导　　师： 尹居良

学科专业： 070103

授予学位： 硕士

作　　者： ;

机构地区： 暨南大学

摘　　要： 本文主要研究带跳随机波动率模型的数值解的性质、数值模拟方法以及在欧氏期权和障碍期权的价格计算上的应用. 第一章介绍研究背景和本文的主要工作. 第二章叙述跳扩散随机微分方程的Ito公式,Euler-Maruyama近似方法,以及一些不等式. 第三章我们给出带跳随机波动率模型的等价形式并且得出了模型解的非负性,给出并证明了模型的矩的表达式,证明了数值解的强收敛性.应当指出,对于波动率所服从的均值回复平方根过程,Higham和Mao给出了数值解的强收敛性,本文将以引理的形式给出这些强收敛性. 第四章分别证明了欧氏看跌期权的期望收益的数值近似和向上敲出看涨期权的期望收益的数值近似是收敛到真实的期望收益的.应当指出,我们的证明思想是受到Higham和Mao的启发,但是他们的资产价格是连续的随机过程,而我们是带跳的,这使得我们又要寻求新的解决方法. 第五章首先简单介绍了Monte Carlo方法,然后对期权的价格进行了数值模拟. In this dissertation, we mainly investigate the numerical solution property of the stochastic volatility with jumps /(SVJ/) model, numerical simulation and some applica-tion on European and barrier options pricing. In the first chapter, we outline the background and the major work. In the second chapter, we state the Ito formula of the stochastic differential equa-tion /(SDE/) with jumps, the method of Euler-Maruyama approximations, and some inequalities. In the third chapter, we give equivalent form of the SVJ model and obtain the nonnegative solution, we then give and prove the moments of the model. We also prove the strong convergence of the numerical solution. It should be pointed out that for the mean-reverting square root process, Higham and Mao give the proof of the strong convergence of the solution. In this dissertation, these strong convergence will be given in the form of lemma. In the fourth chapter, we prove that the numerical approximation of the pay-off for the European put and up-and-out barrier options are convergent. It should be pointed out that our proof idea is inspired by Higham and Mao, but their stock price process is a continuous stochastic process, and ours is with jumps, so we need seeking new methods to solve it. In the fifth chapter, we first introduced the Monte Carlo method, then price of option was simulated.

关 键 词： 随机波动率 跳扩散随机微分方程 均值回复平方根过程 数值方法 强收敛 障碍期权 模拟

领　　域： [经济管理] [经济管理]