Application of RQMC for CDO Pricing with Stochastic Correlations under Nonhomogeneous Assumptions

In consideration of that the correlation between any two assets of the asset pool is always stochastic in the actual market and that collateralized debt obligation (CDO) pricing models under nonhomogeneous assumptions have no semianalytic solutions, we designed a numerical algorithm based on randomized quasi-Monte Carlo (RQMC) simulation method for CDO pricing with stochastic correlations under nonhomogeneous assumptions and took Gaussian factor copula model as an example to conduct experiments. ,e simulation results of RQMC and Monte Carlo (MC) method were compared from the perspective of variance changes. ,e results showed that this numerical algorithm was feasible, efficient, and stable for CDO pricing with stochastic correlation under nonhomogeneous assumptions. ,is numerical algorithm is expected to be extended to other factor Copula models for CDO pricing with stochastic correlations under nonhomogeneous assumptions.


Introduction
Collateralized debt obligation (CDO) is one of the most typical securitized assets in credit derivative markets, and its pricing has always been the focus of scholars. In the pricing process of CDO, the reduced model is widely used for establishing default distribution of a single reference entity, factor Copula methods are the mainstream of joint default distribution of asset portfolio [1,2], and the spread of each tranche of CDO can be calculated according to the principle of arbitrage-free pricing [3].
e Gaussian factor Copula model is usually considered to be the standard model for CDO pricing, but its correlation coefficient is a deterministic parameter. In fact, the stochastic correlation copula is closer to the market [3]. erefore, some scholars extended the constant correlation coefficient to a stochastic one. Burtschell et al. [4] provided a thorough analysis of the three-state stochastic correlation model with homogeneity and obtained the semianalytical pricing formula. Yang et al. [5] proposed a semianalytical method for the credit spreads of each CDO tranche under the conditions of homogeneity and stochastic correlations based on singlefactor Copula model with mixed distribution.
As we can see, the above literature is all based on the assumption of homogeneity. However, it is difficult to meet the requirements of homogeneity in actual markets and the arbitrage-free pricing models under nonhomogeneous assumptions have no semianalytical solutions [6]. erefore, it might be a good idea to resort to numerical methods.
Monte Carlo (MC) is one of the most common numerical methods [7,8]. However, this method has a certain dimension disaster problem. Quasi-Monte Carlo (QMC) [9,10] is an extension of MC, and it is usually superior to MC in dealing with high-dimensional problems. Furthermore, by introducing a randomized component into the QMC method, the randomized quasi-Monte Carlo (RQMC) method can effectively improve the cycle problem [11], which usually has higher convergence order than MC and QMC. Some scholars applied RQMC to option pricing [12][13][14]. Johansson et al. [12] studied the pricing of American options using RQMC and found that RQMC could reduce both the variance and the bias of the option. Amal et al. [13] applied array-RQMC for option pricing under a stochastic volatility process, and the empirical results showed that it could bring very significant variance reductions compared with MC. He [14] found that RQMC had a better rate of convergence in financial option pricing.
Qu et al. [15] introduced RQMC into the CDO pricing based on a single-factor Copula framework and nonhomogeneous hypothesis and achieved good results. However, only the constant correlation coefficient was considered in the model. To our knowledge, there have been no further reports of RQMC being used to solve the CDO pricing issues. In view of this, we intend to design a numerical algorithm for the CDO pricing with stochastic correlations based on RQMC and Gaussian factor Copula model under the conditions of nonhomogeneous assumptions and conduct empirical research. We try to provide some ideas for the pricing problem with stochastic correlations under the nonhomogeneous assumptions in the credit derivative markets. e rest of this paper is organized as follows: Some preliminary knowledge is introduced in Section 2. In the following section, the concrete algorithm process is given based on RQMC. e empirical study is carried out in Section 4. Finally, the conclusion and the discussion are presented in Section 5.

Arbitrage-Free Pricing
Model. First, we give some symbols used in this paper and their meanings are shown in Table 1.
According to the principle of arbitrage-free pricing, for the m-th tranche, the discounted expectation of premium leg (PL) and default leg (DL) should be equal, i.e., In the continuous case, they can be expressed as follows: From equations (1)-(3), we have

Gaussian Factor Copula Model with Stochastic
Correlations. In Gaussian factor Copula model with stochastic correlations, the yield rate X i (i � 1, 2, . . . , n) of the i-th asset is determined by a common factor F and a special factor Z i (i � 1, 2, . . . , n), namely, where F and Z i (i � 1, 2, . . . , n) are independent of each other and they are all subject to N(0, 1); ρ i (i � 1, 2, . . . , n) are stochastic correlation coefficients between X i (i � 1, 2, . . . , n) and F, and it is independent of F and Z i (i � 1, 2, . . . , n); X i (i � 1, 2, . . . , n) is also independent of each other when F and ρ i (i � 1, 2, . . . , n) are conditions. e distribution functions of F, Z i , and X i are denoted as G F , G Z i , and G X i , respectively, where i � 1, 2, . . . , n. Now, we consider the case that the random variable ρ i of equation (5) is in two states. (5) can be written as e probability of this state is p.
In this case, the probability is 1 − p.
We point out that in this paper, we use corresponding superscript symbols (1) and (2) to indicate States 1 and 2, respectively.
For the m-th tranche, according to equation (4), we have Now, let us take State 1 as an example to illustrate. In State 1, according to the reduced model, the accumulative probability of the i-th asset defaulting before t (1) i can be written as where λ (1) i is the default intensity of the i-th asset and T (1) i is default time. For the i-th asset, we know that the default correlation of yield rate X (1) i can be reflected by the correlation of default time T (1) i , so we can assume that the relationship between T (1) i and X (1) i can be described as namely, en, according to equations (10)-(12), t (1) i can be obtained:
Similarly, the corresponding default time t (2) i in State 2 can be obtained as follows: where λ (2) i is default intensity of the i-th asset in State 2. en, s (2) m (m � 1, ..., M) can be obtained. Finally, we have Figure 1 shows the scatter diagrams of pseudo-random sequences used in MC, Sobol sequences used in QMC, and randomized Sobol (Sobol scramble) sequences used in RQMC in high-dimensional cases (125-th dimension and 126-th dimension), respectively, where the number of points are all 1100. It can be seen from Figure 1 that randomized Sobol sequences not only maintain good uniformity but also improve circulation problems of Sobol sequences in high dimension. In this paper, RQMC method based on randomized Sobol sequences is adopted.

Algorithm
In this section, based on the Gaussian factor Copula model, we design the RQMC simulation algorithm for CDO pricing with stochastic correlations of the two states under nonhomogeneous assumptions. First, we draw the algorithm flow chart (concise format), as shown in Figure 2. e meanings of ρ i , η i , p, and ST are shown in Step 1 of the algorithm.
Next, we give the concrete steps of the algorithm.
Step 1. Determine and input the relevant data. Assume that there are only two values ρ i and η i for stochastic correlation coefficients ρ i and corresponding probabilities are p and 1 − p, respectively, where i � 1, . . . , n. In addition, we preset the total number of simulations, denoted as ST.
Step 6. Allocate default losses for the two states.
Let b (1) m and b (2) m be the corresponding positions of the maximum loss that the m-th tranche can bear in L (1) and L (2) , respectively, which means that where a m is a separation point (maximum loss should be taken by the m-th tranche); then, the default information allocated to the m-th tranche in the two states can be represented by the following default matrices: Step 7.

Parameter Values.
In this paper, the values of each parameter are as follows (refer to [3]). Let n � 125, N � 1, the nominal value and return rate of each reference entity be all equal, respectively; that is,  (2) i � λ � 0.0083, and the Copula correlation coefficients of the two states are all equal, respectively; let ρ i � ρ � 0.12, η i � η � 0.21, and the probability of state 1 be p � 0.3. (Note: in the practical application, corresponding values of ρ i and η i are just substituted into equations (6) and (7), respectively; the default intensities λ (1) i and λ (2) i are just substituted into equations (13) and (14), respectively; and other steps in Section 3 are all the same.)

Result Analysis.
In order to investigate the stability of simulation results of the RQMC method and MC method, we calculate the variances of the two simulation methods under different simulation times and observe their changes for each CDO tranche, respectively. We set the variation range of simulation times from 5000 to 50000, with the step size of 5000. For each different simulation times, we all simulate 40 times and then calculate the corresponding variances of MC and RQMC of each CDO tranche. e variance changes of the simulation results of the two Complexity simulation methods for the pricing of each CDO tranche with the increase in simulation times are shown in Figure 3.
On the whole, we can see from Figure 3 that the variances of the two simulation methods in six different tranches are all decreased with the increase in simulation times, and in each subgraph, the gaps between the two curves all reduce gradually with the increase in simulation times. ese show that the results of the two simulation methods are all gradually stable with the increase in simulation times. It can also be seen from Figure 3 that, compared with the MC method, the variance changes of each CDO tranche of the RQMC method are more gentle with the increase in simulation times. Especially, when the simulation time is relatively smaller, with the increase in simulation times, the variances of MC all decrease faster than those of RQMC in each CDO tranche. is indicates that the simulation results of the RQMC method are more stable than those of MC even when the simulation time is not too large. In other words, the RQMC method can obtain more stable results with fewer simulation times. From the local point of view on each tranche diagram, when the simulation times are the same, the variances of the RQMC method are much smaller than those of the MC method, which indicates that the simulation results of the RQMC method are more stable than those of the MC method when the simulation times are the same.
Particularly, in order to compare the simulation times required under the same variance of two simulation methods, we carry out spline interpolations for the variances data of the RQMC method for each subgraph in Figure 3. Without loss of generality, in each subgraph of Figure 3, we take the variances corresponding to 50000 simulation times of the MC method as an example to illustrate. At this time, the simulation times required by the RQMC method are shown in Table 2.
We can see from Table 2, for the same variances, when the total number of simulations of the MC method required is 50000, in addition to the tranche [22%-100%], simulation times of the other five tranches of the RQMC method are all far lower than 50000. Furthermore, the RQMC method can save about 25000 times on average and reduce the cost of the program running effectively.

Conclusion and Discussion
In this paper, a numerical algorithm based on the RQMC simulation method was designed and an empirical study was carried out to solve CDO pricing with the stochastic correlation problem of the two states under nonhomogeneous assumptions. e variances of RQMC simulation results were compared with those of the MC method from the perspective of variance changes. e results showed that the algorithm designed in this paper is more stable and reliable whether from the local view or from the global view, and it is an efficient and stable algorithm to solve CDO pricing with stochastic correlation problems under nonhomogeneous conditions, which would provide theoretical and empirical support for solving similar problems. e algorithm in this paper is only empirically analyzed based on the Gaussian Copula model with stochastic correlations under nonhomogeneous assumptions. We point out that the algorithm can be generalized. For example, under the framework of the algorithm, the Gaussian Copula model in the algorithm can be replaced with other singlefactor Copula models and the algorithm still works. Of course, for different single-factor Copula models, the implementation of the algorithm may be more complex.
In addition, the algorithm can also be extended to CDO pricing with three-state stochastic correlations under nonhomogeneous assumptions.
In the following study, we will further optimize the algorithm to improve its versatility. It is also considered to apply the numerical algorithm to the pricing of other assets in credit derivative markets under nonhomogeneity assumptions.

Data Availability
e data for the empirical study are randomly generated by MATALB code according to the algorithm in this paper. If readers need it, please contact the corresponding author for discussion.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.