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数学建模 Math Modeling 加中金融
3.3 Default correlations
Default correlation measures whether credit risky assets are more likely to default together or separately. It is essential for
estimating the risk of credit portfolios. For the credit derivatives with multiple assets as the underlying asset pool like Collateral
Debt Obligation (CDO), correlation risk is the major risk factor, which are captured via default correlation models such as the
normal copula model.
Most default correlation models can be directly or indirectly linked to correlated asset process and PD mapping through a
threshold, within the Merton model framework.
The most important (and material) application of this concept is the Vasicek model, which is used to compute regulatory capital
[9, 10]. Assume a credit portfolio loss with N assets, with each asset following a one factor process:
3.3 违约相关性
违约相关性衡量信用风险资产是否更有可能一起违约或单独违约。它对于估计信贷组合的风险至关重要。对于以多种资产
为基础资产池的信用衍生品，如抵押债务义务（CDO）等，相关风险是主要的风险因素，通过违约相关模型（如正态
copula 模型）捕捉。
在 Merton 模型框架内，大多数默认关联模型可以通过阈值直接或间接链接到关联的资产流程和 PD 映射。
这一概念最重要（也是最重要）的应用是 Vasicek 模型，该模型用于计算监管资本 [9, 10]。假设具有 N 项资产的信贷组合
损失，每项资产都遵循单因素过程：
= ∑ 1 { } [9]
= + 1 − [10]
} is the obligor default indictor. X_i is the “normalized” asset process with a default threshold of U_i. ρ_i is the
where 1 {
correlation; ε_i is an independent random number with standard normal distribution. S is the one common factor that shared by
all obligors. It can be shown that correlation between two obligors is ⟨X_i,X_j⟩=ρ_i ρ_j.
If we assume the portfolio is large enough that the individual risk is diversified away and we can approximate the percentile loss
with variable percentile of the common factor:
} 是义务人违约指标。 X_i 是默认阈值为 U_i 的“规范化”资产流程。 ρ_i 是相关性； ε_i 是标准正态分布的独
这里 1 {
立随机数。 S 是所有债务人共有的一个公因子。可以证明两个债务人之间的相关性为⟨X_i,X_j ⟩=ρ_i ρ_j。
如果我们假设投资组合足够大以分散个人风险，我们可以用公因子的可变百分位数来近似百分位损失：
( ) ≈ ( [ | ) [11]
= ∑ 1 − ( ) [12]
Based on the Vasicek model, the regulatory capital for a credit portfolio is defined as the unexpected loss at 99.9 percentile over
one-year time horizon.
基于 Vasicek 模型，信贷投资组合的监管资本被定义为一年时间范围内 99.9% 的意外损失。
4 The Latest Development
One latest application of Merton type model is to model climate-related transition risk. A Merton framework is used to link the
climate transition scenarios to credit risk, which is measured by PDs for sector/segment and geography. A detailed description of
the model can be found in Ref. [11].
Another latest development is DRC IMA model in FRTB. FRTB is the new risk-based capital requirements for the trading book,
which was initiated by the Basel Committee on Banking Supervision (BCBS) in the years following 2008 financial crisis [12].
DRC captures the default risk of any exposure to credit risk, replacing current IRC models in Basel 2.5. DRC IMA is measured using
a Value-at-Risk (VaR) model. A multi-factor Monte Carlo simulation model over one-year time horizon is required, with PD
correlations and PD/LGD correlation captured in the model.
CCFA JOURNAL OF FINANCE December 2022
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