Research On Credit Migration Model With Asymmetric Boundary And Its Related Problems

In this dissertation,we study the problem of credit rating migration with fixed asymmetric boundaries under the structural framework by PDE methods.An empirical study on the structural credit migration model is also conducted.The main contents are as follows:1.Establish a fixed asymmetric threshold model of credit migration,and study related mathematical problems.First,a buffer zone is introduced in the asymmetric model by setting different boundaries for upgrades and downgrades,which avoids the high frequent changes of credit ratings that might appear in the single-threshold model.Mathematically,the model turns to a system of partial differential equations coupled on asymmetric boundaries with an overlapping area.Using PDE techniques,the existence of the solution is established through the monotonic iteration method,and the uniqueness of the solution is proved through the maximum principle by contradiction.Then,we introduce a risk discount rate on the asymmetric boundary of the model.When the discount factor is within a certain range,we find an asymptotic traveling wave solution with a buffer,which has a closed form.By establishing two sets of sub and super solution sequences,we prove that the solution of the model converges to the traveling wave solution along a certain direction as t goes to infinity.Further,we consider the relationship between the asymmetric-threshold model and the single-threshold model.We prove that when one asymmetric boundary approaches the other,the solution of the asymmetric model converges to the solution of the corresponding single model.Finally,we give the explicit difference scheme of the partial differential equation system,calibrate the parameters using the real issuer example,and analyze the influence of each parameter on the solution.2.Extend the asymmetric model to different situations.First,we consider an asymmetric model for multiple credit ratings in order to be more in line with market reality.Taking the three-rating model as an example,we discuss the existence and uniqueness of solution and indicate that the corresponding conclusions holds for the general multi-rating model.Multi-rating models have diverse structures,as a result of the different asymmetric boundary settings.Second,an asymmetric model with default boundary is established,which allows bonds to default before maturity.The model has an asymptotic steady-state solution with an explicit expression.A parametric analysis was performed using numerical methods and we find that longer-term zero-coupon bonds are priced lower at higher grades.3.Conduct an empirical analysis on the structural credit migration model and obtain preliminary evidence for the model’s validity.From the perspective of practical application,a steady-state free boundary model is established,with considering the impact of tax and bankruptcy losses on the capital structure of the issuer.Using data from the U.S.corporate bond market,we estimate the credit rating migration boundary based on this model.The empirical results show that the structural model can accurately measure credit risk and monitor credit migration in a way.The main innovations of this dissertation include:1.The idea of introducing buffer areas by setting fixed asymmetric boundaries is proposed,which solves the problem of high frequent changes of credit ratings in the original model.The modeling idea of the asymmetric threshold can also be applied to other fields,such as describing the hysteresis phenomenon in physics,automatic control in engineering,etc.2.Mathematically,we obtain a new partial differential equation problem coupled on two inner boundaries with an overlapping region.The well-posedness of the problem,the traveling wave solution and the asymptotic relationship between this problem and the previous problem are analyzed in detail.Our work may lay a theoretical foundation of this kind of problems.3.The asymmetric model is extended to a variety of situations,making it suitable for different financial scenarios.4.It is the first time that the structural model of credit rating migration is empirically tested from the perspective of application.Not only the preliminary evidence of the model’s validity is provided,but also the method and specific process of model implementation are shown.