The CIR Model With Jump And Delay

As one of the most important achievements in the financial field and the fundamental result of the pricing theory, the Black & Scholes model has had a profound influence. However, due to some simple assumptions of the BS model such as constant volatility and constant interest rates, which do not satisfy the practice in the financial markets. The fluctuations of interest rates play an important role on the stock price, and interest rate is reversely proportional to the stock price. The stock price is down when the interest rate rises, and the stock price is up when the interest rate falls down. Therefore, this thesis analyzes the CIR model about interest rates.In contract with the continuous model, people find that the movement of the security's value sometimes is affected by some sudden impact and shows the behavior of non-continuity. So, it seems to be a good idea to introduce a jump process into the interest rates model. Although the EMH implies that all information is reflected in the present price of the stock and past stock performance gives no information, some statistical studies of stock prices have indicated the dependence on past date. Then, we consider the time delay, and introduce time delay and jump into the CIR model, considering SDDE with jump.In the chapter 1, we clarify the background and recall the domestic development of interest rates.From chapter 2, firstly, we introduce the relative knowledge of interest rates derivatives and some basic definitions of mathematics, including several important inequalities. Secondly, we analyze two kinds of improved models simply, local volatility model and stochastic volatility model. Last, we introduce a stochastic delay differential equation which is prepared for the next chapter.In chapter 3, since stochastic delay differential equations have no explicit solutions generally, then the numerical methods for approximations have become powerful techniques in the valuation of financial quantities. We will concentrate on the EM method for CIR model with delay and jump and prove the existence of the nonnegative solution and the strong convergence of its EM approximate solutions.