Discrete Dynamics in Nature and Society

Volume 2018, Article ID 3481863, 10 pages

https://doi.org/10.1155/2018/3481863

## A New Default Probability Calculation Formula and Its Application under Uncertain Environments

^{1}Postdoctoral Research Base, Henan Institute of Science and Technology, Xinxiang 453003, Henan, China^{2}Postdoctoral Research Station, Henan University, Kaifeng 475000, China

Correspondence should be addressed to Liang Wu; moc.361@gnailuwedin

Received 20 March 2018; Revised 31 May 2018; Accepted 10 June 2018; Published 1 August 2018

Academic Editor: Beatrice Di Bella

Copyright © 2018 Liang Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the real world, corporate defaults will be affected by both external market shocks and counterparty risks. With this in mind, we propose a new default intensity model with counterparty risks based on both external shocks and the internal contagion effect. The effects of the external shocks and internal contagion on a company cannot, however, be observed, as uncertainty in the real world contains both randomness and fuzziness. This prevents us from determining the size of the shocks accurately. In this study, fuzzy set theory is utilized to study a looping default credit default swap (CDS) pricing model under uncertain environments. Following this, we develop a new fuzzy form pricing formula for CDS, the simulation analysis of which shows that all kinds of fuzziness in the market have a significant impact on credit spreads, and that the credit spreads, relative to the degree of external shock fuzziness, are much more sensitive. Nevertheless, for a certain degree of fuzziness in the market, credit spreads, relative to changes in counterparty risk, are much more sensitive. Using random analysis and fuzzy numbers, one can think of even more uncertain sources at play than the processes of looping default and investor subjective judgment on the financial markets, and this broadens the scope of possible credit spreads. Compared to the existing related literature, our new fuzzy form CDS pricing model with counterparty risk can consider more factors that influence default and is closer to the reality of the complexity of the dynamics of default. It can also employ the membership function to describe the fuzzy phenomenon, enable the fuzzy phenomenon to be estimated in two kinds of state, and can simultaneously reflect both the fuzziness and randomness in financial markets.

#### 1. Introduction

Information disclosure plays a role as a bridge of information asymmetry between issuers and investors, for disclosure can reduce information asymmetry. Credit derivatives, as nonstandardized financial derivatives, have the following characteristics: the information of trading environment not open to the public, no guarantee of the performance from stock exchange, high risk of performance, and volatile price floating. As a result, the asymmetry of market information flow that acts as a decisive factor of the sensitive credit risk pricing of financial instruments appears. Moreover, it will lead to obvious fuzziness in the process of characterization of the counterparty risk. For example, when some emergencies are caused by financial institution as stock and stock markets, the adverse information dissemination will lead to counterparty default behavior. Meanwhile, the internal market credit default will be convergent and infectious. The counterparty default is featured by synchronicity, aggregation, fuzziness, and uncertainty. In the credit default swap pricing of counterparty risk, the external market shocks and internal contagion effects on the company’s size cannot be observed; i.e., the precise value cannot be gotten through a set of random values. Similarly, in the process of risk analysis and other derivatives pricing, people will encounter many similar fuzziness due to the lack of cognition on financial markets, which leads to incapability of seizing market information. For instance, in factor copulas model, correlation cannot be observed between the company’s assets. That is to say, people will face the ubiquity of fuzziness in the process of financial risk analysis and its derivatives pricing. As a consequence, the study of fuzziness, hesitation, presentation, and evolution under the condition of asymmetric information is extremely important to control the outbreak, diffusion, and aggregation of the credit risk. In addition, the pricing model by combining fuzziness and randomness under the condition of asymmetric information can strip, transfer, and hedge credit risk more efficiently. Thus, the credit risk analysis in the fuzzy uncertain environment and derivatives pricing model in this paper is of great realistic and theoretical significance.

As pointed out in [1–5], in the real world, corporate defaults will be affected by both external market shocks and counterparty risks. With this in mind, we proposed a new default intensity model with counterparty risks based on both external shocks and the internal contagion effect. However, the effects of external shocks and internal contagion on a company cannot be observed, because uncertainty in the real world contains randomness and fuzziness [6–13]. This prevents us from determining the size of shocks accurately. Therefore, inspired by [14–18], fuzzy set theory is adopted to study a looping default credit default swap (CDS) pricing model under uncertain environments. Following this, we set up a new fuzzy form pricing formula for CDS, the simulation analysis of which shows that all kinds of fuzziness in the market have a significant impact on credit spreads. Using random analysis and fuzzy numbers, one can think of more uncertain available sources than the processes of looping default and investor subjective judgment on the financial markets, which broadens the scope of possible credit spreads. Compared to the existing related references [19–23], our new fuzzy form CDS pricing model with counterparty risk can consider more factors that influence default and is closer to the reality of complex default dynamics. It can also employ the membership function to describe the fuzzy phenomenon, enable the fuzzy phenomenon to be estimated in two kinds of state, and simultaneously reflect the fuzziness and randomness in financial markets.

#### 2. Introducing the Fuzziness to the CDS Pricing Model with Counterparty Risk

In this section, we study the CDS pricing model under fuzzy random environments based on the new looping default model, and the basic concept of fuzzy set theory can refer to literature [24–27].

##### 2.1. The New Looping Default Intensity Model with Counterparty Risk

Firstly, we introduce some basic concepts. Let be a complete probability space. Here, is a càdlàg, , is the time limit for uncertain economic system, and is the equivalent martingale measure. Suppose that there are companies in the market, and stochastic default time of each company is expressed as ; the external shock arrival time is expressed as . According to [1], we define the default time as , where is a unit exponential random variable. Meanwhile, the external shock arrival time is assumed to obey uniform distribution in the time interval . Let represent the information sets. So, the conditional survival probability of company i can be expressed as .

For considering more factors that influence default and that are closer to the reality of the complexity of the dynamics of default, we propose a new model as follows:where , , and represent the initial default intensity, the upward jump ratio, and the counterparty risk, respectively, and is an indicator function of company i; i.e., if the company i defaults, then the function value is 1; otherwise it is 0.

To simplify the discussion, we discuss the following:

Without loss of generality, we assume that before moment t, shock events happen, so (2) and (3) can be written as

In order to begin the operation in the new measure , we have the following lemmas.

Lemma 1 ([28]). *Suppose the parameter is the intensity process of default time relative to the probability space ; if there exist a constant and a nonrandom time , which meets the integrability condition, ; at the same time, we set a nonrandom time ; then the nonnegative processis a uniformly integrable martingale relative to the probability space , and the process is almost always positive in the time interval .*

Lemma 2 ([29]). *Suppose there exists a probability space , and is a new absolutely continuous probability measure relative to the measure , is the derivation process of the corresponding Radon-Nikodym, and if is -measurable, then we have the following conclusion: .*

Theorem 3. *The joint survival probability of different companies and the marginal survival probability of a single company for formulas (4) and (5), are, respectively, as follows: if , we have the following conclusion:and if , we haveThe single company’s marginal survival probability is as follows:** if , we have** and if , we have*

*Proof. *By Lemmas 1 and 2, we have the following: if , where and , then we can derive the following:if , we can derive the following:if , where and , then there areand if , we can derive the following:The proof of the marginal survival probability of a single company only needs to take the value , in formulas (7) and (8), which completes the proof.

##### 2.2. The CDS Pricing Model with Counterparty Risk under Uncertain Environments

Financial data may not be timely or accurately recorded, due to unforeseen circumstances or man-made errors. In order to reflect the influence of fuzziness on the interest rate, we assume that the interest rate is a triangular fuzzy random variable ,That is, the triangular fuzzy random variable is represented by the center , left-width , and right-width (also known as fuzzy degree factor); see Figure 1. We can find that the fuzziness of investors on the market interest rate in the process increases as becomes larger. The -cut of isMeanwhile, the influences of external market shocks and counterparty risks are assumed to be triangular fuzzy numbers, whose membership functions are defined as follows:that is, the fuzzy numbers , and , have a symmetric triangle-type shape, respectively, with centers , and , , and fuzzy degree factors , and , . The rationale behind fuzzy external market shocks and counterparty risk lies in the difficulty of getting a precise estimate of the actual external shock and counterparty risk of default intensity. By modeling the external shocks and counterparty risk as a fuzzy number, one can take the investors’ subjective judgment into account.