Applications of certain non-Gaussian processes in financial mathematics

本研究项目将探讨在更现实的资产价格非高斯Levy模型下的金融工程所产生的一些问题，以及人工智能和大数据在金融中的应用：*1.开发Levy模型下的金融工程的有效数值方法。例如，当资产价格遵循一些更现实的非高斯过程模型时，尝试探索期权价格和最优投资组合的可能的解析或近似分析公式。如果这样的公式不存在或很难找到，那么我将尝试设计有效的蒙特卡罗/准蒙特卡罗(QMC)模拟方法。虽然我已经在更现实的单一从属Levy模型下的期权定价的方差减少(VR)方法上取得了一些显著的结果，但这些VR方法非常依赖于问题。我计划继续研究有效的方差向量机结合QMC方法来处理奇异的多资产期权和具有多个从属因素的资产价格的时变布朗运动(TCBM)模型下的金融工程问题。*2.进一步利用Malliavin微积分在金融中的应用。在过去的二十年里，Malliavin演算在以下方面得到了成功的应用：期权敏感性或希腊字母的估计；美式期权的模拟；最优投资组合的估计等。然而，大多数工作都是针对资产价格的几何布朗运动模型。我在模拟多资产期权的希腊字母和模拟多资产美式期权价格，以及它们在具有单一从属于资产价格的TCBM模型下的希腊字母方面做了一些工作。我计划将我以前的工作扩展到多资产价格服从具有多个从属项的TCBM模型的情况。*3.继续研究资产价格更现实的Levy模型下的投资组合优化(PO)问题。对于PO问题，我已经取得了一些不错的效果。我计划继续在更现实的资产价格模型下研究PO和养老基金投资问题。我将尝试用传统的方法和Malliavin微积分方法来寻找最优投资组合和有效边界等。*4.大多数关于衍生产品定价和投资组合优化的文章都是在假设资产价格服从一定的随机过程的情况下讨论的。这种方法有一些限制：(1)。检验随机过程模型是否能很好地模拟资产价格并非易事。(2)。对于给定的随机模型，特别是对于多资产的情况，估计参数也是困难的。因此，我们计划利用人工智能和大数据分析的优势，尝试一种用于衍生品定价和投资组合优化的“无模型”方法。在这个方向上有很多问题值得探讨。

This research program will explore some problems arising from financial engineering under more realistic non-Gaussian Levy models for asset prices and applications of artificial intelligence and big data in finance:******1. To develop efficient numerical methods for financial engineering under Levy models. For example, try to explore possible analytic or approximate analytic formulas for option prices and optimal portfolios when asset prices follow some more realistic non-Gaussian process models. If such formulas do not exist or are hard to find, then I will try to design efficient Monte Carlo / quasi-Monte Carlo (QMC) simulation methods. Although I have obtained some remarkable results on variance reduction (VR) methods for option pricing under more realistic Levy models with single subordinator, these VR methods are quite problem dependent. I plan to keep working on efficient variance VRs combined with QMC methods for exotic multi-asset options and for financial engineering problems under time-changed Brownian motion (TCBM) models for asset prices with multi-subordinator. ******2. To make further use of Malliavin calculus in finance. In the past twenty years, Malliavin calculus was successfully applied in finance in the following ways: in the estimation of option sensitivities or Greek letters; in the simulation of American style options; in the estimation of optimal portfolios, etc. However, most of the works were done for Geometric Brownian motion models for asset prices. I have done some work on simulations of Greek letters for multi-asset options and simulations of multi-asset American option prices as well as their Greek letters under TCBM models with single subordinator for asset prices. I plan to extend my previous work to the cases where the multi-asset prices follow the TCBM models with multi-subordinators.******3. To continue to work on portfolio optimization (PO) problems under more realistic Levy models for asset prices. I have achieved some good results for PO problems. I plan to continue to work on PO and pension fund investment problems under more realistic asset price models. I will try to find optimal portfolios and efficient frontiers, etc. by both the traditional way and the Malliavin calculus method. ******4. Most papers on derivative pricing and portfolio optimizations are discussed under the assumption that asset prices follow certain stochastic processes. There are some restrictions on this approach: (1). It is not easy to test whether a stochastic process model can model the asset prices very well. (2). It is also hard to estimate parameters for a given stochastic model, especially for the multi-asset case. Thus, we plan to try a "model-free" approach for derivative pricing and portfolio optimizations by using the advantage of artificial intelligence and big data analytics. There are lots of problems in this direction worth to be explored.*****