A Highly Nonlinear Multi-Factor Interest Rate Model And Its Numerical Solution

With the development of economic,interest rates are playing an increasingly important role in the market.However,with economic development,financial markets have become increasingly complex.The uncertainty of interest rate changes has also brought many risks.In order to better assess these risks,scholars have proposed a series of interest rate term structure models,such as Vasicek,CIR,Ait-Sahalia,and Fong-Vasicek models etc.This thesis summarizes the importance of interest rate model research,development history and important results achieved during the period.After introducing a series of classic interest rate models,the author further extended the Fong-Vasicek model,using the nonlinear characteristics of the Ait-Sahalia model to describe the fluctuation of spot interest rates,and proposed a highly nonlinear multi-factor spot interest rate model as follows.dVt=(?-1Vt-1-?0+?1Vt-?2Vt2)dt+(?0-?1Vt+?2Vt?3)dWt where Rt represents the interest rate,and Vt represents the volatility.For this model,the following four aspects of research has mainly done:1?Using the local Lipschitz condition and the linear growth condition,the existence and uniqueness of the local positive solution of the volatility equations are proved.Then using the techniques of stopping time proves the existence and uniqueness of the global positive solution of the volatility equation.Finally,by using the constant variability formula,the existence and uniqueness of the global positive solution of the model are obtained.2?Using polynomial boundedness and Gronwall inequality,etc.,it is proved that the first and second moments of the global solution of this model are bounded.3?Numerical solution is obtained by Euler-Maruyama numerical method.It is proved that the Euler-Maruyama numerical solution converges to the analytical solution in probability by using techniques such as stopping time.Then the numerical simulation was carried out using Matlab,which illustrate the feasibility of the model in the interest rate market.4?It is proved that the Euler-Maruyama numerical solution can be used to fit the changes in the pricing of bonds and options,which shows the practical significance of the proposed model in the financial market.