跳扩散市场下的几类期权定价问题研究

发布时间：2018-07-03 09:11

本文选题：美式期权 + 俄式期权　；参考：《中国石油大学（华东）》2013年硕士论文

【摘要】：期权定价一直以来都是金融数学研究的热点和前沿问题，对其研究有着深刻的理论和现实意义。论文以最优停时为主线，运用鞅方法和微分方程自由边界问题方法，分别研究了扩散市场和跳扩散市场下美式期权、俄式期权和博弈期权的定价问题。主要研究工作包括：对于美式期权定价问题，分别讨论了无限平美式期权（或称永久美式期权）定价问题和有限水平美式期权定价问题。对于无限水平美式期权定价问题，首先给出了定价问题的鞅表示模型，据此给出了值函数满足的自由边界问题。应用待定系数法求解得到了值函数和最优停止边界值。对于有限水平的美式期权定价问题，首先给出了定价问题的鞅表示模型，据此给出了值函数满足的抛物型自由边界问题，即自由边界是一条需要求解的移动边界。对于抛物型方程的自由边界问题，应用最优停时理论研究了最优停时边界的正则性质，并把这一理论分析方法用于分析美式期权最优实施边界的正则性分析，得到了较好的结果。从数学上来讲，扩散市场下的美式期权定价问题归结为抛物型方程的自由边界问题，而跳扩散市场下的美式期权定价问题归结为抛物型微分-积分方程的自由边界问题，其本质区别就是跳扩散过程产生的无穷小生成元具有积分算子部分，这对问题的建模和求解都带来本质性的困难。俄式期权和美式期权的主要区别就是收益函数不同，其基本的处理方法类似。在俄式期权定价问题这一部分，论文研究了永久俄式期权的鞅表示模型和值函数的求解，讨论了有限水平俄式期权定价问题的变换简化方法。对于简化的一维问题，，给出了对应的自由边界模型，并研究了值函数的相关性质。对于博弈期权，论文给出了一般的鞅表示模型，对永久博弈期权定价问题进行的详细的求解，得到值函数和停止边界。对于具有障碍的的博弈期权给出了对应微分方程模型，并进行了求解。最后研究了跳扩散市场下的永久博弈期权的模型和求解。
[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.
【学位授予单位】：中国石油大学（华东）
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：O211.6;F830

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