本文選題：美式期權(quán) + 俄式期權(quán)��； 參考：《中國石油大學（華東）》2013年碩士論文

【摘要】：期權(quán)定價一直以來都是金融數(shù)學研究的熱點和前沿問題，對其研究有著深刻的理論和現(xiàn)實意義。論文以最優(yōu)停時為主線，運用鞅方法和微分方程自由邊界問題方法，分別研究了擴散市場和跳擴散市場下美式期權(quán)、俄式期權(quán)和博弈期權(quán)的定價問題。主要研究工作包括： 對于美式期權(quán)定價問題，分別討論了無限平美式期權(quán)（或稱永久美式期權(quán)）定價問題和有限水平美式期權(quán)定價問題。對于無限水平美式期權(quán)定價問題，首先給出了定價問題的鞅表示模型，據(jù)此給出了值函數(shù)滿足的自由邊界問題。應(yīng)用待定系數(shù)法求解得到了值函數(shù)和最優(yōu)停止邊界值。對于有限水平的美式期權(quán)定價問題，首先給出了定價問題的鞅表示模型，據(jù)此給出了值函數(shù)滿足的拋物型自由邊界問題，即自由邊界是一條需要求解的移動邊界。對于拋物型方程的自由邊界問題，應(yīng)用最優(yōu)停時理論研究了最優(yōu)停時邊界的正則性質(zhì)，并把這一理論分析方法用于分析美式期權(quán)最優(yōu)實施邊界的正則性分析，得到了較好的結(jié)果。從數(shù)學上來講，擴散市場下的美式期權(quán)定價問題歸結(jié)為拋物型方程的自由邊界問題，而跳擴散市場下的美式期權(quán)定價問題歸結(jié)為拋物型微分-積分方程的自由邊界問題，其本質(zhì)區(qū)別就是跳擴散過程產(chǎn)生的無窮小生成元具有積分算子部分，這對問題的建模和求解都帶來本質(zhì)性的困難。 俄式期權(quán)和美式期權(quán)的主要區(qū)別就是收益函數(shù)不同，其基本的處理方法類似。在俄式期權(quán)定價問題這一部分，論文研究了永久俄式期權(quán)的鞅表示模型和值函數(shù)的求解，討論了有限水平俄式期權(quán)定價問題的變換簡化方法。對于簡化的一維問題，，給出了對應(yīng)的自由邊界模型，并研究了值函數(shù)的相關(guān)性質(zhì)。 對于博弈期權(quán)，論文給出了一般的鞅表示模型，對永久博弈期權(quán)定價問題進行的詳細的求解，得到值函數(shù)和停止邊界。對于具有障礙的的博弈期權(quán)給出了對應(yīng)微分方程模型，并進行了求解。最后研究了跳擴散市場下的永久博弈期權(quán)的模型和求解。
[Abstract]:Option pricing has always been a hot topic and frontier problem in financial mathematics, which has profound theoretical and practical significance. In this paper, the pricing problems of American option, Russian option and game option in diffusion market and jump diffusion market are studied by means of martingale method and differential equation free boundary problem method. The main research work includes: for the pricing of American option, we discuss the pricing problem of infinite equal American option (or permanent American option) and the pricing problem of finite level American option respectively. For the pricing problem of infinite level American option, the martingale representation model of pricing problem is given, and the free boundary problem of value function satisfying is given. The value function and the optimal stop boundary value are obtained by using the undetermined coefficient method. For the American option pricing problem of finite level, the martingale representation model of the pricing problem is first given, and then the parabolic free boundary problem satisfying the value function is given, that is, the free boundary is a moving boundary that needs to be solved. For the free boundary problem of parabolic equations, the canonical properties of the optimal stopping time boundary are studied by using the optimal stopping time theory, and the method is applied to the analysis of the regularity of the optimal executive boundary of American option. Good results have been obtained. Mathematically speaking, the American option pricing problem in the diffusion market is reduced to the free boundary problem of parabolic equation, while the American option pricing problem in the jump diffusion market is reduced to the free boundary problem of the parabolic differential-integral equation. The essential difference is that the infinitesimal generator produced by the jump diffusion process has integral operator part, which brings essential difficulties to the modeling and solving of the problem. The main difference between Russian option and American option is that the income function is different. In the part of the Russian option pricing problem, the martingale representation model and the solution of the value function of the permanent Russian option are studied, and the transformation simplification method of the finite level Russian option pricing problem is discussed. For the simplified one-dimensional problem, the corresponding free boundary model is given, and the related properties of the value function are studied. For game options, the paper gives a general martingale representation model, and gives a detailed solution to the option pricing problem of permanent game, and obtains the value function and stop boundary. The corresponding differential equation model for the options with obstacles is given and solved. Finally, the model and solution of permanent game options in jump diffusion market are studied.
【學位授予單位】：中國石油大學（華東）
【學位級別】：碩士
【學位授予年份】：2013
【分類號】：O211.6;F830

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本文編號：2093139