Numerical Solutions Of Highly Sensitive Mean-reverting Stochastic Differential Equations With Markovian Switching And Applications To Finance

In the past thirty years, with the rapid development of financial mathematics, stochastic differential equation theory is widely applied to interest rate model, optional pricing, risky investment and other financial fields. In 1973, Fischer Black and Myron Scholes used no-arbitrage pricing theory and stochastic differential equations theory to get the famous Black- Scholes European options pricing formula, which lays a key foundation to the financial engineering and begins an era of financial quantitative analysis.A highly sensitive mean-reverting stochastic differential equation is a kind of widely used financial mathematical model. In this article, a Markovian switching mechanism is added into this model, and resulting in general financial model(RS CKLS for short).The purpose of the article is to study numerical solutions of RS CKLS and apply RS CKLS to study financial issues.More specifically, Firstly, we use Lyapunov functions to prove that there exists a unique global positive solution for a RS CKLS under local Lipschitz conditions. Then it shows four types of numerical solution to a RS CKLS converges to its real solution in the sense of probability. Finally a RS CKLS is applied to conduct empirical analysis of 7-day Shanghai interbank offered rates from January 2007 to February 2014. Maximum likelihood method is used to estimate model parameters, and hypothesis test results show that the RS CKLS has a better performance in fitting 7-day Shanghai interbank offered rates compared with a CKLS. Compared with effectiveness of four numerical solutions, Backward Euler numerical solution and stochastic theta numerical solution are better than Euler numerical solution and Split-Step Backward Euler numerical solution.