The Crash-NIG factor model

Abstract

It is well known that the one-factor copula models are very useful for risk management and measurement applications involving the generation of scenarios for the complete universe of risk factors and the inclusion of CDO structures in a portfolio context. For this objective, it is necessary to have a simple and fast model that is also consistent with the scenario simulation framework. In this paper we present three extensions of the NIG one-factor copula model which jointly have not been considered so far: (1) tranches with different maturities modeled in a consistent way, (2) a portfolio with different rating buckets, relaxing the assumption of a large homogeneous portfolio, and (3) different correlation regimes. The regime-switching component of the proposed Crash-NIG factor model is especially important in view of the current credit crisis. We also introduce liquidity premiums into the Crash-NIG factor model and show that the actual credit crisis is substantially driven by liquidity effects.

Appendices

Appendix A: Term structure extension of the NIG factor copula model

The Gaussian factor copula model as well as the NIG factor copula model assume the distributions of the factors to be the same over arbitrary time horizons up to maturity of the tranches. In particular, for pricing CDO tranches the loss distributions for each payment date (i.e. quarterly for iTraxx tranches) are needed for valuation. Thus, applying the model to the long-dated tranches is not consistent with the short-dated ones.

The appropriate process for the factors with NIG-distributed increments (see [10] for more details on NIG-distribution) is given by N_(s)(t) with a scaling factor s and independent increments $d {\hbox{N}}_{(s)}(t) \sim {\mathcal{NIG}}\left(s \alpha, s \beta, -s \frac{\beta\gamma^{2}}{\alpha^{2}}dt, s \frac{\gamma^{3}}{\alpha^{2}}dt \right), \gamma=\sqrt{\alpha^2-\beta^2},$ and has the following properties:

• (i) the increments d N_(s)(t) have zero mean and variance dt;

• (ii) the process N_(s)(t) has zero mean, variance t, skewness $3\frac{\beta}{s \gamma^2 \sqrt{t}}$ and kurtosis $3 + 3\left(1 + 4\left( \frac{\beta}{\alpha} \right)^2 \right) \frac{\alpha^2}{s^2 \gamma^4 t};$

• (iii) ${\hbox{N}}_{(s)}(t) \sim {\mathcal{NIG}}\left(s \alpha, s \beta, -s \frac{\beta \gamma^2}{\alpha^2}t, s \frac{\gamma^3}{\alpha^2}t \right).$

Thus, we can define the processes of the common (non-normalized) market factor M and of the idiosyncratic (non-normalized) factor X _i as

$$ X_i(t) = {\hbox{N}}_{\left(\frac{\sqrt{1-a^2}}{a}\right)}(t), M(t) = {\hbox{N}}_{(1)}(t) $$

with independent processes X _i and M. Due to the scaling and convolution properties of the NIG distribution (see, e.g. [10]), the asset return processes $A_{i}(t)=a M(t)+\sqrt{1-a^{2}}X_{i}(t)$ are also processes of the same kind, namely ${\hbox{N}}_{(\frac{1}{a})}(t).$ Normalizing the processes

$$ {\widetilde{X}}_{i}(t)=\frac{X_i(t)}{\sqrt{t}}, {\widetilde{M}}(t)=\frac{M(t)}{\sqrt{t}}, {\widetilde{A}}(t)=\frac{A(t)}{\sqrt{t}} $$

(6.1)

we do not loose the time dependence in the distribution as it is the case for the Gaussian distribution. For example, the distribution of the normalized market factor would be:

$$ {\widetilde{M}}(t) \sim {\mathcal{NIG}}\left(\alpha \sqrt{t}, \beta \sqrt{t}, -\frac{\beta \gamma^2}{\alpha^2} \sqrt{t}, \frac{\gamma^3}{\alpha^2} \sqrt{t}\right) $$

with a zero mean, variance 1, skewness $3\frac{\beta}{\gamma^2 \sqrt{t}}$ and kurtosis $3 + 3\left(1 + 4\left( \frac{\beta}{\alpha} \right)^2 \right) \frac{\alpha^2}{\gamma^4 t}.$

In contrast to the Gaussian distribution with zero skewness and kurtosis 3, the time component now influences the skewness and kurtosis of the NIG distribution. Note that the skewness converges to zero and the kurtosis to 3 with infinitely large t. According to the central limit theorem the sum of a large number of independent returns is approximately normally distributed.

Appendix B: Proof of Proposition 1

Proof We start with a general NIG distribution for the factors ${\widehat{M}}$ and ${\widehat{X}}_{ij}$:

$$ d{\widehat{M}}(t) \sim {\mathcal{NIG}}\left( \alpha^{CM} , \beta^{CM}, \mu^{CM}, \delta^{CM} \right), $$

(7.1)

$$ d{\widehat{X}}_{ij}(t) \sim {\mathcal{NIG}}\left( \alpha^{CX} , \beta^{CX}, \mu^{CX}, \delta^{CX} \right). $$

(7.2)

It follows from the requirement (A2) and the convolution property of the NIG distribution, that the first two parameters of the distributions must be equal to those in the first state:

$$ \alpha^{CM}= \alpha, \beta^{CM}= \beta, \alpha^{CX}= \frac{\sqrt{1-a_j^2}}{a_j} \alpha, \beta^{CX}= \frac{\sqrt{1-a_j^2}}{a_j} \beta. $$

(7.3)

Besides, the requirement (A1) means that:

$$ \mu^{CM}= - \delta^{CM}\frac{\beta}{\gamma}, \mu^{CX}= - \delta^{CX}\frac{\beta}{\gamma}. $$

(7.4)

Now, we consider the distribution of the asset return in the second state. This is the distribution of the sum of two NIG random variables:

$$ {\widehat{a}}_jd{\widehat{M}}(t) \sim {\mathcal{NIG}}\left( \frac{\alpha}{{\widehat{a}}_j}, \frac{\beta}{{\widehat{a}}_j},-{\widehat{a}}_j\delta^{CM}\frac{\beta}{\gamma}, {\widehat{a}}_j\delta^{CM} \right), $$

(7.5)

$$ \begin{aligned}\sqrt{1-\widehat{a}_j^2}d\widehat{X_{ij}}(t) &\sim {\mathcal{NIG}}\left( \frac{\sqrt{1-a_j^2}}{a_j\sqrt{1-\widehat{a}_j^2}}\alpha , \frac{\sqrt{1-a_j^2}}{a_j\sqrt{1-\widehat{a}_j^2}} \beta , \right.\\ &\quad\left. - \sqrt{1-\widehat{a}_j^2}\delta^{CX}\frac{\beta}{\gamma}, \sqrt{1-\widehat{a}_j^2}\delta^{CX} \right).\end{aligned} $$

(7.6)

The both distributions are stable under convolution if the first two parameters are equal. This is the case when

$$ \frac{1}{\widehat{a}_j}=\frac{\sqrt{1-a_j^2}}{a_j\sqrt{1-\widehat{a}_j^2}} $$

or equivalent by

$$ \frac{\sqrt{1-\widehat{a}_j^2}}{\widehat{a}_j}=\frac{\sqrt{1-a_j^2}}{a_j}. $$

Hence,

$$ \widehat{a}_j=a_j. $$

(7.7)

The distribution of the asset return in the second state is thus given by

$$ dA_{ij}(t)\sim {\mathcal{NIG}}\left(\frac{\alpha}{a_j}, \frac{\beta}{a_j} , -\left( a_j \delta^{CM}+\sqrt{1-a_j^2}\delta^{CX} \right)\frac{\beta}{\gamma}, a_j \delta^{CM}+\sqrt{1-a_j^2}\delta^{CX}\right). $$

(7.8)

According to assumption (A4), the distribution must be the same as in the first state, i.e. the third and the fourth parameter must satisfy:

$$ -\left( a_j \delta^{CM}+\sqrt{1-a_j^2}\delta^{CX} \right)\frac{\beta}{\gamma} = -\frac{1}{a_j} \frac{\beta \gamma^2}{\alpha^2} dt, $$

(7.9)

$$ a_j \delta^{CM}+\sqrt{1-a_j^2}\delta^{CX} = \frac{1}{a_j} \frac{\gamma^3}{\alpha^2} dt $$

(7.10)

The two equations are actually the same:

$$ \delta^{CM}=\frac{1}{a_j^2} \frac{\gamma^3}{\alpha^2} dt-\frac{\sqrt{1-a_j^2}}{a_j}\delta^{CX}. $$

According to assumption (A3), δ^CM has to be independent of a _j . Therefore, it has to be of the form

$$ \delta^{CM}=k \frac{\gamma^3}{\alpha^2} dt $$

for some constant k > 0. Then corresponding δ^CX is given by

$$ \delta^{CX}=\frac{1-k a_j^{2}}{1-a_j^2}\frac{\sqrt{1-a_j^2}}{a_j} \frac{\gamma^3}{\alpha^2} dt, $$

and we come up with the distributions in Eq. (2.5) that completes the proof. □

Appendix C: Unconditional moments of the Crash-NIG factor

Proposition 2 Denoting the variance by ${{\mathbb V}, }$ the skewness by ${{\mathbb S}}$ and the kurtosis by ${{\mathbb K}, }$ the moments of the unconditional distribution of the factor M(t) are:

$$ \begin{aligned} {\mathbb E}\left(M(t)\right) &= 0 \\ {\mathbb V}\left(M(t)\right) &= {\mathbb E}\left( T^r_1(t)+\lambda^2 T^r_2(t) \right) \\ {\mathbb S}\left(M(t)\right) &= \frac{3\beta}{\gamma^2} {\mathbb E}\left(\frac{1}{ \sqrt{T^r_1(t)+\lambda^2 T^r_2(t)} }\right) \\ {\mathbb K}\left(M(t)\right) &= 3 + 3\left( 1 +4 \left( \frac{\beta}{\alpha}\right)^2\right){\mathbb E}\left(\frac{1}{ T^r_1(t)+\lambda^2 T^r_2(t)} \right) \frac{\alpha^2}{\gamma^4}. \end{aligned} $$

The moments of the unconditional distribution of the factor X _ij (t) are:

$$ \begin{aligned} {\mathbb E}\left(X_{ij}(t)\right) &= 0 \\ {\mathbb V}\left(X_{ij}(t)\right) &= \frac{t-a_j^2 {\mathbb E}\left( T^r_1(t)+\lambda^2 T^r_2(t)\right)}{1-a_j^2} \\ {\mathbb S}\left(X_{ij}(t)\right) &= \frac{3\beta a_j}{\gamma^2} {\mathbb E}\left(\frac{1}{ \sqrt{t-a_j^2\left( T^r_1(t)+\lambda^2 T^r_2(t)\right)} }\right) \\ {\mathbb K}\left(X_{ij}(t)\right) &= 3 + 3\left( 1 +4 \left( \frac{\beta}{\alpha}\right)^2\right){\mathbb E}\left(\frac{1}{ t-a_j^2\left(T^r_1(t)+\lambda^2 T^r_2(t)\right)} \right) \frac{a_j^2 \alpha^2}{\gamma^4}. \end{aligned} $$

Proof For the i-th central moment of the unconditional distribution of M(t) we have:

$$ \begin{aligned} \underset{-\infty}{\overset{\infty}{\int}}x^i f_{M(t)}(x)dx =& \underset{-\infty}{\overset{\infty}{\int}} \underset{[0,t]^2}{\int} x^i f_{{\mathcal{NIG}}}\left(x; \alpha , \beta , -\left(z_1+\lambda^2 z_2\right)\frac{\beta \gamma^2}{\alpha^2}, \left(z_1+\lambda^2 z_2\right) \frac{\gamma^3}{\alpha^2} \right) \\ &\times f_{T^r(t)}(z_1,z_2)d(z_1,z_2)\,dx \\ =&\underset{[0,t]^2}{\int} \left[ \underset{-\infty}{\overset{\infty}{\int}} x^i f_{{\mathcal{NIG}}}\left(x; \alpha , \beta , -\left(z_1+\lambda^2 z_2\right)\frac{\beta \gamma^2}{\alpha^2}, \left(z_1+\lambda^2 z_2\right) \frac{\gamma^3}{\alpha^2} \right)\,dx\right] \\ &\times f_{T^r(t)}(z_1,z_2)d(z_1,z_2). \end{aligned} $$

So we must just integrate the moments of M(t)|T ^r(t), i.e. the moments of a NIG distribution, over the distribution of the duration stays. Since the expectation of the conditional distribution of M(t)|T ^r(t) is zero, the unconditional expectation is zero as well, i.e. ${{\mathbb E}(M(t)) = 0. }$ Further, the variance of M(t) is:

$$ {\mathbb V}\left(M(t)\right)={\mathbb E}\left( \left(T^r_1(t)+\lambda^2 T^r_2(t)\right) \frac{\gamma^3}{\alpha^2} \frac{\alpha^2}{\gamma^3} \right)={\mathbb E}\left( T^r_1(t)+\lambda^2 T^r_2(t) \right). $$

The skewness of M(t) is:

$$ {\mathbb S}(M(t)) = {\mathbb E}\left(\frac{3\beta}{\alpha \sqrt{\gamma} \sqrt{\left(T^r_1(t)+\lambda^2 T^r_2(t)\right) \frac{\gamma^3}{\alpha^2}}} \right) = \frac{3\beta}{\gamma^2} {\mathbb E}\left(\frac{1}{ \sqrt{T^r_1(t)+\lambda^2 T^r_2(t)} }\right). $$

And finally, the kurtosis of M(t) is:

$$ \begin{aligned} {\mathbb K}\left(M(t)\right) &= 3+ 3\left( 1+4\left(\frac{\beta}{\alpha} \right)^2 \right){\mathbb E}\left( \frac{1}{\gamma \left(T^r_1(t)+\lambda^2 T^r_2(t)\right) \frac{\gamma^3}{\alpha^2}}\right)\\ &= 3 + 3\left( 1 +4 \left( \frac{\beta}{\alpha}\right)^2\right){\mathbb E}\left(\frac{1}{ T^r_1(t)+\lambda^2 T^r_2(t)} \right) \frac{\alpha^2}{\gamma^4}. \end{aligned} $$

The derivation of the moments for X _ij (t) is analogue and straightforward.□

Now we consider the moment matching exemplarily for the market factor. If we want to approximate M(t) with a NIG distribution by moment matching:

$$ M(t)\simeq {\mathcal{NIG}}\left(\widehat{\alpha}(t), \widehat{\beta}(t), \widehat{\mu}(t), \widehat{\delta}(t)\right), $$

(8.1)

then, setting $\widehat{\gamma}(t)=\sqrt{\widehat{\alpha}^2(t)-\widehat{\beta}^2(t)},$ the following system of four equations has to be solved for $\widehat{\alpha}(t), \widehat{\beta}(t), \widehat{\mu}(t), \widehat{\delta}(t)$:

$$ \begin{aligned} \widehat{\mu}(t) + \widehat{\delta}(t)\frac{\widehat{\beta}(t)}{\widehat{\gamma}(t)} &= 0 \\ \widehat{\delta}(t)\frac{\widehat{\alpha}^2(t)}{\widehat{\gamma}^3(t)} &= {\mathbb E}\left( T^r_1(t)+\lambda^2 T^r_2(t) \right) \\ \frac{3\widehat{\beta}(t)}{\widehat{\alpha}(t)\sqrt{\widehat{\delta}(t)\widehat{\gamma}(t)}} &= \frac{3\beta}{\gamma^2} {\mathbb E}\left(\frac{1}{ \sqrt{T^r_1(t)+\lambda^2 T^r_2(t)} }\right) \\ \left( 1 +4 \left( \frac{\widehat{\beta}(t)}{\widehat{\alpha}(t)}\right)^2\right)\frac{1}{\widehat{\delta}(t)\widehat{\gamma}(t)} &= \left( 1 +4 \left( \frac{\beta}{\alpha}\right)^2\right){\mathbb E}\left(\frac{1}{ T^r_1(t)+\lambda^2 T^r_2(t)} \right) \frac{\alpha^2}{\gamma^4}. \end{aligned} $$

In general, this system of equations cannot be solved analytically. In the special case of β = 0, however, the system is easy to solve. The parameters for M(t) are:

$$ \begin{aligned} \widehat{\beta}(t)&=0, \hbox { }\widehat{\mu}(t)=0 \hbox { and, using } \widehat{\gamma}(t)=\sqrt{\widehat{\alpha}^2(t)-\widehat{\beta}^2(t)}=\widehat{\alpha}(t),\\ \widehat{\alpha}(t)&= \frac{\alpha}{\sqrt{{\mathbb E}\left( T^r_1(t)+\lambda^2 T^r_2(t)\right) {\mathbb E}\left(\frac{1}{ T^r_1(t)+\lambda^2 T^r_2(t) }\right)}},\\ \widehat{\delta}(t)&=\alpha \cdot \sqrt{\frac{{\mathbb E}\left( T^r_1(t)+\lambda^2 T^r_2(t)\right)}{{\mathbb E}\left(\frac{1}{ T^r_1(t)+\lambda^2 T^r_2(t)} \right)}}. \end{aligned} $$

The expectation ${{\mathbb E}\left( T^r_1(t)+\lambda^2 T^r_2(t)\right)}$ can be easily computed as:

$$ {\mathbb E}\left( T^r_1(t)+\lambda^2 T^r_2(t)\right)= {\mathbb E}\left( T^r_1(t)\right)+\lambda^2{\mathbb E}\left(T^r_2(t)\right)=h^r_1(t)+\lambda^2 h^r_2(t), $$

with

$$ \begin{aligned} h^r_i(t)&={\mathbb E}\left(T^r_i(t)\right)={\mathbb E}\left(\underset{0}{\overset{t}{\int}}1_{\{\rm{state\,i\,at\,time\,s}\}}\,ds\right)=\underset{0}{\overset{t}{\int}}{\mathbb Q}\left[\rm{state\,i\,at\,time\,s}\right]\,ds \\ &=\underset{0}{\overset{t}{\int }}(exp(sO))_{r,i}\,ds\end{aligned} $$

(8.2)

for i = 1, 2, where $O:=\underset{h\downarrow 0}{lim}\frac{P(h)-I}{h}$ is the intensity matrix of the transition function P(h), i.e. $P(h)=exp\{hO\}=\sum\nolimits_{k=0}^{\infty}\frac{h^k}{k!}O^k$ (see, e.g., [4, 12]).