Modeling and Numerical Methods for financial risk based on Stochastic Differential Equations (SDE) -Taking interest rates, volatility and extreme Events as examples

In view of the strong volatility and complexity of modern financial markets, dynamic mathematical models are often needed as technical support in actual financial risk scenarios. Stochastic differential equations (SDE) provide a theoretical framework for risk assessment by describing the continuous evolution and jump behavior of interest rates and volatility. This paper investigates the application of SDE to modeling interest rates and volatility, as well as the stability of its numerical solution methods. The content covers the mathematical basis, explicit and implicit numerical solutions of SDE. In particular, the Euler-Maruyama method is compared with the θ-Scheme in terms of simulation accuracy and stability, and the case data show that the θ-Scheme has more advantages in dealing with nonlinear terms and ensuring long-term stability, as well as the case study of the financial valuation model. In addition, this paper also discusses the future application prospects and improvement directions of SDE, emphasizing that robust and efficient numerical solutions are crucial to capture extreme market dynamics, which can significantly improve the accuracy of risk assessment and help financial institutions cope with extreme market events.