金融中的Levy模型及其仿真

近20年来，金融中Levy模型与蒙特卡洛仿真技术日益受到重视. 在连续时间过程的金融建模中带跳跃的Levy模型相比于连续轨道的布朗运动模型能很好地刻画市场的跳跃，更好地拟合金融数据的统计特征，更准确地对衍生品定价. 但是，相较于经典的Black-Scholes模型，用Levy模型对衍生品定价以及求解对冲策略的计算复杂度大大增加. 蒙特卡洛仿真成为Levy模型计算中最重要的方法之一. 首先详细地介绍了Levy模型引入的背景，并引出仿真方法在其中重要的应用价值. 最后，简要地给出了Levy过程仿真及其梯度估计的基本方法.

Simulation of Levy-driven models and its application in finance

Levy processes have been widely used to model financial assets since the 1990s. The reason of their widespread applications is mainly due to the fact that they provide more realistic models that capture discontinuous behaviors and stylized empirical statistical characteristics of time series data in economy and finance. However, when applied to derivative pricing, very few analytical results are available except for European options. Therefore, one usually has to resort to numerical methods such as Monte Carlo simulation method. The simulation method is so attractive that it is very general and can also handle high dimensional problems very well. In this short survey paper, we first provide an overview on Levy processes. We then introduce Monte Carlo simulation method for Levy processes. Finally, we discuss the two main simulation based gradient estimation methods: perturbation analysis and likelihood ratio method.