【作者】 王晓娟；

【导师】 殷宝健；

【作者基本信息】 华中科技大学 ， 概率论与数理统计， 2011， 硕士

【摘要】 随机微分方程在金融经济学中的应用越来越普遍,常被用于描述动态的资产价格。但是,应用于金融中的大多数随机微分方程都没有显示解,所以,在评估一些金融模型时,数值解就是一个非常有用的方法,如果我们能够控制其数值解的误差,这个方法就可以更好的描述一些金融数量。作为一个重要的数值方法,蒙特卡罗模拟被广泛用于计算资产的期望值,本文主要考虑的是基于Euler-Maruyama方法的蒙特卡罗模拟。Ait-Sahalia在他的开创性论文中提出了金融中的一个高度非线性的模型,这个金融模型是一个非线性的随机微分方程,它的漂移系数及扩散系数均不满足线性增长条件,并且这个非线性的随机微分方程是没有显示解的。除了通常的外部扰动,金融资产还常常受到突发事件的影响,越来越多的实证研究表明用跳跃-扩散过程的模型去模拟资产价格、利率或随机波动更为合适。所以,以Ait-Sahalia的模型为基础,这篇论文专注于研究此类带跳跃过程的高度非线性模型的Euler-Maruyama方法,并用此方法得到在有限的时间间隔下明确的相对误差界,这些误差界表明当步长趋于零时的强收敛性。为了考虑随机波动性,许多的文章都在基本的价格过程中相应的引入了跳跃过程,本文同样将把此类带跳跃过程的高度非线性模型作为波动性过程考虑,研究此带相关跳的随机波动过程的Euler-Maruyama方法及其误差界的强收敛性。最后由得到的收敛性结果表明Euler-Maruyama方法可以应用于计算一些具体的金融数量,如债券、路径依赖期权等。更多还原

【Abstract】 Stochastic differential equations are more and more applicable in financial economics, and they are often used to describe the dynamics of asset’s price. However, most of stochastic differential equations in finance do not have explicit solutions. Hence, numerical methods is a very useful method, if we can control the error bound of its numerical simulations, then this method can describe some financial quantities. As one of the most important numerical methods, Monte Carlo simulations have received more and more attention on the valuation of financial quantities. In this paper, we mainly consider the Euler-Maruyama method of Monte Carlo simulations.Ait-Sahalia in his seminal paper proposed a highly nonlinear model in finance which is a nonlinear stochastic differential equation, where both the drift and diffusion coefficients do not obey the classical linear growth condition. Of course, this nonlinear stochastic differential equation do not have explicit solution either. In addition to the usual external disturbance, financial assets are also often subject to the impact of unexpected events, more and more empirical evidence shows that the jump-diffusion process is more ideal than the general stochastic differential equations to model an asset’s price, the interest rate or stochastic volatility. Consequently, by the Ait-Sahalia’s model, we in this paper concentrate on investigating the Euler-Maruyama method for a highly nonlinear model with jump and deriving explicitly comparable error bounds over finite time intervals. These error bounds imply strong convergence as the time step tends to zero.For considering the stochastic volatility, many papers introduce jump into the underlying price process. In this paper, we also consider the highly nonlinear model with jump as stochastic volatility, then study the Euler-Maruyama method and the strong convergence of error bounds under stochastic volatility with correlated jumps. Finally, from those convergence results, we show that the Euler-Maruyama method can be applied to compute some financial quantities, for example, bonds, path-dependent option and so on.更多还原