t){\beta }_X-{\lambda }_C(t){\beta }_X\right)dt+\left(\exp \left(Y_X(t-)\right)-1\right)d{N}_X(t)\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\sigma }_X(t)\left({\rho }_{Xf}d{W}_f(t)+a_{Xd}d{W}_d^1(t)+a_{SX}d{W}_S^1\right.\\ \hspace{1em}\hspace{1em}\hspace{1em} \left.+\sqrt{1-{\rho }_{Xf}^2-a_{Xd}^2-a_{SX}^2}d{W}_X^1(t)\right),\end{array}$$

(17)

$$d{r}_d(t)={\kappa }_d(t)(a_d(t)-r_d(t))dt+{\sigma }_d(t)\sqrt{r_d(t)}\left({\rho }_{df}d{W}_f(t)+\sqrt{1-{\rho }_{df}^2}d{W}_d^1\right) ,$$

(18)

$$d{r}_f(t)={\kappa }_f(t)(a_f(t)-r_f(t))dt+{\sigma }_f(t)\sqrt{r_f(t)}d{W}_f(t),$$

(19)

where

$$\begin{array}{l}{a}_{Sd}={\displaystyle \frac{{\rho }_{Sd}-{\rho }_{Sf}{\rho }_{df}}{\sqrt{1-{\rho }_{df}^2}}}\\ a_{Xd}={\displaystyle \frac{{\rho }_{Xd}-{\rho }_{Xf}{\rho }_{df}}{\sqrt{1-{\rho }_{df}^2}}}\\ a_{SX}={\displaystyle \frac{{\rho }_{SX}-{\rho }_{Xf}{\rho }_{Sf}-\left({\rho }_{Xd}-{\rho }_{Xf}{\rho }_{df}\right)\left({\rho }_{Sd}-{\rho }_{Sf}{\rho }_{df}\right)/\left(1-{\rho }_{df}^2\right)}{\sqrt{1-{\rho }_{Sf}^2-{\rho }_{Sd}^2}}}\end{array}$$

and $W_f\perp W_d^1\perp W_S^1\perp W_X^1$.

3.2. Valuing Euro-Convertible Bonds Under the MMCJD-MMCIR Model

For evaluating the ECB prices, where the risk-neutral asset price dynamics follow the stochastic differential Equations (16)–(19), the sample paths are generated by LSMC simulations (Lian et al. [39]). We then determine the expected holding value at each time $t_{\tau -1},t_{\tau -2},\dots ,t_1$ in a backward manner. We know the striking value at maturity date $t_{\tau }$ and at time $t_{\tau -1}$. If the ECB price is larger than the striking price, we calculate the expected holding value using the following regression: ${\rm Y}=a+bH(t_{\tau -1})+c{H}^2(t_{\tau -1})+{\epsilon }_{t_{\tau -1}}$, where ${\rm Y}$ represents the discounted cash flow of an ECB at time $t_{\tau }$. If the simulated path is large enough, we can estimate the coefficients a, b, and c through the ordinary least-squares approach. Let ECB (w, s;t, T) denote the sample path w of the cash flow generated by an ECB, conditional on the ECB being unconverted at or before time t but terminable at each time s ∈ (t, T ]. The backward induction is preceded from the last to the latest and step by step to calculate the derivative price. If we want to estimate the holding value of the ECB at time $t_{\tau -1}$, we conduct a regression of $H\left(\omega ,t_{\tau };t_{\tau -1},T\right)=E_{t_{\tau -1}}\left[{\rm Y}\right]={\widehat{a}}_1+{\widehat{b}}_{1}H(t_{\tau -1})+{\widehat{c}}_1{H}^2(t_{\tau -1})$ using the cash flow at time $t_{\tau }$, discounted back to $t_{\tau -1}$, under the paths of in-the-money at time $t_{\tau -1}$, to estimate the holding value. Furthermore, we compare this value with the terminated value at time $t_{\tau -1}$ to determine whether to exercise the ECB. Through a similar procedure, we can estimate the holding value at $t_{\tau -2}$ by a regression of $H\left(\omega ,t_{\tau },t_{\tau -1};t_{\tau -2};T\right)=E_{t_{\tau -2}}\left[{\rm Y}\right]={\widehat{a}}_2+{\widehat{b}}_{2}H(t_{\tau -2})+{\widehat{c}}_2{H}^2(t_{\tau -2})$ using the cash flow at time $t_{\tau -1}$ and $t_{\tau }$ discounted back to $t_{\tau -2}$ under the paths of in-the-money at time $t_{\tau -2}$. We then compare the estimated holding value with the terminated value at time $t_{\tau -2}$ to determine whether to exercise the ECB. The optimal stopping point is determined using the estimator function with $H\left(\omega ;t_{\tau },t_{\tau -1},\dots ,t_2;t_1,T\right)=E_{t_1}\left[{\rm Y}\right]={\widehat{a}}_{\tau }+{\widehat{b}}_{\tau }H(t_1)+{\widehat{c}}_{\tau }{H}^2(t_1)$ after the previously mentioned procedure is repeated for each simulated path. According to the option pricing theory, an optimal exercise policy generates exactly one cash flow for each path. The ECBs are then priced by discounting the resulting cash flows to zero and averaging the discounted cash flows over all paths. To more precisely illustrate the application of LSMC to the ECB valuation in this study, Appendix A includes pseudo-code describing the valuation process.

4. Numerical Illustrations

This section presents the Monte Carlo simulation and a numerical experiment for ECB pricing under the MMCJD-MMCIR model. We further compare the numerical results of the ECB price implied by the MMCJD-MMCIR model with those implied by the MMBS-MMCIR model (i.e., λS (t) = λX (t) = λC (t) = 0). The numerical simulations assume that the Markov chain ξ(t) has two states—namely, “boom” and “recession” states—for the domestic economy. We assume that the transition probability matrix of the two-state Markov chain is given by

$$\left[\begin{array}{cc}{p}_{11}& p_{12}\\ p_{21}& p_{22}\end{array}\right]=\left[\begin{array}{cc}0.7& 0.3\\ 0.2& 0.8\end{array}\right].$$

(20)

Furthermore, the face value and conversion price for the case is 100 and 110, respectively. In following, we adopt the sample values for the model parameters listed in

Table 1. The parameter values.

Parameter Name	Value in Boom State	Value in Recession State
CIR domestic initial interest rate (rd)	0.02	0.01
CIR foreign initial interest rate (rf)	0.03	0.015
Stock price volatility (σS)	0.2	0.1
FX rate volatility (σX)	0.3	0.2
Idiosyncratic jump intensity of S (λS)	4	2
Idiosyncratic jump intensity of X (λX)	6	3
Cojump intensity of S and X (λC)	2	1
Jump amplitude of stock price (uS)	0.002	0.002
Jump amplitude of FX rate (uX)	0.001	0.001
Volatility of stock price jump (δS)	0.01	0.01
Volatility of FX rate jump (δX)	0.02	0.02
Initial stock price (S (0))	100	100
Initial FX rate (X(0))	32	32
Correlation coefficient between S and X (rSX)	0.001	0.001
Correlation coefficient between S and rd (rSd)	0.002	0.002
Correlation coefficient between S and rf (rSf)	0.003	0.003
Correlation coefficient between X and rd (rXd)	0.0015	0.0015
Correlation coefficient between X and rf (rXf)	0.002	0.002
Correlation coefficient between rd and rf (rdf)	0.003	0.003
CIR domestic interest rate volatility (σd)	0.0015	0.001
CIR foreign interest rate volatility (σf)	0.002	0.001
CIR mean-reverting rate of domestic interest rate (kd)	0.04	0.02
CIR mean-reverting rate of foreign interest rate (kf)	0.03	0.015
CIR domestic long-term average interest rate (ad)	0.06	0.03
CIR foreign long-term average interest rate (af)	0.05	0.025

and apply the Euler discretization method for Monte Carlo simulation. In this method, the initial stock price S (0) = 100; the initial FX rate X(0) = 32; and the initial state of the Markov chain ξ(0) = 1. When one assumes that there are 252 trading days in one year, the discretization interval is 1/252; we conduct 100,000 simulations to calculate each ECB price.

Figure 1 depicts the plot of ECB prices under different times to maturity. From this plot, we can find the mixed effect of the cojump risk. The stochastic volatility implied by the proposed MMCJD-MMCIR seems to have much more impact on the option prices than the stochastic volatility, as the increasing magnitudes of the ECB price on the MMCJD-MMCIR are much larger than those of MMBS-MMCIR under different times to maturity. Moreover, for the fixed maturity year T = 1, we take a range of stock volatility from 0.2 to 0.25 with an increment of 0.005. We then observe that the mixed effect becomes more apparent and pushes the MMCJD-MMCIR model farther away from MMBS-MMCIR. Meanwhile, the greater the stock price volatility, the higher the price of ECBs, as illustrated in Figure 2. At last, we investigate how the ECB prices change with the correlation coefficient between the stock price and the FX rate under the MMCJD-MMCIR model. For the fixed maturity year T = 1, we take a range of the correlation coefficient between the stock price and the FX rate from 0.01 to 0.03 with an increment of 0.002. Figure 3 indicates that the larger the correlation coefficient between the stock price and the FX rate, the higher is the EBC price. Furthermore, in the comparison of Figure 2 and Figure 3, we observe that the volatility risk is more influential than the correlation coefficient. By contrast, the distance between the lines in Figure 2 is greater than that in Figure 3. Our findings show an asymmetric impact of the volatility and the correlation coefficient on the ECB price.

5. Conclusions

In this study, we evaluated the ECB prices by providing an MMCJD-MMCIR model. Under the no-arbitrage framework of the MMCIR model, we employed an MMCJD model for stock price and FX rate modeling from a physical measure to a risk-neutral measure. Compared with most existing Markov-modulated models, the main advantage of the proposed model is that we simultaneously incorporated stochastic volatilities, cojumps, jump intensity switching, and Markov-modulated interest rates into the asset price dynamics. Under an incomplete market setting, we employed the LSMC method for pricing ECBs. The numerical results reveal that under the model settings of MMCJD, compared with the MMBS model, the price of ECBs with jumps is relatively higher. Furthermore, the higher the stock price volatility and correlation coefficient between the stock price and the FX rate, the higher is the price of ECBs. Therefore, in this study, we conclude that the stochastic volatilities (of stock prices and FX rates) and cojumps significantly influence ECB prices. While the credit quality and ratings issues represent significant aspects deserving future investigation, these issues are not explicitly included within our current model framework. Nonetheless, these considerations suggest valuable directions for subsequent extensions of this study.