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

用于金融衍生品定价的著名的Black-Scholes模型中的一个假设是，资产的对数回报遵循正态分布或高斯分布。虽然在实践中得到了广泛的应用，但该模型存在严重的缺陷，如重尾、波动面/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驱动过程取代潜在资产的驱动过程。征费过程提供了各种各样的分配来满足这些目的。实证研究表明，一类特殊的基于广义双曲分布的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研究。然而，新分布的使用导致缺乏封闭公式和需要探索的更复杂的问题。在数值方面，迫切需要蒙特卡罗/拟蒙特卡罗方法等有效的数值方法。我的研究有两个主要目的。首先是构造新的准随机序列，也称为低差序列。二是研究更现实模型下的金融衍生品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.