信用风险定价新模型：负相关恢复与CDS

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"这篇研究论文探讨了信用风险定价模型中的隐含恢复概念，特别是在连续期限结构的背景下。传统的信用违约掉期（CDS）模型基于历史恢复率假设，但作者提出这种假设可能并不准确，因为它忽略了恢复率与违约率之间的潜在负相关性。通过建立一个新的CDS定价模型，并使用偏微分方程来描述CDS的路径，论文旨在揭示连续隐含恢复率和违约强度的期限结构。通过蒙特卡罗模拟，作者提取了这些结构，并用它们来评估历史恢复假设对信用价值调整（CVA）计算的影响。结果表明，当考虑负相关时，恒定恢复模型可能导致CVA估计过高或过低。" 这篇硕士论文由Nathan Juliano Meibergen撰写，于2015年在荷兰代尔夫特理工大学的电气工程、数学和计算机科学学院的代尔夫特应用数学研究所完成。指导教授是Prof. Dr. C.W. Oosterlee，其他委员会成员包括Dr. ir. J.H.M. Anderluh和Dr. D. Fokkema。论文采用了创作共享署名-相同方式共享4.0国际许可，可在SSRN上找到电子版。论文的核心贡献在于提出了一个改进的CDS定价框架，该框架考虑了恢复率的动态变化，这使得模型能更准确地反映市场实际情况。传统的信用风险模型通常假设恢复率为历史平均值，但这种简化假设可能无法充分捕捉到信贷市场的复杂性。负相关性的引入使得模型能够更好地估计信用事件的真实成本，进而影响金融机构的CVA计算。CVA是金融机构因对手方信用风险而可能遭受损失的现值，其准确估计对于风险管理至关重要。在实际应用中，理解并量化隐含恢复率的期限结构可以帮助金融机构更精确地评估和对冲信用风险，特别是在设计和交易信用衍生产品时。此外，这个模型对于监管机构制定更为稳健的资本充足率要求和市场参与者进行风险对冲策略的制定也具有重要意义。这篇论文通过构建新的信用风险定价模型，为理解和改善金融市场的风险评估提供了重要的理论基础和实践工具。通过深入研究恢复率与违约率之间的关系，它为信用风险的量化提供了一种更为细致入微的方法，从而有助于提高整个金融系统的稳健性。

Contents XIII

I Matlab Code 164

I.1 D&H model of Das&Hanouna . . . . . . . . . . . . . . . . . . . . . . . . . 165

I.1.1 Main code for Calibration and pricing of CDSs . . . . . . . . . . . 165

I.1.2 Calculation of multiple spreads . . . . . . . . . . . . . . . . . . . . 166

I.1.3 Calculation of spread given input variables and parameters . . . . 168

I.1.4 Reaching probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 169

I.2 Main optimization ﬁle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

I.2.1 Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

I.2.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

I.2.3 Dummy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

I.2.4 Monte Carlo for retrieving the term structure . . . . . . . . . . . . 180

Bibliography 183

Introduction 3

1.1 The correct measure

When calculating certain probabilities or expectations we are always implicitly basing this

upon a certain measure. A measure can be understood as the distribution of probabilities

related to events. In measure M

event A may have probability 0.3, whereas in measure

this same event may have a probability of 0.4.

The understanding of the usage of measures is especially important in the world of ﬁnance.

Formulas and deductions may be simpliﬁed if a convenient measure is applied, whereas

results may only be valid under the correct measure.

We won’t get into the details of measure theory as this a very large and speciﬁc ﬁeld within

mathematics. What we do want, is to give some intuition for the two most frequently

used measures within ﬁnance. This intuition is often enough to understand which one to

use when.

Two measures are of great importance, namely the physical measure, which contains the

probabilities of certain events occurring in our real world: if ﬁrms go bankrupt with

probability 0.01, then on the long term 1% of the ﬁrms should have gone bankrupt. The

other measure is the risk-neutral measure, this measure sets probabilities in such way that

they agree with our model and the market expectations: even though on the long term

1% of all ﬁrms go bankrupt, the risk-neutral measure might tell us that the probability of

a bankruptcy equals 0.015. We will explain both and their diﬀerences more thoroughly.

1.1.1 Risk-neutrality: the Q-measure

The risk-neutral measure describes a complete world in which there are no risks related

to any ﬁnancial product. All contingent attainable claims are assumed to move according

to a risk free rate (see (B.7)). Thus suppose a contingent attainable claim V

matures at

time T , let the interest rate r be a known constant, then

= E

−r(T −t)

. (1.1)

As the risk-neutral world induced by Q-measure assumes there to be no risks, the terminal

payoﬀ V

is risk-free and by a no-arbitrage argument it should be discounted by the risk

free rate. The convenience of this measure is twofold:

1. The discounting term is always set to the risk-free interest rate r, as all products

are risk-free. In reality this payoﬀ doesn’t have to be risk-free such that deciding on

the discounting term is a complicated issue.

2. Everything in this world increases at a risk-free rate and we thus don’t have to decide

upon the correct drift term. The most straightforward example of this convenience

Introduction 4

is seen in modeling the movement of stock prices, which in this world have drift

term r, such that a simple Monte Carlo approach can be used to determine prices.

In appendix B it is shown to some extend how the risk-neutral measure is constructed

as the measure which makes contingent attainable claims a martingale under the money

market account numeraire.

As (1.1) tells us, under the risk neutral measure, all that is required to price a product

is its resulting claim. And as it is very complicated to quantify risk premiums we prefer

determining the claims of products such that discounted by the risk-neutral interest rate

we retrieve the fair price. Usually however, we cannot directly ﬁll in the Right Hand Side

(rhs) of (1.1) as it will depend upon unknown parameters, think for example about a

probability of default. Instead we come up with a model for the claim and then set the

unknown parameters such that the expectation provides us with the correct price. This

process, in which parameters are adjusted to ﬁt the market prices is called calibration. The

resulting parameters are called their implied counterparts, e.g. the implied probability of

default.

As we will show in section 1.1.3, implied parameters do not represent their real world

counterparts due to the lack of risk premiums.

1.1.2 Physical measure: the P-measure

The P-measure corresponds to observed realisations of underlying properties. For exam-

ple: the probability of default in the P-measure corresponds to analysing the probability

of default from historical realisations, thus by observing for example an empirical distri-

bution of defaults.

The P-measure being related to historical observables provides the misleading understand-

ing

that: ‘anything that is historical is strictly related to the P-measure, whereas anything

that is forward looking is in the Q-measure.’ It is true that observables are P-measure,

thus historical prices are P-measure prices. We may however use these to ﬁnd historical

implied parameters. The process of calibration can easily be applied to historical data

such that we can get historical implied data, which may sound as a contradiction. An

example of such approach can be found in [SUH14]).

1.1.3 The diﬀerence between the P- and Q-measure

At the beginning of this section we stated that probabilities in both measures might

diﬀer. We shall now show you where this diﬀerence comes from. In short the answer is:

I have personally frequently heard this faulty understanding, even from people with a decent amount

of experience.

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