Chapter 3 Methodology
3.3 Asymptotic Single Factor Models
3.3 Asymptotic Single Factor Models
Definition (One factor Gaussian copula). Consider a portfolio of m credit instruments. The standardized asset return up to time t of the i th issuer in the portfolio, 𝐴𝑖(𝑡) , is assumed to be of the form: time 𝜏𝑖 of the i th issuer using a percentile-to-percentile transformation, i.e. the issuer i defaults before time t when
where 𝑄𝑖(𝑡) denotes the probability of the issuer i to default before time t
i i
Q t Q t (3.18)
The risk-neutral probabilities are implied from the observable market prices of credit default instruments (e.g. bonds or CDS).
The factor M can be interpreted as the systematic common market factor and 𝑋𝑖 as the idiosyncratic factors. Correlation between the asset returns of the issuers i and j equals 𝑎𝑖𝑎𝑗 . Conditionally on M, the asset returns of the different issuers are independent.
According to (3.15), the i th issuer defaults up to time t if
Then the probability that the i th issuer defaults up to time t, conditional on the factor
‧
Under the assumption of LHP, the default thresholds 𝐶(𝑡) of all issuers are the same as well and the default probability of all issuers in the portfolio conditional on M is given by
By the theorem, the loss distribution of an infinitely large homogeneous portfolio with the asset returns following a one factor Gaussian copula model is given by
1 2Φ 1
with x ∈ [0,1] the percentage portfolio loss.
Definition (Double-t copula). Consider a portfolio of m credit instruments. The
One natural extension of the LHP approach is to use a distributional assumption that produces heavy tail. The double t one factor model proposed by Hull and White assumes Student t distributions for the common market factor M as well as for the individual factors 𝑋𝑖. Then the loss distribution 𝐹∞ in (13) becomes:
1 2 1
,
‧
where T denotes the Student t distribution function with degrees of freedom. In
general, the degrees of freedom 𝑣𝑋 and 𝑣𝑀 can be different. Unfortunately, it is not possible to solve the integral in (5) analytically using this loss distribution and one has to use some numerical integration method.
The asset returns Ai do not follow necessarily Student-t distributions since the Student-t distribution is not stable under convolution. The distribution function 𝐻𝑖 of 𝐴𝑖 must be computed numerically. Afterwards, it is possible to calculate the default thresholds 𝐶𝑖 by 𝐻𝑖−1(𝑞𝑖). This procedure is quite time consuming and it makes the double t model too slow for Monte Carlo based risk management applications.
Definition (One factor NIG copula model). Consider a homogeneous portfolio of m credit instruments. The standardized asset return up to time t of the i th issuer in the portfolio, 𝐴𝑖(𝑡) , is assumed to be of the form: i th issuer using a percentile-to-percentile transformation. The issuer i defaults before time t if
‧
where 𝑄(𝑡) denotes the risk-neutral probability of the instruments to default before time t .
By the theorem, considering an infinitely large homogeneous portfolio with the asset returns following the one factor NIG copula model. Then the distribution of the portfolio loss before recovery is given by
probability of each issuer in the portfolio.The correlation parameters were assumed to be constant over time so far. It is no problem for a pricing application since the correlation parameter can be updated every day by a new calibration. However, it is not realistic to assume the correlations being constant for a simulation application. Especially the financial crisis has shown an extreme correlation shift. For this reason, we want to integrate the possibility of different correlation regimes to the NIG model. First, we consider two regimes: the first is the regime of an usual correlation and the second of a high correlation.
‧
Definition (NIG-Regime Switch copula model). The asset return of the i th issuer, 𝐴𝑖(𝑡), is assumed to be of the form: and a (2 × 2) transition function{𝑃(ℎ)}ℎ≥0. The distribution of the increment of the asset return is d𝐴𝑖𝑗(𝑡) = 𝑁(1
Proposition. Consider an asset return of the i th issuer, 𝐴𝑖(𝑡), as defined in Definition.
Assume, the process Λ𝑡 was in state 𝑟 ∈ {1,2} at the time 0. Let 𝑇𝑟(𝑡) ≔
Then the distributions of 𝑀(𝑡) and 𝑋𝑖(𝑡), the cumulated returns on[0, 𝑡], conditional on the realization of 𝑇𝑟(𝑡), are NIG with the following parameters:
‧
Proposition. The moments of the unconditional distribution of the factor 𝑀(𝑡) are:
M t
0The moments of the unconditional distribution of the factor 𝑋𝑖𝑗(𝑡) are:
M t
0‧
fits the first two moments of the exact distributions. The third and the fourth moments of the approximate distribution are not higher than those of the exact distribution. In the special case of a non-skewed distributions, i.e. 𝛽 = 0, the skewness is zero for the approximate and the exact distributions.Now all distributions necessary to describe the portfolio loss distribution are available. The approximate loss distribution of an infinitely large homogeneous portfolio with the asset returns following a NIG-Regime Switch copula model is given by The default thresholds are computed as
‧
The risks of CDO tranches are mainly from the uncertainties of loss. If the risks of CDO are measured by the basis of its loss distribution function, then it can better reflect the loss uncertainties from investors of tranches. The followings are the definitions of CDO’s risk measurements.
(1) Expected loss as the risk indicator
We can retrieve the expected loss of tranches from the loss distribution of targeted portfolio directly. However, since expected loss represents the absolute loss amount, we can’t use it as the indicator for measuring relative risks. We can adopt the risk indicators that are based on expected loss, such as the expected loss rate and the leverage of expected loss. Expected loss rate is the expected loss of tranche to total nominal principal by definition. Leverage of expected loss is the expected loss rate of tranches to expected loss rate of targeted portfolio ,which is expressed as the following:
Expected loss rate of tranches = ELi D C
D C size of reference portfolio