本文選題：即期利率 + 隨機(jī)微分方程��； 參考：《華中科技大學(xué)》2012年碩士論文

【摘要】：本文對即期利率的基本含義、研究的重要性、發(fā)展歷程及其取得的重要成果進(jìn)行了概述，詳細(xì)介紹了利用隨機(jī)微分方程建立的利率模型。其中的眾多模型是線性的形式，具有良好的性質(zhì)，滿足線性增長條件和(局部)Lipschitz條件，，方程滿足解的存在唯一性條件。雖然有些方程并非滿足線性增長條件或(局部)Lipschitz條件，如CIR模型，但是其非負(fù)解的存在唯一性等解析性質(zhì)也得到了證明。Wu還證明了均值回歸-過程的非負(fù)解的存在唯一性、有界性、EM數(shù)值解的收斂性。另外，也有非線性的形式，比較有代表性的就是Ait-Sahalia模型，方程的漂移系數(shù)與擴(kuò)散系數(shù)均為非線性，該模型的有關(guān)解析性質(zhì)至今也得到了較好的證明。 在CIR模型、均值回歸-過程的基礎(chǔ)上，本文建立了一個更一般化的高度非線性隨機(jī)微分方程形式的即期利率模型：顯然，這個方程是包含CIR模型、均值回歸-γ過程的。在此模型的基礎(chǔ)上，本文主要做了以下兩方面的工作： 第一，選取美國聯(lián)邦儲備系統(tǒng)公布的2002年1月2日至2012年2月17日的短期國債收益率估計模型中的參數(shù)，估計結(jié)果顯示參數(shù)α㧐1，β㧐1。 第二，在上述估計的參數(shù)結(jié)果下，證明了方程的非負(fù)解的存在唯一性、解的隨機(jī)有界性，還特別指出了方程的EM近似解在時間步長充分小時是依概率1收斂于方程的真解，收斂的結(jié)果還說明了基于EM方法的蒙特卡洛模擬可以用來計算期權(quán)等金融產(chǎn)品的預(yù)期收益問題。
[Abstract]:In this paper, the basic meaning of spot interest rate, the importance of research, the development of spot interest rate and its important achievements are summarized, and the interest rate model based on stochastic differential equation is introduced in detail. Many of the models are linear, have good properties, satisfy the linear growth condition and (local Lipschitz condition), the equation satisfies the existence and uniqueness conditions of solution. Although some equations do not satisfy the linear growth condition or (local Lipschitz condition, such as CIR model), the existence and uniqueness of the nonnegative solution are also proved. Wu also proves the existence and uniqueness of the non-negative solution of the mean regression-process. Boundedness and convergence of EM numerical solutions. In addition, there is also a nonlinear form. The Ait-Sahalia model is more representative. The drift coefficient and diffusion coefficient of the equation are both nonlinear. The analytical properties of the model have been well proved. On the basis of mean regression-process, a more general model of spot interest rate in the form of highly nonlinear stochastic differential equation is established: obviously, this equation includes CIR model, mean regression-緯 process. On the basis of this model, this paper mainly does the following two aspects of work: first, select the parameters of the short-term Treasury bond yield estimation model published by the United States Federal Reserve system from January 2, 2002 to February 17, 2012. The estimation results show that the parameters 偽 1, 尾 1. Secondly, the existence and uniqueness of the nonnegative solution of the equation and the stochastic boundedness of the solution are proved under the above estimated parameter results. In particular, it is pointed out that the EM approximate solution of the equation converges to the true solution of the equation according to probability 1 when the sufficient time step is small. The convergence results also show that Monte Carlo simulation based on EM method can be used to calculate the expected returns of financial products such as options.
【學(xué)位授予單位】：華中科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2012
【分類號】：F224;F820

【參考文獻(xiàn)】

相關(guān)碩士學(xué)位論文 前1條

1 張娟;隨機(jī)利率模型下的期權(quán)定價研究[D];國防科學(xué)技術(shù)大學(xué);2006年

本文編號：1987254