2. A general pricing formula for CDSs

One of the most important factors affecting the accuracy in the determination of the CDS price is the assumption on the default model. As illustrated before, Merton’s assumption [Reference Merton23] that the default would only occur at expiry is certainly not appropriate, and a more realistic assumption is adopted hereafter that the default can take place at any time during the lifespan of the CDS contract. With the default mechanics being determined, we can now proceed to the pricing of CDSs. Unlike most financial derivatives, whose price is defined as the value of the corresponding contract, the price of a CDS refers to the spread, that is, the regular fee that the buyer pays to the seller, and is quoted as the ratio of the face value of the reference asset. This implies that we need to analyse the cash flows between buyer and seller.

On the one hand, the CDS buyer needs to regularly pay a protection fee to the seller before the default occurs or expiry. Specifically, if we denote by c and M the target CDS spread per unit time that needs to be determined and the face value of the reference asset respectively, the buyer needs to pay $\displaystyle cMdt$ to the seller if there is no default before time t. According to the risk-neutral pricing rule, the current value of the cash flow made by the buyer (denoted by $V_1$ ) is the expectation of all the discounted future payments, which can be expressed as

$$ \begin{align*} \displaystyle V_1=\sum_{t}[e^{-rt}cMp(t)\,dt]=cM\int_{0}^{T}e^{-rt}p(t)\,dt, \end{align*} $$

where the current time is 0, T is the expiry, r is the risk-free interest rate, and $p(t)$ represents the probability of no default before time t. On the other hand, if a default does occur, the seller has to pay $\displaystyle (1-R)M$ , with R being the recovery rate specified in the contract, to compensate the loss of the buyer, implying that the current value of the cash flow made by the seller (denoted by $V_2$ ) depends on the probability that the default takes place. In particular, the probability of the default taking place between time t and $t+dt$ can be directly computed as

$$ \begin{align*} \displaystyle [1-p(t+dt)]-[1-p(t)]=p(t)-p(t+dt)=-dp(t), \end{align*} $$

with which the risk-neutral pricing rule yields

$$ \begin{align*} \displaystyle V_2=\sum_{t}\{e^{-rt}(1-R)M[-dp(t)]\}=(1-R)M\int_{0}^{T}-e^{-rt}dp(t). \end{align*} $$

Thus, to be fair to both parties, the current value of the cash flow from the buyer should be equal to that from the seller, leading to

$$ \begin{align*} \displaystyle cM\int_{0}^{T}e^{-rt}p(t)\,dt=(1-R)M\int_{0}^{T}-e^{-rt}dp(t), \end{align*} $$

which can be simplified as

(2.1) $$ \begin{align} c&=\frac{(1-R)\int_{0}^{T}-e^{-rt}dp(t)}{\int_{0}^{T}e^{-rt}p(t)\,dt}\nonumber\\ &=\frac{(1-R)[-e^{-rt}p(t)\bigg|_{0}^{T}-r\int_{0}^{T}e^{-rt}p(t)\,dt]}{\int_{0}^{T}e^{-rt}p(t)\,dt}\nonumber\\ &=\frac{(1-R)[1-e^{-rT}p(T)]}{\int_{0}^{T}e^{-rt}p(t)\,dt}-r(1-R). \end{align} $$

The general pricing formula presented in (2.1) only contains one unknown term, the no-default probability $p(t)$ , solving which would yield a completely analytical solution. If $S_t$ denotes the reference asset price at time t, and D represents the default level, $p(t)$ is clearly the probability of the minimal value of the reference asset within the time period $[0,t]$ being higher than D, or, more specifically,

(2.2) $$ \begin{align} \displaystyle p(t)=P\big(\mathop{\min}\limits_{0\leq u\leq t}S_{u}>D\big), \end{align} $$

with $p(0)=0$ as the CDS contract would never be entered if the default occurs at the current time. If we rewrite (2.2) as

$$ \begin{align*} \displaystyle p(t)=E[I_{\{\mathop{\min}\limits_{0\leq u\leq t}S_{u}>D\}}], \end{align*} $$

with I being the indicator function, it is not difficult to find that $\displaystyle e^{-rt}p(t)$ is in fact the price of a down-and-out binary option having time to expiry t, which can be formulated as

$$ \begin{align*} \displaystyle e^{-rt}p(t)\equiv P(S,t)=e^{-rt}E[\Pi(S,t)], \end{align*} $$

where the payoff function is

$$ \begin{align*} \displaystyle \Pi(S,t)=\left\{ \begin{array}{@{}lll} 1 & & \mathop{\min}\limits_{0\leq u\leq t}S_{u}>D, \nonumber\\ 0 & & \mathrm{otherwise}. \nonumber \end{array} \right. \nonumber \end{align*} $$

In this case, the pricing formula (2.1) can be rearranged as

(2.3) $$ \begin{align} \displaystyle c=\frac{(1-R)[1-P(S,T)]}{\int_{0}^{T}P(S,t)\,dt}-r(1-R). \end{align} $$

It is now clear that once the price of the down-and-out binary option written on the reference asset, $P(S,t)$ , has been derived, we are able to determine the CDS price through (2.3). Thus, a necessary step is to first specify the dynamic of the reference asset, and then we are able to consider this particular option pricing problem. These details will be presented in the next section.

3. Analytical approximation of a down-and-out binary option

In this section the reference asset is assumed to follow a regime-switching model, and the down-and-out binary option price is then determined with a two-step process; the first step deals with the option pricing problem conditional upon all the future information of the Markov chain, based on which the target option price is written in a Fourier cosine series in the second step.

3.1. The regime-switching model

As mentioned earlier, there is a lot of empirical evidence suggesting the existence of regime switching in real markets. Therefore, we would like to incorporate the effect of regime switching in the pricing of CDS contracts. It should be pointed out that the extra uncertainty introduced by regime switching makes our market incomplete. This issue has been discussed in [Reference Guo11], where a risk premium is introduced related to the cost of switching. We shall not repeat this argument, but directly assume that we are already working under a risk-neutral measure $\mathbb {Q}$ with zero risk premium associated with regime switching, like many other authors in the literature [Reference Buffington and Elliott4, Reference Zhu, Badran and Lu27], as our focus here is on how we can efficiently determine the price of a CDS contract under the regime-switching model without losing too much accuracy.

In particular, the reference asset of the CDS contract, denoted by $S,$ is assumed to follow a regime-switching model under a risk-neutral measure $\mathbb {Q}$ as

(3.1) $$ \begin{align} \displaystyle \frac{dS_u}{S_u}=r\,dt+\sigma_{X_u}dW_u, \end{align} $$

where r is the risk-free interest rate, $W_u$ is a standard Brownian motion and $X_u$ is a two-state Markov chain, independent of the Brownian motion with its state space being $\{e_1,e_2\}$ , where the jth component of the column vector $e_i$ is the Kronecker delta $\delta _{ij}$ for all $i,j=1,2$ . It should be pointed out here that the Markov chain is assumed to contain two states for illustration purposes, but the extension to arbitrary but finite states is very straightforward. It should also be remarked that the dynamics of the Markov chain is actually generated by a transition rate matrix $\displaystyle A=(a_{ij})_{i,j=1,2}$ [Reference Elliott, Aggoun and Moore8], where $a_{ij}$ is the transition rate of $X_t$ from state i to j satisfying $a_{ij}\geq 0, i\neq j$ and $\sum _{j=1}^{2}a_{ij}=0$ . In this case, if we assume that $a_{12}=\lambda _{12}$ and $a_{21}=\lambda _{21}$ , then the transition rate matrix A here is defined as

$$ \begin{align*} \displaystyle A=\left(\!\begin{array}{ccc} -\lambda_{12} & \lambda_{12}\\ \lambda_{21} & -\lambda_{21} \end{array}\!\right). \end{align*} $$

In this case, $\displaystyle \sigma _{X_u}$ would have two different values, denoted by $\sigma _1$ and $\sigma _2$ , according to the current state of the reference asset, and it can be represented as

$$ \begin{align*} \displaystyle \sigma_{X_u}=\langle\bar{\sigma},X_u\rangle, \end{align*} $$

if $\langle \cdot ,\cdot \rangle $ is the inner product of two vectors, and $\bar {\sigma }=(\sigma _1,\sigma _2)^T$ .

3.2. The Fourier cosine series solution

This subsection is devoted to deriving the analytical approximation of the down-and-out binary option under the regime-switching model $P(S,t,X_0)$ , which is dependent on $X_0$ as we are working under the regime-switching model, where the Markov chain is also a random process. The main result is provided in the following theorem, followed by its detailed proof.

Theorem 3.1. If the underlying price S follows the dynamic specified in (3.1), the price of a down-and-out binary option $P(S,t,X_0)$ can be approximated by a Fourier cosine series as

(3.2) $$ \begin{align} \displaystyle P(S,t,X_0)=\frac{1}{2}A_0(X_0)V_0+\sum_{k=1}^{+\infty}A_k(X_0)V_k, \end{align} $$

where

$$ \begin{align*} \displaystyle A_k(X_0)&=\frac{2}{b-a}\mathrm{Real}\bigg[g\bigg(\frac{k\pi}{b-a}\bigg)e^{-jka\pi/(b-a)}\bigg], \quad k=0,1,2,\ldots ,\nonumber\\ V_k&=\int_a^bW({h})\cos\bigg(k\pi\frac{{h}-a}{b-a}\bigg)\,d{h},\quad k=0,1,2,\ldots,\nonumber\\ g(\phi)&=\langle e^{A^T\cdot t+B}X_0,I\rangle,\quad I=(1,1)^T,\nonumber\\ W({h})&=e^{-rt}N\bigg[\frac{\ln(S/D)+rt-h/2}{\sqrt{{h}}}\bigg]-e^{-rt+(h-2rt)/h\ln(S/D)}N\bigg[-\frac{\ln(S/D)-rt+h/2}{\sqrt{{h}}}\bigg],\nonumber\\ A&=\left(\!\begin{array}{ccc} -\lambda_{12}& \lambda_{12}\\ \lambda_{21} & -\lambda_{21}\\ \end{array}\!\right), \quad B=\left(\!\begin{array}{ccc} j\phi\sigma_1^2t & 0\\ 0 & j\phi\sigma_2^2t \\ \end{array}\!\right). \nonumber \end{align*} $$

Here, $a=\min (\sigma _1^2,\sigma _2^2)t,~b=\max (\sigma _1^2,\sigma _2^2)t$ , the current time is $0$ , t is the expiry, $A^T$ denotes the transpose of the matrix A, D is the knock-out barrier, and $N(\cdot )$ is the standard normal distribution function.

Proof. The existence of regime switching in the dynamics of the reference asset poses an obstacle in analytically evaluating the target option. To overcome this difficulty, a two-step solution procedure is applied. As a prior step, we firstly assume that all the future information of the Markov chain up to the expiry time t is known at the current time so that the regime-switching volatility becomes only a time-dependent parameter. It should be pointed out that due to this particular assumption made in the first step here, the target option price should follow a time-dependent Black–Scholes model without the effect of the Markov chain since all its future information is given. Its impact will be recovered in the second step when this assumption is relaxed. In this case, if the current time u is assumed to be within $[0,t]$ , the price of the down-and-out binary option conditional upon all the information of the Markov chain, $U(y,\tau |X_t)$ with $\displaystyle y=\ln (S/D)$ and $\tau =t-u$ , should satisfy the following partial differential equation (PDE) system:

(3.3) $$ \begin{align} \displaystyle \left\{ \begin{array}{@{}lll} \displaystyle \frac{\partial U}{\partial\tau}=\frac{1}{2}\sigma_{\tau}^2\frac{\partial^2 U}{\partial y^2}+\bigg(r-\frac{1}{2}\sigma_{\tau}^2\bigg)\frac{\partial U}{\partial y}-rU,\quad y>0,& & \\ U(y,0|X_t)=I_{\{y>0\}}, & & \\ U(0,\tau|X_t)=0. & & \\ \end{array} \right. \end{align} $$

Here, $\sigma _{\tau }=\langle \bar {\sigma },X_{\tau }\rangle $ is a known value for all $\tau \in [0,t]$ . Following Lo et al. [Reference Lo, Lee and Hui20] and He and Chen [Reference He and Chen14], we make the transformation $\displaystyle z=y-y^*(\tau )$ , where

$$\begin{align*}y^*(\tau )=-c_1(\tau )-\beta c_2(\tau )\end{align*}$$

with

$$ \begin{align*}\displaystyle c_1(\tau)=r\tau-\frac{1}{2}\int_0^{\tau}\sigma_s^2\,ds, \quad \displaystyle c_2(\tau)=\frac{1}{2}\int_0^{\tau}\sigma_s^2\,ds,\end{align*} $$

and $\beta $ being an adjusted parameter to be determined later. If we further denote $\displaystyle \tilde {U}(z,\tau |X_t)=e^{r\tau }U(y,\tau |X_t)$ and $\displaystyle \bar {U}(z,\xi |X_t)=e^{-\beta z/2+\beta ^2 \xi /4}\tilde {U}(z,\tau |X_t)$ with $\xi =c_2(\tau )$ , the PDE system in (3.3) can be transformed into

(3.4) $$ \begin{align} \displaystyle \left\{ \begin{array}{@{}lll} \displaystyle \frac{\partial \bar{U}}{\partial\xi}=\frac{\partial^2\bar{U}}{\partial z^2},\quad z>-y^*(\tau), \\ \displaystyle \bar{U}(z,0|X_t)=U(z,0|X_t)e^{-\beta z/2}, \\ \displaystyle \bar{U}(-y^*(\tau),\xi|X_t)=0.\end{array} \right. \end{align} $$

Note that solving system (3.4) is still not straightforward, though its PDE is a standard heat equation, since the fixed absorbing boundary is now a moving absorbing boundary. Since such a difficulty arises from the moving absorbing boundary, one possible approach is to find an appropriate value for $\beta $ , such that the moving boundary is a good simulation of a certain fixed boundary. Considering the definition of $\displaystyle y^*(\tau )$ , what we adopt here is

(3.5) $$ \begin{align} \displaystyle \beta=-\frac{c_1(t)}{c_2(t)}, \end{align} $$

with which we use the fixed boundary 0 to approximate $\displaystyle y^*(\tau )$ for all $\tau \in [0,t]$ . In this case, the solution to (3.4) can be approximated by the solution to the following PDE system:

(3.6) $$ \begin{align} \displaystyle \left\{ \begin{array}{@{}lll} \displaystyle \frac{\partial \bar{U}}{\partial\xi}=\frac{\partial^2\bar{U}}{\partial z^2},\quad z>0 & &\\ \displaystyle \bar{U}(z,0|X_t)=U(z,0|X_t)e^{-\beta z/2}, & &\\ \displaystyle \bar{U}(0,\xi|X_t)=0. & & \end{array} \right. \end{align} $$

Clearly, the closer $y^*(\tau ),~\tau \in [0,t]$ is to 0, the more accurate such an approximation will be. If the volatility becomes a constant, such an approximation will degenerate to the exact down-and-out binary option price under the BS model since in this case $y^*(\tau )=0$ for all $\tau \in [0,t]$ . Thus, it is of interest to calculate the magnitude of the distance between $y^*(\tau )$ and 0, which is

$$ \begin{align*} |y^*(\tau)-0|&=r\tau \frac{|\tau^{-1}\int_0^{\tau}\sigma_{s}^2\,ds-\tau^{-1}\int_0^{t}\sigma_{s}^2\,ds|}{\tau^{-1}\int_0^{t}\sigma_{s}^2\,ds}\nonumber \\ &\leq r\tau\bigg(\frac{\max_{0\leq s\leq\tau}\sigma_s^2}{\min_{0\leq s\leq\tau}\sigma_s^2}-1\bigg). \end{align*} $$

This is an upper bound, which indicates that the accuracy of the approximation actually depends on the extent of fluctuation of the underlying volatility. If the difference between the maximum and minimum values of the volatility during the lifetime of the contract is small, one could expect this approximation to be very accurate. Of course, it is also interesting to see whether the approximation would still provide reasonably accurate results when the difference is enlarged, which will be tested in the next section.

With the method of images, it is not difficult to derive the solution to (3.6) as

$$ \begin{align*} \displaystyle \bar{U}(z,\xi|X_t)=\int_{0}^{+\infty}\frac{1}{\sqrt{4\pi\xi}}e^{-(z-v)^2/4\xi}\bar{U}(v,0)\,dv-\int_{-\infty}^{0}\frac{1}{\sqrt{4\pi\xi}}e^{-(z-v)^2/4\xi}\bar{U}(-v,0)\,dv,\end{align*} $$

based on which we can obtain

$$ \begin{align*} \displaystyle U(y,\tau|X_t)=e^{-r\tau}N\bigg[\frac{y+c_1(\tau)}{\sqrt{2c_2(\tau)}}\bigg]-e^{-r\tau+\beta[y+c_1(\tau)+\beta c_2(\tau)]}N\bigg[-\frac{y+c_1(\tau)+2\beta c_2(\tau)}{\sqrt{2c_2(\tau)}}\bigg].\end{align*} $$

Setting $u=0$ finally yields $P(S,t|X_t)=U(y,t|X_t)$ , and thus we have

(3.7) $$ \begin{align} \displaystyle P(S,t|X_t)=e^{-rt}N\bigg[\frac{\ln(S/D)+c_1(t)}{\sqrt{2c_2(t)}}\bigg]-e^{-rt+\beta\ln(S/D)}N\bigg[-\frac{\ln(S/D)-c_1(t)}{\sqrt{2c_2(t)}}\bigg], \end{align} $$

the simplification of which relies on the use of $c_1(t)+\beta c_2(t)=0$ .

Hitherto, the conditional option price has been presented in (3.7) with all the information of the Markov chain. However, one would never be able to predict the future information of the Markov chain at the current time, which forces us to conduct an extra step to work out the expectation of $P(S,t|X_t)$ with respect to the Markov chain, that is,

$$ \begin{align*} \displaystyle P(S,t;X_0)=E_{X_t}[P(S,t|X_t)|X_0].\end{align*} $$

Note that analytically evaluating the expectation in the above formula is usually not possible, given the convoluted expression for $P(S,t|X_t)$ . Although He and Chen [Reference He and Chen14] developed a semi-Monte Carlo approach to obtain the value of the expectation, derivative pricing involving generating sample paths is often time-intensive, and analytical solution, if exists, is much more favoured in practice. To achieve this, the $\beta $ presented in (3.5) has been chosen to be different from the one used in [Reference He and Chen14, Reference Lo, Lee and Hui20], and this is actually essential, paving the way for the analytical derivation of the target expectation.

If we denote $ h_t=\int _0^{t}\langle \bar {\sigma }^2,X_{s}\rangle ds$ , then we have $c_1(t)=rt-h_t/2$ , $c_2(t)=h_t/2$ , $\beta =(h_t-2rt)/h_t$ , and thus (3.7) can be rewritten as

$$ \begin{align*} \displaystyle W(h_t)&\triangleq P(S,t|X_t) \\ &=e^{-rt}N\bigg[\frac{\ln(S/D)+rt-h_t/2}{\sqrt{h_t}}\bigg]-e^{-rt+\ln(S/D)(h_t-2rt)/h_t }N\bigg[-\frac{\ln(S/D)-rt+h_t/2}{\sqrt{h_t}}\bigg]. \end{align*} $$

By noticing that $h_t$ is the only random variable contained in $P(S,t|X_t)$ , the price of the down-and-out option can be formulated as

(3.8) $$ \begin{align} \displaystyle P(S,t;X_0)=E[V(h_t)|X_0]=\int_a^bW({h})f({h}|X_0)\,d{h}, \end{align} $$

where $f({h}|X_0)$ is the probability density function of $h_t$ defined on $[a,b]$ , conditional upon the current state of the Markov chain. It should be pointed out here that if $f(h|X_0)$ were available in closed form, we would already obtain an analytical pricing formula for the target option. Unfortunately, this is not the case here and we have to find an alternative way to derive $P(S,t;X_0)$ . In particular, $f({h}|X_0)$ can be alternatively expressed as

(3.9) $$ \begin{align} \displaystyle f({h}|X_0)=\frac{1}{2}A_0(X_0)+\sum_{k=1}^{+\infty}A_k(X_0)\cos\bigg(k\pi\frac{{h}-a}{b-a}\bigg), \end{align} $$

with

(3.10) $$ \begin{align} \displaystyle A_k(X_0)=\frac{2}{b-a}\int_a^bf({h}|X_0)\cos\bigg(k\pi\frac{{h}-a}{b-a}\bigg)\,d{h}. \end{align} $$

Such an expansion is always possible since any real function has a Fourier cosine expansion when it is finitely supported [Reference Boyd3], implying that its theoretical convergence is guaranteed. Thus, substituting (3.9) into (3.8) yields

$$ \begin{align*} \displaystyle P(S,t;X_0)=\int_a^bW(h)\bigg[\frac{1}{2}A_0(X_0)+\sum_{k=1}^{+\infty}A_k(X_0)\cos\bigg(k\pi\frac{{h}-a}{b-a}\bigg)\bigg]\,d{h}.\end{align*} $$

After interchanging the summation and the integration, the option price can finally be derived as

(3.11) $$ \begin{align} \displaystyle P(S,t,X_0)=\frac{1}{2}A_0(X_0)V_0+\sum_{k=1}^{+\infty}A_k(X_0)V_k, \end{align} $$

where

(3.12) $$ \begin{align} \displaystyle V_k=\int_a^bW({h})\cos\bigg(k\pi\frac{{h}-a}{b-a}\bigg)\,d{h},\quad k=0,1,2,\ldots. \end{align} $$

Equation (3.11) is almost analytical and explicit, except that $A_k(X_0)$ defined in (3.10) still remains unknown as it involves the density function $f({h}|X_0)$ . However, we have managed to derive $A_k(X_0)$ , by making use of the relationship between the characteristic function and the density function. If $g(\phi )$ is the characteristic function of $h_t$ , it actually has a closed-form expression [Reference Elliott and Lian9] given by

(3.13) $$ \begin{align} \displaystyle g(\phi)=\int_a^bf({h}|X_0)e^{j\phi {h}}d{h}=\langle e^{A^T\cdot t+B}X_0,I\rangle,\end{align} $$

from which one can easily represent $A_k(X_0)$ as

(3.14) $$ \begin{align} \displaystyle A_k(X_0)=\frac{2}{b-a}\text{Real}\bigg[g\bigg(\frac{k\pi}{b-a}\bigg)e^{-jka\pi/(b-a)}\bigg],\quad k=0,1,2,\ldots. \end{align} $$

This completes the proof.

Clearly, the CDS pricing formula (2.3) is now completely analytical, after the substitution of the down-and-out binary option price presented in (3.2). It should particularly be emphasized that computing the option price through equation (3.2) or the CDS price with equation (2.3) does not involve the density function $f(h|X_0)$ . Instead, all we need is to calculate $V_k$ and $A_k(X_0)$ through equations (3.12) and (3.14) respectively, which are only dependent on the characteristic function $g(\phi )$ in equation (3.13).

Of course, there are still several issues that need to be addressed. First of all, as the solution is written in a series form, its speed of convergence needs to be checked as this is an important factor in practice, especially in the recent trend of algorithmic trading. The accuracy of the formula should also be considered to ensure that the approximation used is appropriate and there are no algebraic errors in the derivation process. Of course, it is also of interest to demonstrate whether the introduction of regime switching into the dynamic of the reference asset would have a significant impact on the CDS price. These issues will be discussed in the next section.

4. Numerical examples and discussion

In this section numerical experiments are carried out to show the efficiency and accuracy of the approximation, after which the effect of regime switching on the CDS price is demonstrated. It should be pointed out that although any mathematical model needs to go through a calibration process before it can be applied in practice, the main purpose of this paper is to derive an efficient pricing formula for CDS contracts with a high degree of accuracy under the well-known regime-switching model that can facilitate the calibration process. Therefore, rather than calibrating the underlying model with real market data, it suffices for us to test the efficiency and accuracy of our approximation formula with artificial data.

As it was shown in the previous section that the accuracy of our approximation depends on the difference between $\sigma _1$ and $\sigma _2$ , we consider two cases when conducting numerical experiments: case 1 sets $\sigma _1$ and $\sigma _2$ as 0.1 and 0.2 respectively, while case 2 uses 0.1 and 0.4 for $\sigma _1$ and $\sigma _2$ respectively. Other parameters include $\lambda _{12}=\lambda _{21}=20$ , $T=1$ , $D=12$ , $S=15$ and $r=0.1$ . The current state is assumed to be 1.

The speed of convergence for a series solution is always an important factor to demonstrate its computational efficiency. To check the efficiency, it suffices to use only the down-and-out binary options since the series solution is only introduced when deriving the option pricing formula. Thus, with N being the number of terms used in the formula, the absolute difference between the down-and-out binary option prices calculated with $(N+1)$ terms and those produced with N terms is shown in Figures 1(a) and 2(a) corresponding to case 1 and case 2, respectively. It is clear that option prices in both figures converge very fast for different reference asset prices as the absolute difference decreases sharply when we increase the number of terms, and they are almost zero even when fewer than 10 terms are used. A clearer pattern for the two cases is respectively shown in Figures 1(b) and 2(b), where option prices produced by 15 and 16 terms are plotted against different time to expiry. Option prices in both figures are pointwise close to each other, with the maximum absolute difference being of the order of $1\times 10^{-7}$ for Figure 1(b) and $2\times 10^{-5}$ for Figure 2(b). For the convenience of the reader in repeating the experiments, numerical values in producing Figures 1 and 2 are provided in Tables 1 and 2.

Figure 1

Speed of convergence for equation (3.2) with $\sigma _1=0.1,~\sigma _2=0.2$ .

Figure 2

Speed of convergence for equation (3.2) with $\sigma _1=0.1,~\sigma _2=0.4$ .

Table 1

Numerical values for Figures 1(a) and 2(a).

Table 2

Numerical values for Figures 1(b) and 2(b).

Having demonstrated the rapid speed of convergence for equation (3.2), the accuracy of our approach is another aspect that needs to be checked as approximation is involved in the derivation process. Thus, CDS prices calculated with our approach (our prices), are compared with those obtained using the approach in [Reference He and Chen14] (He–Chen prices) and those from Monte Carlo simulation (Monte Carlo prices) in Figures 3 and 4 corresponding to case 1 and case 2, respectively. It can be easily observed from Figures 3(a) and 4(a) that our approximation is almost the same as the He–Chen approximation, both of which are very close to the Monte Carlo price. Figures 3(b) and 4(b) exhibit the relative errors between the two approximation prices and Monte Carlo prices, and the error of our price is slightly lower than that of the He–Chen price. Furthermore, our maximum relative error being less than 0.91% for both cases is certainly evidence demonstrating the accuracy of our approach. One should also note that although enlarging the difference between $\sigma _1$ and $\sigma _2$ increases the relative error between our price and the Monte Carlo price, which confirms what we have shown in the previous section, our approximation is still quite accurate even when

$$ \begin{align*} \frac{\max_{0\leq s\leq\tau}\sigma_s^2}{\min_{0\leq s\leq\tau}\sigma_s^2}=16. \end{align*} $$

Figure 3

Accuracy with $\sigma _1=0.1,~\sigma _2=0.2$ .

Figure 4

Accuracy with $\sigma _1=0.1,~\sigma _2=0.4$ .

Again, for the convenience of the reader in repeating the numerical experiments, Table 3 presents the numerical values in producing Figures 3 and 4.

Table 3

Numerical values for Figures 3 and 4.

As our approach provides similar accuracy to the He–Chen approach, it is of interest to compare the computational efficiency of the two approaches, and the CPU time cost to compute one CDS price is listed in Table 4. It is expected that Monte Carlo simulation, as the benchmark, is the most time-intensive approach of the three, as it needs to simulate both the Brownian motion and the Markov chain. The He–Chen approach has an advantage over Monte Carlo simulation, since it avoids generating sample paths for the Brownian motion. Our approach, on the other hand, involving no simulations, is the most efficient one, and it only requires around 0.13 seconds to derive one price, almost 100 times less than the time cost by the He–Chen approach, even when 100 terms are used for the series solution.

Table 4

Comparison of CPU times (seconds).

Having established the advantages of our approach, we are now ready to study how regime switching would affect the CDS prices. Depicted in Figure 5 is the change in CDS prices under the regime-switching model with different transition rates, with the prices under the BS model as a benchmark. Firstly, our price of state 1 (2) is exactly the same as the BS price, with its volatility being equal to $\sigma _1$ ( $\sigma _2$ ) when the transition rates are 0 for both cases, which is expected as there is no actual regime switching and the volatility value will remain as its initial value. With the increase in the transition rates, the transition of the volatility between the two states would become more frequent, increasing (decreasing) the average of the volatility of the reference asset starting in state 1 (2), and this leads to a higher (lower) risk, as a result of which both prices of the two states are moving away from the corresponding BS prices while at the same time getting closer to each other. The corresponding numerical values to produce Figure 5 are shown in Table 5.

Figure 5

Effect of regime switching.

Table 5

Numerical values for Figure 5.