Levy processes and (quasi-)Monte Carlo methods in finance

在金融衍生品定价中使用的著名的布莱克-斯科尔斯模型的假设之一是，资产的对数收益服从正态分布或高斯分布。虽然该模型在实际中得到了广泛的应用，但它存在着严重的缺陷，如厚尾、波动率微笑面等。BS框架下的随机波动率模型可以在一定程度上改善结果。另一种完全不同的方法是用更现实的方法来取代基础资产的驱动过程。Levy过程提供了各种各样的分布来服务于这些目的。实证研究表明，基于广义双曲分布的一类特殊Levy过程比Black-Scholes模型更能拟合真实的金融数据。近年来，人们对广义双曲分布下的衍生品定价问题做了大量重要的工作。然而，使用新的分布导致缺乏封闭的公式和更复杂的问题，需要探索。在数值方面，迫切需要有效的数值方法，如Monte Carlo/拟Monte Carlo方法。 我的研究有两个主要目标。第一种方法是构造新的准随机序列，也称为低偏差序列。第二部分是在更现实的模型下研究金融衍生产品的定价和风险管理问题及其数值计算方法。 这项研究应该引起学术界和工业界的高度兴趣。

One of the assumptions in the famous Black-Scholes model used in financial derivative pricing is that the log returns of an asset follow the normal or Gaussian distribution. Although it is widely used in practice, this model has serious drawbacks, such as heavy tail, volatility smile / surface, etc. The stochastic volatility models under the BS framework can improve the results to some extent. A quite different approach is to replace the driving processes for the underlying assets by more realistic ones. Levy processes provide a large variety of distributions to serve such purposes. Empirical studies showed that a special class of Levy process based on the generalized hyperbolic distributions can fit the real financial data much better than the Black-Scholes model. A lot of important work on derivative pricing under the generalized hyperbolic distributions has been done in recent years. However, the use of new distributions results in a lacking of closed formulas and more complex problems that need to be explored. On numerical aspects, efficient numerical methods such as Monte Carlo/quasi-Monte Carlo methods are desperately needed.   There are two main objectives of my  research. The first one is to construct new quasi-random sequences, also known as the low-discrepancy sequences. The second one is to study problems related to financial derivative pricing and risk management under more realistic models and their numerical computation methods.   This research should be of high interest for both academics and industry.