Page 38 - CCFA Journal - Ninth Issue

数学建模 Math Modeling 加中金融
3 Extensions and applications of the Merton Model
The simplicity of the Merton model relies on applying the Black-Scholes-Merton formula of pricing the European options to value
firm’s equity and debt. However, this comes at the cost of too simplistic assumptions regarding the asset value process, interest
rate, and the capital structure. Since the Merton model was developed, there has been many efforts to improve the model, which
has been proved to be challenging.

3 Merton 模型的扩展与应用

Merton 模型的简单性依赖于应用 Black-Scholes-Merton 欧洲期权定价公式来评估公司的股权和债务。然而，这是以对资产
价值过程、利率和资本结构的过于简单化的假设为代价的。自 Merton 模型被开发以来，人们进行了许多改进模型的努力，
这被证明是具有挑战性的。

However, the concept of the Merton model has become fundamental to the credit risk modeling and has found many applications
in the financial industry. Three major industry application of the structured credit model is discussed in this section.

然而，Merton 模型的概念已成为信用风险建模的基础，并在金融业中得到广泛应用。本节讨论结构化信贷模型的三大行
业应用。

3.1 The implied PD and credit migration models

The structured credit model, which has a widespread practical usage, is Moody’s KMV model [2]. There are few changes made to
the Merton model in the KMV model. First, Instead of the Gaussian distribution, the EDF in KMV model is replaced by some
decreasing function that is empirically estimated; Secondly the liabilities is replaced by one that better reflects the firm’s liabilities.
An important quantity of the KMV model is the so-called distance to default (DD):

3.1 隐含的 PD 和信用迁移模型

具有广泛实际应用的结构化信用模型是穆迪的 KMV 模型[2]。 KMV 模型中对 Merton 模型的改动很少。首先，KMV 模型中
的 EDF 不是高斯分布，而是用一些经验估计的递减函数代替；其次，负债被更能反映公司负债的负债所取代。KMV 模型
的一个重要量是所谓的违约距离 (DD)：

( )
( . )( ))
DD = [5]
√
Another widely used structured credit model in the industry is the one proposed by CreditMetrics [3], in which both default and
credit quality, measured by rating, are linked to the firm’s asset value, as shown in Figure 2. The PD and migration can be
modeled as

另一种在业界广泛使用的结构化信用模型是由 CreditMetrics [3] 提出的模型，其中违约和信用质量（通过评级衡量）与公
司的资产价值相关联，如图 2 所示。PD 和迁移可以被建模为

( ) = < = [6]

( ) = < < = − [7]

It can be used to model both rating migration and default risk, which is required in the Incremental Risk Charge (IRC) in Basel 2.5
[4] and valuation/risk/capital model in Credit Valuation Adjustment (CVA). The model can also be used for credit derivatives
pricing (For example, see [5, 6]).

它可用于对评级迁移和违约风险建模，这是巴塞尔协议 2.5 [4] 中的增量风险(IRC) 和信用估值调整 (CVA) 中的估值/风险/资
本模型所要求的。该模型还可用于信用衍生品定价（例如，参见 [5, 6]）

Figure 2

Source: CreditMetrics - Technical Document: The Benchmark for Understanding Credit Risk

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