本文選題：美式期權(quán) + 體制轉(zhuǎn)換��； 參考：《閩南師范大學》2017年碩士論文

【摘要】：期權(quán)定價理論作為金融領域中最重要的發(fā)展之一,是目前金融數(shù)學研究的重點問題。期權(quán)價格作為影響買賣雙方收益的直接因素,成為期權(quán)定價理論的重點研究對象。期權(quán)分為歐式期權(quán)和美式期權(quán),與歐式期權(quán)相比,美式期權(quán)要比歐式期權(quán)復雜很多。其中,美式期權(quán)的研究也因此成為期權(quán)定價理論的核心問題。近年來金融的很多領域應用體制轉(zhuǎn)換模型,并且在對相關的金融數(shù)學研究時應用體制轉(zhuǎn)換模型取得了更好的結(jié)果。在前人研究的基礎上,本文著重討論求解基于體制轉(zhuǎn)換模型的美式期權(quán)定價問題的一種高精度緊致有限差分格式。首先,本文考慮基于Black-Scholes模型的美式期權(quán)定價問題,先對Black-Scholes方程做代數(shù)變換,消除方程中對空間上的一階導數(shù),得到美式期權(quán)定價問題的數(shù)學模型,進而提出高精度的緊致有限差分格式。然后,應用離散能量法分析了該格式的穩(wěn)定性和收斂性。最后,給出兩個數(shù)值算例,其數(shù)值結(jié)果證實了理論分析及該格式的應用可行性。其次,本文考慮基于體制轉(zhuǎn)換模型的美式期權(quán)定價問題,該問題滿足由M個自由邊界值問題組成的方程組,這使得問題很難得到解決。我們先對最初體制轉(zhuǎn)換下美式期權(quán)滿足的方程組作代數(shù)變換,消除方程組中對空間上的一階導數(shù),得到體制轉(zhuǎn)換模型下美式期權(quán)定價問題的數(shù)學模型,進而提出高精度的緊致有限差分格式。然后,應用離散能量法分析了該格式的穩(wěn)定性和收斂性。最后,給出了幾個數(shù)值算例,其數(shù)值結(jié)果證實了理論分析及該格式的應用可行性。
[Abstract]:As one of the most important developments in the field of finance, option pricing theory is one of the most important problems in financial mathematics.Option price, as a direct factor affecting the income of buyers and sellers, has become an important research object of option pricing theory.Options are divided into European options and American options. Compared with European options, American options are much more complex than European options.Among them, the research of American option becomes the core problem of option pricing theory.In recent years, the system transformation model has been applied in many fields of finance, and better results have been obtained in the study of financial mathematics.On the basis of previous studies, this paper focuses on a compact finite-difference scheme with high precision for solving American option pricing problems based on system transformation model.First of all, this paper considers the American option pricing problem based on Black-Scholes model. Firstly, the Black-Scholes equation is algebraic transformed to eliminate the first derivative in the space of the equation, and the mathematical model of American option pricing problem is obtained.Then a compact finite difference scheme with high accuracy is proposed.Then, the stability and convergence of the scheme are analyzed by using the discrete energy method.Finally, two numerical examples are given. The numerical results verify the theoretical analysis and the feasibility of the scheme.Secondly, this paper considers the American option pricing problem based on the system transformation model, which satisfies the equations composed of M free boundary value problems, which makes it difficult to solve the problem.First, we make algebraic transformation of the equations satisfied by American options under the initial system transformation, eliminate the first derivative of the equations on the space, and obtain the mathematical model of the pricing problem of American options under the system transformation model.Then a compact finite difference scheme with high accuracy is proposed.Then, the stability and convergence of the scheme are analyzed by using the discrete energy method.Finally, several numerical examples are given. The numerical results confirm the theoretical analysis and the feasibility of the scheme.
【學位授予單位】：閩南師范大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.3;F830.93

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本文編號：1735487