Structural Credit Risk Model with Jumps Based on Uncertainty Theory

. Traditional finance studies of credit risk structured models are based on the assumption that the price of the underlying asset obeys a stochastic differential equation. However, according to behavioral finance, the price of the underlying asset is not entirely stochastic, and the credibility of financial investors also plays a very important role in asset prices. In this paper we introduce uncertainty theory to describe these credibility of investors and propose a new credit risk structured model with jumps based on the assumption that the underlying asset is described by an uncertain differential equation with jumps. The company default belief degree formula, zero coupon bond value and stock value formula are formulated. Company bond credit spread and credit default swap (CDS) pricing are studied as applications of the proposed model in uncertain markets.


Introduction
Credit risk is the risk that a party to a contract will not be able to perform or fulfill its stated obligations during the transaction due to bankruptcy or other serious economic problems. For a long time, credit risk has been considered as one of the main negative factors affecting the development of the financial. After the outbreak of the global financial crisis in 2008, the management and control of credit risk has become the most important topic of quantitative finance research.
The belief degree of default is a measure of credit risk. There are two major schools of thought in the study of credit risk today, one is the structured model and the other is the parsimonious model. In this paper, the structured model is used to study the company credit risk. Merton(1974) used the Black-Scholes(1973) formula to create the first credit structured model. This approach has now evolved into a large system of models that occupy an important position in bond pricing and default probability measurement in bond pricing and default belief degree measures. The KMV model(2000), developed by Moody based on a structured model, is now one of the most popular credit risk management models in the financial market. Merton model assumes that the value of a company's asset satisfied geometric Brownian motion. It can only describe the continuous changes in the company's asset value, but ignore the abnormal changes in assets caused by a series of abnormal conditions such as the Brexit vote and the epidemic. Xue and Wang(2008) investigated the structured model of credit risk driven by Lévy process. Based on the theory of stochastic analysis of Lévy process, they analyze the default belief degree, bond value and credit spread of company and obtain their analytical expressions. Assumed that the underlying stock price satisfies a geometric double-exponential jump-diffusion process, and the default occurs when the company stock price falls below the default threshold for the first time, Wu, Tian and Chen(2008) proved a structured model for pricing equity default swaps.
The above studies on structured models of credit risk are conducted in the framework of belief degree theory, which we can use only when the belief degree distribution is close enough to the true frequency. In order to get the actual frequency we need to have sufficient sample data and use statistical methods to infer the belief degree distribution. However, due to market or technical reasons, the problems we face in practice sometimes do not allow us to obtain sample data, many investors usually take the advice of experts as their degree of belief in some financial events and use this as the basis for decision-making. Beliefs play an important role in actual financial practice.
In order to portray this belief reasonably, Liu(2010Liu( , 2021 proposed uncertainty theory in 2007 and refined into a branch of mathematics based on normality, duality, subadditivity and the product axiom in 2010. In 2009, Liu(2009 defined the uncertain process and uncertain differential equation, the uncertain differential equation corresponds to the stochastic differential equation in the B-S model. Using this uncertain differential equation to describe stock price, Liu established an uncertain stock price model and gave the pricing formula of standard European options under uncertainty theory. The uncertainty theory was quickly applied to practice, and it also attracted the attention of scholars from all over the world. Chen(2011) proved American option pricing formulas based on the uncertain stock price model above. Peng and Yao(2011) proved a stock model with mean-reverting process and the corresponding option pricing formulas for uncertain markets. Sun and Su(2017) proposed a mean-reverting stock model with floating interest rate in the uncertain financial markets and then employed it to the European option and American option in the uncertain markets. Wu, Zhuang(2018) introduced uncertain theory into credit derivatives, and developed an uncertain formula for credit default swap. Gao, Liu(2021) used uncertain differential equation to dispose of the foreign exchange rate, and investigated an American barrier option of currency model.
To describe a discontinuous uncertain system, Liu(2008) defined an uncertain renewal process in 2008. Yao(2012Yao( ,2021 established an uncertain calculus on the uncertain renewal process, and proposed a uncertain differential equation with jumps, then gave a sufficient condition for this equation having a unique solution and some stability theorems. Yu(2012) proposed a stock model with jumps for uncertain financial markets, and derived the pricing formulas for European call and put options with jumps. By means of uncertain differential equation with jumps, Ji and Zhou(2015) proposed a stock model which contains both the positive jumps and the negative jumps and they also proved European option pricing formulas. Based on these European option pricing formulas with jumps this paper will discuss the measurement of credit risk, where the company's assets are described by uncertain differential equation with jumps.
This paper is organized as follows. In section 2, we introduce some useful concepts of uncertain process and uncertain differential as needed. Then a structural credit risk model with jumps is proposed in section 3. In section 4, as an application of the credit risk structuring model, we discuss the pricing of credit derivatives, including credit spreads and credit default swap.
Section 5 is the conclusion of this paper.

Preliminaries
Uncertainty theory was founded by Liu [6] to provide a mathematical model for dealing uncertain phenomena in human system. Let   , , L M  be an uncertainty space with an uncertain measure M , and the uncertain measure is a set function satisfied the normality, self-duality, countable sub-additivity, and product measure axioms. An uncertain variable  is a measurable function from an uncertainty space to the set of real numbers.
Definition 1 [6] . The uncertainty distribution Definition 2 [6] . Let be an uncertain variable. Then the expected value of  is defined by And an uncertain process is a sequence of uncertain variables indexed by time and space, In this paper we will use the most important two uncertain processes: canonical Liu process and renewal process.
Definition 3 [6] . An uncertain process t C is called a canonical Liu process if (i) 0 =0 C and almost all sample paths are Lipschitz continuous; (ii) t C has stationary and independent increments; C is a normal uncertain variable with expected value 0 and variance 2 t ,and the following uncertainty distribution Definition 5 [7] . Let t X be an uncertain process and t C be a canonical Liu process. For any . Let t X be an uncertain process and t N be a renewal process. For any partition Next, we introduce uncertain differential with respect to both t C and t N .
Definition 7 [16] . Let t C be a canonical Liu process, t N a renewal process and t Z an uncertain process, if there exist uncertain processes , ,    s s s such that And if , f g and h are some given functions, then is called an uncertain differential equation with jumps.
For example let ,   and  be real numbers, we will use the following uncertain differential equation with jumps in the rest of this paper And it is easy to verify that this equation has a solution x represent the maximal integer less than or equal to x .

Default belief degree and Default Distance
Let t p be the bond price, and t V the company value price. Assume that the company value price follows a geometric canonical process with jumps. Then we propose an uncertain company value model as follows here r is the risk-less interest rate, ,   and 0   is the company value process drift, diffusion and jump coefficients. t C is a standard canonical Liu process, t N is a renewal process with positive uncertain inter-arrival times 1 2 3 , ,     . Assume that t C and t N are independent process, 1 2 3 , ,     are iid with distribution function ( ).

 x
It is easy to verify the company value process in uncertain market t Theorem 1. Assume that t N is an uncertain renewal process with independent identical distribution uncertain inter-arrivals 1 2 3 , ,     whose uncertainty distribution is ( ).
  If the company asset is described as (3), and assume that the company's default time can only be T , then the company default belief degree p and default distance d in uncertain market are and   0 ln ln 1 .
where ( )   is a standard normal uncertainty distribution Proof. According to the definition of company default belief degree and company value process (3), we can get which is called default distance of company and then get the following company default belief The prove is completed.

Risk
It means that stock value is a call option of the company asset and the company zero coupon bond value is a non-default bond minus the value of a put option with company asset as the subject matter. Option pricing requires financial markets to be free of arbitrage opportunities, i.e.
"fair prices". According to the no-arbitrage theorem proposed by , the sufficient and necessary condition for uncertain stock model (3) to be no-arbitrage is that the company value process drift coefficient  is equal to the risk-free rate r .That is, we need a risk-neutral uncertainty Let  be the expected rate of return on t V , then the following equation holds 0 ( ) , Combining equations (10) and (11) ,we can derive the following theorem for expectation of t V under the risk-neutral measure.
Theorem 2. Assume company asset value process is described by (3), then expected rate of return on company asset  is and the expectation of t V under the risk-neutral measure is where ( )   is a standard normal uncertainty distribution (7), and Next we prove equation (12) holds. From Equation (11) , it follows that then we can easily get the expected rate of return of t V   Combining equations (10) and (11), we get The prove is completed.

Company Stock Value and Zero Coupon Bond Value
According to Black-Scholes formula, we get the following theorem 3 and 4 about the value of company stock and company zero coupon bond .
Theorem 3. Assume company asset value process is described by (3), the value of company stock at time 0 is where  is the expected rate of t V with expression(12), ( )   is a standard normal uncertainty distribution (7) (14), and ( )   is a standard normal uncertainty distribution (7).
The prove is completed.
Similarly, we can obtain the following formula for company zero coupon bond.
Theorem 4. Assume company asset value process is described by (3)