Multi-dimensional Self-Exciting NBD Process and Default Portfolios

Abstract

In this study, we apply a multidimensional self-exciting negative binomial distribution (SE-NBD) process to default portfolios with 13 sectors. The SE-NBD process is a Poisson process with a gamma-distributed intensity function. We extend the SE-NBD process to a multidimensional process. Using the multidimensional SE-NBD process (MD-SE-NBD), we can estimate interactions between these 13 sectors as a network. By applying impact analysis, we can classify upstream and downstream sectors. The upstream sectors are real-estate and financial institution (FI) sectors. From these upstream sectors, shock spreads to the downstream sectors. This is an amplifier of the shock. This is consistent with the analysis of bubble bursts. We compare these results to the multidimensional Hawkes process (MD-Hawkes) that has a zero-variance intensity function.

Appendices

Appendix 1. Proof of the Impact Calculation

In this section, we present the calculation of Eq. (21). We consider an event at $t=0$ and the number of affected events. We consider the sequence $a_t$ that is the effect at t and $v_t=\sum _t a_t$. The sequence is $a_0=1$ and

$$\begin{aligned} a_{t+1}=\frac{M_0}{L_0}\sum _{s=0}^{t}{\hat{d}}_{t-s} a_s= \frac{M_0}{L_0}\sum _{s=0}^{t}r^{t-s} a_s. \end{aligned}$$

(37)

After solving this recurrence formula, we obtain

$$\begin{aligned} a_{t+1}=\frac{M_0}{L_0}\left( \frac{M_0}{L_0}+r\right) ^t. \end{aligned}$$

(38)

Subsequently, the impact is,

$$\begin{aligned} v_t=\frac{M_0}{L_0}\frac{1-\left( r+\frac{M_0}{L_0}\right) ^t}{1-r-\frac{M_0}{L_0}}. \end{aligned}$$

(39)

In the limit $t\rightarrow \infty$, we can obtain Eq. (21).

Appendix 2. Impact and Branching Process

In this section, we consider the relationships between impact and branching process to clarify the mean of the impact analysis based on the single-line case. We consider the case of ${\hat{d}}_1=1$ and ${\hat{d}}_s=0$, $s=2,3,\ldots$ for Eqs. (6) and (9), which denote the discrete NBD and Hawkes processes, respectively. In this case, we obtain ${\hat{T}}=1$. This corresponds to an event that affects only the next term.

In Eqs. (6) and (9), $M_0$ denotes the birth probability of the 0-th generation, which corresponds to the parent events, and the second term corresponds to the birth of the offspring events.

We focus on a single parent event and its offspring events. $Y_s$ denotes the number of offspring events. The parent event occurs at t and the offspring events occur at the s-th generation at $t+s$. Here, we set the parent event to the 0-th generation.

We define $p_k$ as the offspring distribution for k children. With probability $p_0$, there are no offspring events. Using Eqs. (6) and (9), the offspring distribution becomes $\text{ NBD }(K_0/L_0,M_0/K_0 L_0)$, while the Poisson distribution becomes $\text{ Poisson }(M_0/L_0)$. Because these distributions have a reproductive function, the sum of the distributions becomes Eqs. (6) and (9). This corresponds to the decomposition of the SE-NBD and Hawkes processes, focusing on a single parent. This is a discrete-time branching process. An image of the process is shown in Fig. 7.

In summary, the process has the following properties:

• $Y_0=1$ this is the parent event.

• When $Y_n=i$, where $i=0,1,2,\ldots$, if $i=0$, $Y_{n+1}=0$. This corresponds to the extinction of the offspring event.

• If $i>0$, new offspring events occur independently with the probability $\text{ NBD }(K_0/L_0, M_0/K_0)$ or $\text{ Poisson }(M_0/L_0)$ at the next term.

Note that the impact is considered as the sum of the offspring events $\sum _k Y_k-1$. This impact does not include the parent event.

Fig. 7

Parent and offspring events. We focus on a single parent event and its offspring events. The dotted circle denotes the parent event, while the black and white circles denote the offspring and other events, respectively

Here, we consider the extinction probability $\epsilon$ and probability of $Y_n=0$ for any n, considering the phase transition. After this term, there are no offspring events for the parent event. We can obtain the self-consistent equation

$$\begin{aligned} x=\sum _{k=0}^{\infty }p_k x^k, \end{aligned}$$

(40)

and its solution is $x=\epsilon$. Here, we introduce the moment-generating function

$$\begin{aligned} f(x)=\sum _{k=0}^{\infty }p_k x^k. \end{aligned}$$

(41)

We rewrite Eq. (40) using the generating function,

$$\begin{aligned} x=f(x). \end{aligned}$$

(42)

Equation (42) has the solution $x=1$ because $f(1)=\sum _k p(k)=1$. If $f'(1)>1$, then there is only one solution, $x=1$; accordingly, the probability is 1. If $f'(x)<1$, then there are two solutions, where the smaller solution denotes the extinction probability $\epsilon$. The transition point is $f'(1)=1$. This is known as the branching process phase transition. Note that the order parameter of the phase transition is the survival probability $1-\epsilon$.

We change the variable from x to $y=1-x$ and rewrite Eq. (42) around $y\sim 0$,

$$\begin{aligned} 1-y=f(1-y)=1-f'(1)y+f''(1)y^2/2-\cdots . \end{aligned}$$

Then, we can obtain the critical exponent $\beta =1$ at the transition point $f'(1)=M_0/L_0=1$,

$$\begin{aligned} y\sim 2(f'(1)-1)/f''(1), \end{aligned}$$

where $f''(1)>0$.

We consider the transition point from an impact analysis standpoint. The impact is the expected value of the number of offspring events, $v_{\infty }=\sum _{k=1}^{\infty } (M_0/L_0)^k=M_0/L_0\cdot 1/(1-M_0/L_0)$ for the SE-NBD and Hawkes processes determined using Eq. (21), where we assume that ${\hat{d}}_1=1$ and ${\hat{d}}_s=0$, $s=2,3,\ldots$. Hence, we can confirm that the transition point is $M_0/L_0=1$. At this limit, the survival probability is 1.

Appendix 3. Correlation Functions of MD-SE-NBD Process

In this section, we consider the correlation functions for the MD-SE-MBD process.

The covariance density function is defined as

$$\begin{aligned} Cov\left[ X_t^{(i)},X_{t'}^{(j)}\right]= & {} E\left[ X_t^{(i)},X_{t'}^{(j)}\right] -E\left[ X_t^{(i)}\right] E\left[ X_{t'}^{(j)}\right] =E\left[ X_t^{(i)},X_{t'}^{(j)}\right] -{\bar{v}}_i{\bar{v}}_j\Delta ^2\nonumber \\= & {} C_{ij}^{*}\left( t, t'\right) \Delta ^2, \end{aligned}$$

(43)

where $C^{*}_{(ij)}(t, t')$ denotes the covariance density function.

We can obtain for $t=t'$,

$$\begin{aligned} E\left[ X_t^{(i)},X_{t'}^{(i)}\right] =E\left[ \left( X_t^{(i)}\right) ^2\right] =\left( 1+\omega '_i\right) {\bar{v}}_i\Delta . \end{aligned}$$

(44)

and

$$\begin{aligned} E\left[ X_t^{(i)},X_{t'}^{(j)}\right] ={\bar{v}}_i{\bar{v}}_j\Delta ^2, \end{aligned}$$

(45)

because the processes in lines i and j at the same time are independent. Note that the difference between the SE-NBD and Hawkes processes is only the correlation at the same time $t=t'$. Hence, we can decompose the covariance density function as

$$\begin{aligned} C^*_{(ij)}\left( t,t'\right) =\left( 1+\omega '_i\right) {\bar{v}}_i\delta _{ij}\delta \left( t-t'\right) +C_{(ij)}\left( t,t'\right) , \end{aligned}$$

(46)

where $C_{ij}(t,t')$ is the correlation function, $\delta _{ij}$ is the Kronecker’s delta, and $\delta (x)$ is the delta function.

If we set $\tau =t'-t$, we can rewrite Eq. (43) using Eq. (46),

$$\begin{aligned} E\left[ X_t^{(i)},X_{t+\tau }^{(j)}\right] =\left[ C_{ij}(\tau )+{\bar{v}}_i{\bar{v}}_j+\left( 1+\omega '_i\right) {\bar{v}}_i\delta _{ij}\delta (\tau )\right] \Delta ^2. \end{aligned}$$

(47)

Here we define $g_{ij}(x)={\tilde{\omega }}_{ij}{\hat{d}}_t^{(i)}(x)$ and set $\tau >0$.

We can calculate the following correlations:

$$\begin{aligned} E\left[ X_t^{(i)} X_{t+\tau }^{(k)}\right]= & {} \left[ C_{ik}(\tau )+{\bar{v}}_i{\bar{v}}_k\right] \Delta ^2 \\= & {} \theta _0^{(k)}{\bar{v}}_i\Delta ^2+\sum _{j=1}^{D}\lim _{\Delta \rightarrow 0}\sum _{s=0}^{\infty }g_{kj}(s\Delta )E\left[ X_t^{(i)} X_{t+\tau -(s+1)\Delta }^{(j)}\right] \Delta , \\= & {} \theta _0^{(k)}{\bar{v}}_i\Delta ^2+\sum _{j=1}^{D}\lim _{\Delta \rightarrow 0}\sum _{s=0}^{\infty }g_{kj}(s\Delta )\\&\quad \left[ C_{ij}\left( \tau -(s+1)\Delta )+{\bar{v}}_i{\bar{v}}_j+(\omega '_i+1){\bar{v}}_i\delta _{ij}\delta (\tau -(s+1)\Delta \right) \right] \Delta ^3, \\= & {} \theta _0^{(k)}{\bar{v}}_i\Delta ^2+\Delta ^2\sum _{j=1}^{D}\int _{0}^{\infty }g_{kj}(w)\left[ C_{ij}(\tau -w)+{\bar{v}}_i{\bar{v}}_j\right. \\&\left. \quad +\left( \omega '_i+1\right) {\bar{v}}_i\delta _{ij}\delta (\tau -w)\right] \mathrm{d}w, \\= & {} \theta _0^{(k)}{\bar{v}}_i\Delta ^2+\Delta ^2\left( \omega '_i+1\right) {\bar{v}}_i g_{ii}(\tau )+\Delta ^2\sum _{j=1}^{D}\int _{0}^{\infty }g_{kj}(w)\left[ C_{ij}(\tau -w)+{\bar{v}}_i {\bar{v}}_j)\right] \mathrm{d}w, \\= & {} {\bar{v}}_i{\bar{v}}_k\Delta ^2+\Delta ^2\left( \omega '_i+1\right) {\bar{v}}_i g_{ii}(\tau )+\Delta ^2\sum _{j=1}^{D}\int _{0}^{\infty }g_{kj}(w)C_{ij}(\tau -w)\mathrm{d}w \end{aligned}$$

where we used the mean-field approximation defined in Eq. (19),

$$\begin{aligned} {\bar{v}}_k=\theta _0^{(k)}+ \sum _{j=1}^{D}\int _{0}^{\infty }g_{kj}(w){\bar{v}}_j \mathrm{d}w. \end{aligned}$$

(48)

We can obtain the integral equation for $\tau >0$ as

$$\begin{aligned} C_{ik}(\tau )=\left( \omega '_i+1\right) {\bar{v}}_i g_{ii}(\tau )+\sum _{j=1}^{D}\int _{0}^{\infty } g_{kj}(w)C_{ij}(\tau -w)\mathrm{d}w. \end{aligned}$$

(49)

This is the integral equation for the correlation functions. The difference between the SE-NBD and Hawkes processes is only the first term, which signifies the effect of correlation at the same time.

We can rewrite Eq. (49) as

$$\begin{aligned} C_{ik}(\tau )= & {} \left( \omega '_i+1\right) {\bar{v}}_i g_{ii}(\tau )+\sum _{j=1}^{D}\int _{0}^{\tau } g_{kj}(w)C_{ij}(\tau -w)\mathrm{d}w\nonumber \\+ & {} \sum _{j=1}^{D}\int _{0}^{\infty } g_{kj}(\tau +w)C_{ji}(w)\mathrm{d}w, \end{aligned}$$

(50)

where we use the relation $C_{ij}(\tau )=C_{ji}(-\tau )$.

In the single-line case Eq. (50) becomes

$$\begin{aligned} C(\tau )=\left( \omega '+1\right) {\bar{v}}g(\tau )+\int _{0}^{\tau } g(w)C(\tau -w)\mathrm{d}w+\int _{0}^{\infty } g(\tau +w)C(w)\mathrm{d}w. \end{aligned}$$

(51)

Here, we set the kernel function $g(\tau )=ab\exp (-b\tau )$. We can solve this equation using Laplace transformations,

$$\begin{aligned} C(\tau )=\frac{ab\left( \omega '+1\right) {\bar{v}}}{2(1-a)}\exp \left[ -b(1-a)\tau \right] . \end{aligned}$$

(52)

Based on these results, the difference between the SE-NBD and Hawkes processes is only the intensity of the autocorrelation function. In the case of $\omega '=0$, it corresponds to the correlation function of the Hawkes process.