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

【摘要】：期權(quán)定價(jià)一直以來(lái)都是金融數(shù)學(xué)研究的熱點(diǎn)和前沿問(wèn)題，對(duì)其研究有著深刻的理論和現(xiàn)實(shí)意義。論文以最優(yōu)停時(shí)為主線，運(yùn)用鞅方法和微分方程自由邊界問(wèn)題方法，分別研究了擴(kuò)散市場(chǎng)和跳擴(kuò)散市場(chǎng)下美式期權(quán)、俄式期權(quán)和博弈期權(quán)的定價(jià)問(wèn)題。主要研究工作包括： 對(duì)于美式期權(quán)定價(jià)問(wèn)題，分別討論了無(wú)限平美式期權(quán)（或稱永久美式期權(quán)）定價(jià)問(wèn)題和有限水平美式期權(quán)定價(jià)問(wèn)題。對(duì)于無(wú)限水平美式期權(quán)定價(jià)問(wèn)題，首先給出了定價(jià)問(wèn)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(wèn)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(duì)于有限水平的美式期權(quán)定價(jià)問(wèn)題，首先給出了定價(jià)問(wèn)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(wèn)題，即自由邊界是一條需要求解的移動(dòng)邊界。對(duì)于拋物型方程的自由邊界問(wèn)題，應(yīng)用最優(yōu)停時(shí)理論研究了最優(yōu)停時(shí)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(shí)施邊界的正則性分析，得到了較好的結(jié)果。從數(shù)學(xué)上來(lái)講，擴(kuò)散市場(chǎng)下的美式期權(quán)定價(jià)問(wèn)題歸結(jié)為拋物型方程的自由邊界問(wèn)題，而跳擴(kuò)散市場(chǎng)下的美式期權(quán)定價(jià)問(wèn)題歸結(jié)為拋物型微分-積分方程的自由邊界問(wèn)題，其本質(zhì)區(qū)別就是跳擴(kuò)散過(guò)程產(chǎn)生的無(wú)窮小生成元具有積分算子部分，這對(duì)問(wèn)題的建模和求解都帶來(lái)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(jià)問(wèn)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(jià)問(wèn)題的變換簡(jiǎn)化方法。對(duì)于簡(jiǎn)化的一維問(wèn)題，，給出了對(duì)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(duì)于博弈期權(quán)，論文給出了一般的鞅表示模型，對(duì)永久博弈期權(quán)定價(jià)問(wèn)題進(jìn)行的詳細(xì)的求解，得到值函數(shù)和停止邊界。對(duì)于具有障礙的的博弈期權(quán)給出了對(duì)應(yīng)微分方程模型，并進(jìn)行了求解。最后研究了跳擴(kuò)散市場(chǎng)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(xué)位授予單位】：中國(guó)石油大學(xué)（華東）
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2013
【分類號(hào)】：O211.6;F830

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