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

Authors: Shuanghong Qu; Lingxian Meng; Hua Li
Journal: Complexity
 (2022)

---

## Abstract

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. The simulation results of RQMC and Monte Carlo (MC) method were compared from the perspective of variance changes. The results showed that this numerical algorithm was feasible, efficient, and stable for CDO pricing with stochastic correlation under nonhomogeneous assumptions. This numerical algorithm is expected to be extended to other factor Copula models for CDO pricing with stochastic correlations under nonhomogeneous assumptions.


---

## Body

## 1. 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].The 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]. Therefore, 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 single-factor 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]. Therefore, 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 [12–14]. 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.The rest of this paper is organized as follows: Some preliminary knowledge is introduced in Section2. In the following section, the concrete algorithm process is given based on RQMC. The empirical study is carried out in Section 4. Finally, the conclusion and the discussion are presented in Section 5.


## 2. Methodology

### 2.1. Arbitrage-Free Pricing Model

First, we give some symbols used in this paper and their meanings are shown in Table1.Table 1
Symbols and their meanings.
SymbolMeaningNotenNumber of reference entities in the asset pool of CDONiNominal value of thei-th reference entityi=1,2,…,nNTotal nominal valueN=∑i=1nNiRiRecovery rate of thei-th reference entityi=1,2,…,nliLoss of thei-th reference entityli=Ni1−Ri,i=1,2,…,nLtAccumulative default loss at timetLt=∑i=1nli1Ti≤t,1Ti≤t=1,ifTi≤t0,ifTi>trfRisk-free interest ratesTTerm of CDOUnit: yearτjPayment of time nodesj=1,2,…,J,τJ=TΔτjPayment intervalΔτj=τj−τj−1,j=1,2,…,J,τ0=0MTotal number of tranchesam−1,amThem-th tranchem=1,2,…,MLmtLoss suffered by them-th tranche at time tLmt=maxLt−Nam−1,0−maxLt−Nam,0,m=1,2,…,MEmtResidual value of them-th tranche at time tEmt=Nam−Nam−1−Lmt,m=1,2,…,MsmSpread of them-th tranchem=1,2,…,MsmAmDiscounted value of normal payment for promotion of them-th tranchesmBmAccrual payment of them-th tranche when default occursCmDiscounted value of compensation of them-th trancheAccording to the principle of arbitrage-free pricing, for them-th tranche, the discounted expectation of premium leg (PL) and default leg (DL) should be equal, i.e.,(1)EPLam−1,am=EDLam−1,am,m=1,2,…,M.In the continuous case, they can be expressed as follows:(2)EPLam−1,am=E∫0TsmEmte−rftdt≈smAm+Bm,(3)EDLam−1,am=E∫0Te−rftdLmt≈Cm.From equations (1)–(3), we have(4)sm=CmAm+Bmm=1,2,…,M.


### 2.2. Gaussian Factor Copula Model with Stochastic Correlations

In Gaussian factor Copula model with stochastic correlations, the yield rateXii=1,2,…,n of the i-th asset is determined by a common factor F and a special factor Zii=1,2,…,n, namely,(5)Xi=ρ˜iF+1−ρ˜iZi,i=1,2,…,n,where F and Zii=1,2,…,n are independent of each other and they are all subject to N(0, 1); ρ˜ii=1,2,…,n are stochastic correlation coefficients between Xii=1,2,…,n and F, and it is independent of F and Zii=1,2,…,n; Xii=1,2,…,n is also independent of each other when F and ρ˜ii=1,2,…,n are conditions. The distribution functions of F,Zi, and Xi are denoted as GF,GZi, and GXi, respectively, where i=1,2,…,n.Now, we consider the case that the random variableρ˜i of equation (5) is in two states.State 1.
Whenρ˜i=ρii=1,2,…,n, equation (5) can be written as(6)Xi=ρiF+1−ρiZii=1,2,…,n.
The probability of this state isp.State 2.
Whenρ˜i=ηii=1,2,…,n, we have(7)Xi=ηiF+1−ηiZii=1,2,…,n.
In this case, the probability is1−p.
We point out that in this paper, we use corresponding superscript symbols (1) and (2) to indicate States1 and 2, respectively.
For them-th tranche, according to equation (4), we have(8)sm1=Cm1Am1+Bm1,m=1,2,…,M,(9)sm2=Cm2Am2+Bm2,m=1,2,…,M.
Now, let us takeState 1 as an example to illustrate.
InState 1, according to the reduced model, the accumulative probability of the i-th asset defaulting before ti1 can be written as(10)QTi1ti1=1−e−λi1ti1,where λi1 is the default intensity of the i-th asset and Ti1 is default time. For the i-th asset, we know that the default correlation of yield rate Xi1 can be reflected by the correlation of default time Ti1, so we can assume that the relationship between Ti1 and Xi1 can be described as(11)PXi1≤xi1=PTi1≤ti1,namely,(12)GXi1xi1=QTi1ti1.Then, according to equations (10)–(12), ti1 can be obtained:(13)ti1=QTi1−1GXi1xi1=−ln1−GXi1xi1λi1.
By substitutingti1 into equations (2) and (3), sm1m=1,…,M can be obtained by equation (8).
Similarly, the corresponding default timeti2 in State 2 can be obtained as follows:(14)ti2=QTi2−1GXi2xi2=−ln1−GXi2xi2λi2,where λi2 is default intensity of the i-th asset in State 2. Then, sm2m=1,...,M can be obtained.
Finally, we have(15)sm=psm1+1−psm2,m=1,…,M.


### 2.3. Random Sobol Sequences

Figure1 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.Figure 1
Scatter plots of different sequences.
(a)(b)(c)It can be seen from Figure1 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.In this paper, we call command equation (16) in MATLAB to generate (n + 1)-dimensional Sobol sequences and (n + 1)-dimensional randomized Sobol sequences.(16)P=sobolsetn+1;P=scrambleP,‘Matousek Affine Owen’.


## 2.1. Arbitrage-Free Pricing Model

First, we give some symbols used in this paper and their meanings are shown in Table1.Table 1
Symbols and their meanings.
SymbolMeaningNotenNumber of reference entities in the asset pool of CDONiNominal value of thei-th reference entityi=1,2,…,nNTotal nominal valueN=∑i=1nNiRiRecovery rate of thei-th reference entityi=1,2,…,nliLoss of thei-th reference entityli=Ni1−Ri,i=1,2,…,nLtAccumulative default loss at timetLt=∑i=1nli1Ti≤t,1Ti≤t=1,ifTi≤t0,ifTi>trfRisk-free interest ratesTTerm of CDOUnit: yearτjPayment of time nodesj=1,2,…,J,τJ=TΔτjPayment intervalΔτj=τj−τj−1,j=1,2,…,J,τ0=0MTotal number of tranchesam−1,amThem-th tranchem=1,2,…,MLmtLoss suffered by them-th tranche at time tLmt=maxLt−Nam−1,0−maxLt−Nam,0,m=1,2,…,MEmtResidual value of them-th tranche at time tEmt=Nam−Nam−1−Lmt,m=1,2,…,MsmSpread of them-th tranchem=1,2,…,MsmAmDiscounted value of normal payment for promotion of them-th tranchesmBmAccrual payment of them-th tranche when default occursCmDiscounted value of compensation of them-th trancheAccording to the principle of arbitrage-free pricing, for them-th tranche, the discounted expectation of premium leg (PL) and default leg (DL) should be equal, i.e.,(1)EPLam−1,am=EDLam−1,am,m=1,2,…,M.In the continuous case, they can be expressed as follows:(2)EPLam−1,am=E∫0TsmEmte−rftdt≈smAm+Bm,(3)EDLam−1,am=E∫0Te−rftdLmt≈Cm.From equations (1)–(3), we have(4)sm=CmAm+Bmm=1,2,…,M.


## 2.2. Gaussian Factor Copula Model with Stochastic Correlations

In Gaussian factor Copula model with stochastic correlations, the yield rateXii=1,2,…,n of the i-th asset is determined by a common factor F and a special factor Zii=1,2,…,n, namely,(5)Xi=ρ˜iF+1−ρ˜iZi,i=1,2,…,n,where F and Zii=1,2,…,n are independent of each other and they are all subject to N(0, 1); ρ˜ii=1,2,…,n are stochastic correlation coefficients between Xii=1,2,…,n and F, and it is independent of F and Zii=1,2,…,n; Xii=1,2,…,n is also independent of each other when F and ρ˜ii=1,2,…,n are conditions. The distribution functions of F,Zi, and Xi are denoted as GF,GZi, and GXi, respectively, where i=1,2,…,n.Now, we consider the case that the random variableρ˜i of equation (5) is in two states.State 1.
Whenρ˜i=ρii=1,2,…,n, equation (5) can be written as(6)Xi=ρiF+1−ρiZii=1,2,…,n.
The probability of this state isp.State 2.
Whenρ˜i=ηii=1,2,…,n, we have(7)Xi=ηiF+1−ηiZii=1,2,…,n.
In this case, the probability is1−p.
We point out that in this paper, we use corresponding superscript symbols (1) and (2) to indicate States1 and 2, respectively.
For them-th tranche, according to equation (4), we have(8)sm1=Cm1Am1+Bm1,m=1,2,…,M,(9)sm2=Cm2Am2+Bm2,m=1,2,…,M.
Now, let us takeState 1 as an example to illustrate.
InState 1, according to the reduced model, the accumulative probability of the i-th asset defaulting before ti1 can be written as(10)QTi1ti1=1−e−λi1ti1,where λi1 is the default intensity of the i-th asset and Ti1 is default time. For the i-th asset, we know that the default correlation of yield rate Xi1 can be reflected by the correlation of default time Ti1, so we can assume that the relationship between Ti1 and Xi1 can be described as(11)PXi1≤xi1=PTi1≤ti1,namely,(12)GXi1xi1=QTi1ti1.Then, according to equations (10)–(12), ti1 can be obtained:(13)ti1=QTi1−1GXi1xi1=−ln1−GXi1xi1λi1.
By substitutingti1 into equations (2) and (3), sm1m=1,…,M can be obtained by equation (8).
Similarly, the corresponding default timeti2 in State 2 can be obtained as follows:(14)ti2=QTi2−1GXi2xi2=−ln1−GXi2xi2λi2,where λi2 is default intensity of the i-th asset in State 2. Then, sm2m=1,...,M can be obtained.
Finally, we have(15)sm=psm1+1−psm2,m=1,…,M.


## 2.3. Random Sobol Sequences

Figure1 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.Figure 1
Scatter plots of different sequences.
(a)(b)(c)It can be seen from Figure1 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.In this paper, we call command equation (16) in MATLAB to generate (n + 1)-dimensional Sobol sequences and (n + 1)-dimensional randomized Sobol sequences.(16)P=sobolsetn+1;P=scrambleP,‘Matousek Affine Owen’.


## 3. 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 Figure2. The meanings of ρi, ηi, p, and ST are shown in Step 1 of the algorithm.Figure 2
Algorithm flow chart (concise format).Next, we give the concrete steps of the algorithm.Step 1.
Determine and input the relevant data.
Assume that there are only two valuesρi and ηi 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 2.
Generate randomized Sobol sequences.
(n + 1)-dimensional randomized sequences ε0,ε1,…,εn are generated by command equation (16); then, we can get corresponding sequences GF−1ε0,GZ1−1ε1,…,GZn−1εn, which is a set of values F,Z1,…,Zn, denoted as y,z1,…,zn.Step 3.
Simulate a path of yields in the two states, respectively.
Substitutey,zii=1,…,n obtained in Step 2 and ρii=1,…,n determined in Step 1 into equation (6) to get the corresponding values xi1i=1,…,n of yield rates Xi1i=1,…,n in State 1, and substitute y,zii=1,…,n and ηii=1,...,n into equation (7) to get the corresponding values xi2i=1,...,n of yield rates Xi2i=1,…,n in State 2.Step 4.
Generate default times in the two states, respectively.
Substitutexi1andxi2i=1,…,n obtained in Step 3 into equations (13) and (14); then, we can obtain ti1,ti2i=1,...,n.Step 5.
Find out the actual default time in the two states.
Find out the actual default time for the two states:t˜k1|t˜k1∈ti1|ti1≤T,i=1,…,n,k=1,2,…,K1 and t˜k2|t˜k2∈ti2|ti2≤T,i=1,…,n,k=1,…,K2, where K1K1≤n and K2K2≤n are the total number of real default assets in the two states, respectively. Here, we agree that t˜k1k=1,2,…,K1 and t˜k2k=1,2,…,K2 have been arranged in an order from the smallest to the largest, i.e.,(17)t˜k1≤t˜k+11k=1,2,…K1−1,t˜k2≤t˜k+12k=1,2,…K2−1.
We introduce default matricesL1andL2:(18)L1=t˜11t˜21t˜31…t˜K11h11h21h31…hK11H11H21H31…HK11,L2=t˜12t˜22t˜32…t˜K22h12h22h32…hK22H12H22H32…HK22.where hk1k=1,2,…K1 and hk2k=1,2,…K2 are the corresponding nominal values of the k-th real default in the two states, respectively, and Hk1k=1,2,…K1 and Hk2k=1,2,…K2 are the corresponding cumulative nominal values of the k-th real default by time t˜k1 and t˜k2, respectively.Step 6.
Allocate default losses for the two states.
Letbm1andbm2 be the corresponding positions of the maximum loss that the m-th tranche can bear in L1andL2, respectively, which means that Hbm111−R=am,Hbm221−R=am, where am 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:(19)Lm1=t˜bm−1+11t˜bm−1+21…t˜bm1hbm−1+11hbm−1+21…hbm1Hbm−1+11Hbm−1+21…Hbm1,Lm2=t˜bm−1+12t˜bm−1+22…t˜bm2hbm−1+12hbm−1+22…hbm2Hbm−1+12Hbm−1+22…Hbm2.Step 7.
CalculateAm1,Bm1,Cm1 and Am2,Bm2,Cm2.
For them-th tranche am−1,amm=1,…,M, the initial nominal values are E0m1=E0m1=Nam−Nam−1 and we can calculate the residual values of States 1 and 2 at τj:(20)Ejm1=Ej−1m1−∑k∈Ω11−Rhk1,j=1,…,J1Ejm2=Ej−1m2−∑k∈Ω21−Rhk2,j=1,…,J2.
Here,Ω1=k|τj−1≤t˜k1≤τj,bm−11+1≤k≤bm1 and Ω2=k|τj−1≤t˜k2≤τj,bm−12+1≤k≤bm2. Then, we have(21)Am1=∑j=1J1Δτ×Ejm1×e−rfτj,Bm1=∑k=bm−11+1bm1t˜k1−t∗1×hk1×1−R×e−rft˜k1,Cm1=∑k=bm−11+1bm1hk1×1−R×e−rft˜k1,Am2=∑j=1J2Δτ×Ejm2×e−rfτj,Bm2=∑k=bm−12+1bm2t˜k2−t∗2×hk2×1−R×e−rft˜k2,Cm2=∑k=bm−12+1bm2hk2×1−R×e−rft˜k2,where t∗1 and t∗2 are the last coupon payment time periods nearest to t˜k1 and t˜k2 in the two states, respectively.
Note: at this point, the values ofAm1,Bm1,Cm1 and Am2,Bm2,Cm2 of the m-th coupon on a path have been calculated, where m=1,…,M.
Repeat Steps2–7 until the total number of simulations reaches ST.Step 8.
Calculatesmm=1,…,M.
Form = 1, 2, …, M, we calculate the average values of Am1,Bm1,Cm1 and Am2,Bm2,Cm2, respectively, and denote them as Am1¯,Bm1¯,Cm1¯ and Am2¯,Bm2¯,Cm2¯ in sequence. Then, we substitute Am1¯,Bm1¯,Cm1¯ into equation (8) to obtain sm1 and substitute Am2¯,Bm2¯,Cm2¯ into equation (9) to obtain sm2. Finally, we plug sm1m=1,…,Mandsm2m=1,…,M, and the value of p into equation (15) to get smm=1,…,M.
This is the end of the algorithm.


## 4. Empirical Study

### 4.1. 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, Ni=1/n, Ri=R=0.4, and payment intervals are all Δτ=0.25. Let rf=0.035 and duration T=5 (year). CDO tranches are [0, 3%], [3%, 6%], [6%, 9%], [9%, 12%], [12%, 22%], and [22%, 100%]. We assume that the default intensity is equal, that is, λi1=λi2=λ=0.0083, and the Copula correlation coefficients of the two states are all equal, respectively; let ρi=ρ=0.12,ηi=η=0.21, and the probability of state 1 be p = 0.3. (Note: in the practical application, corresponding values of ρi and ηi are just substituted into equations (6) and (7), respectively; the default intensities λi1andλi2 are just substituted into equations (13) and (14), respectively; and other steps in Section 3 are all the same.)


### 4.2. 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. The variance changes of the simulation results of the two simulation methods for the pricing of each CDO tranche with the increase in simulation times are shown in Figure3.Figure 3
Variance changes of CDO pricing in each tranche with the increase in simulation times.
(a)(b)(c)(d)(e)(f)On the whole, we can see from Figure3 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. These 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. This 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 Figure3. 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.Table 2
Comparison of simulation times required of MC and RQMC at the same variance (taking 50000 times of simulation of MC as an example).
TranchesVarianceSimulation times required (RQMC)Simulation times required (MC)[0%–3%]5.042e − 074386050000[3%–6%]1.273e − 0728030[6%–9%]5.11e − 086054[9%–12%]2.833e − 821400[12%–22%]2.95e − 925170[22%–100%]4.923e − 1249998We can see from Table2, 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.


## 4.1. 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, Ni=1/n, Ri=R=0.4, and payment intervals are all Δτ=0.25. Let rf=0.035 and duration T=5 (year). CDO tranches are [0, 3%], [3%, 6%], [6%, 9%], [9%, 12%], [12%, 22%], and [22%, 100%]. We assume that the default intensity is equal, that is, λi1=λi2=λ=0.0083, and the Copula correlation coefficients of the two states are all equal, respectively; let ρi=ρ=0.12,ηi=η=0.21, and the probability of state 1 be p = 0.3. (Note: in the practical application, corresponding values of ρi and ηi are just substituted into equations (6) and (7), respectively; the default intensities λi1andλi2 are just substituted into equations (13) and (14), respectively; and other steps in Section 3 are all the same.)


## 4.2. 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. The variance changes of the simulation results of the two simulation methods for the pricing of each CDO tranche with the increase in simulation times are shown in Figure3.Figure 3
Variance changes of CDO pricing in each tranche with the increase in simulation times.
(a)(b)(c)(d)(e)(f)On the whole, we can see from Figure3 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. These 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. This 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 Figure3. 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.Table 2
Comparison of simulation times required of MC and RQMC at the same variance (taking 50000 times of simulation of MC as an example).
TranchesVarianceSimulation times required (RQMC)Simulation times required (MC)[0%–3%]5.042e − 074386050000[3%–6%]1.273e − 0728030[6%–9%]5.11e − 086054[9%–12%]2.833e − 821400[12%–22%]2.95e − 925170[22%–100%]4.923e − 1249998We can see from Table2, 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.


## 5. 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. The variances of RQMC simulation results were compared with those of the MC method from the perspective of variance changes. The 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.The 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 single-factor 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.



---

*Source: 3243450-2022-07-31.xml*