Study Of The Default Problems On The Several Kinds Of Credit Risk

In the 21st century, with the rapid development of financial markets, investors' risk-bearing capacities have been gradually increasing. However, with the outbreak of the financial crisis, people feel the pinch of risky assets in the market, such as corporate bonds, defaultable stock options, and credit card loans. And all of those risky assets have a common character-credit risk. One of the major forms of credit risk is known as default risk, which means lenders, investors or counterparties suffer a loss when borrowers, bond issuers or counterparties are unwilling or unable to act according to the contracts. But how to characterize the credit risk and how to minimize the default risk are the focuses of individual and institutional investors in the community. In this paper, on the basis of Black-Scholes model and the partial differential equation method, we study the defaulting-model of corporate bonds, defaultable stock options and credit card loans. The equations of model arc solved and the financial meanings of the default risk arc discussed.There are many reasons of default risk, mainly divided into two kinds: one is borrowers' inability to execute the contract, and the other is borrowers' unwillingness to execute the contract, which are usually decided by the characters of the borrowers. The pricing of credit risk and default risk arc usually difficult to estimate. Currently, researchers deal with these problems often by using the reduced form method and the structured form method in the world. In the first chapter, above all, we introduce the emergence, development and recent status of credit risk and the theory of default probability. Consequently, we point out the problem to be solved of this paper.In recent years, an important avenue for corporate finance is through bond issuance. However, corporate bond, as a kind of investment products, also has a credit risk. In the second chapter of this paper, using the method of partial differential equation, we study the pricing problems of the parent corporate bonds when the subsidiary defaults. Under the assumption of stochastic interest rate, the corporate bond of the time-interval has been analyzed by tjie reduced form method. The mathematical model of the parent corporate bonds and the explicit expression are obtained. Finally, the financial meaning of the default correlation is discussed.In the third chapter, on the basis of the jump-differential model, we study the pricing problems of stock options. By the same method of partial differential equation, we study the pricing problems of the stock option when the subsidiary defaults. With the jump-differential assumption, the acceleration of maturity of the option will be considered. The stock option of the time-interval has been analyzed by the structured form method. The mathematical model of the stock option and the expression of the solution are obtained. Sequentially, the pricing model of the stock option is also obtained under the assumption of stochastic interest rate.With the widely use and promotion, credit card has been one of the major businesses that each bank wants to expand in the retailing industry. Therefore, credit card loan, whether effectively repaid or not, has become the major focus in the banking society. Although, credit card is a highly profitable business, it also has the highest risk among all the businesses of credit consumptions. In the fourth chapter, by the method of partial differential equation, we study the default probability of individual's credit card loans, when the proportion of an individual's credit card loans to the total value of personal assets is a constant. Consequently, based on the structured method, the mathematical model and the partial differential equation of the default probability are obtained. Sequentially, by using the difference method, the solution of equation is approximated, and the stability of the solution is also discussed.Lastly, a summary of this paper is made and further research directions are put forward.