Applications of certain non-Gaussian processes in financial mathematics

本研究计划将探讨在更现实的非高斯Levy模型下资产价格和人工智能和大数据在金融中的应用的金融工程中出现的一些问题：**1。发展Levy模型下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解析或近似解析公式。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Monte Carlo / quasi-Monte Carlo（QMC）模拟方法。虽然我已经得到了一些显着的结果，方差减少（VR）方法的期权定价更现实的Levy模型与单一从属，这些VR方法是相当依赖于问题。我计划继续研究有效的方差VR与QMC方法相结合，用于奇异的多资产期权和具有多从属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演算在金融学中的应用。在过去的20年里，Malliavin演算在金融领域的应用主要表现在以下几个方面：期权灵敏度或希腊字母的估计;美式期权的模拟;最优投资组合的估计等。然而，大多数的工作都是针对资产价格的几何布朗运动模型进行的。我做了一些工作，模拟希腊字母的多资产期权和模拟多资产的美式期权价格以及他们的希腊字母在TCBM模型与单一从属的资产价格。我计划将我以前的工作扩展到多资产价格遵循具有多个从属者的TCBM模型的情况。3.继续研究在更现实的资产价格Levy模型下的投资组合优化问题。我在PO问题上取得了一些不错的成绩。我计划在更现实的资产价格模型下继续研究PO和养老基金投资问题。我将尝试用传统的方法和Malliavin演算方法来寻找最优投资组合和有效边界等。4.大多数关于衍生品定价和投资组合优化的论文都是在资产价格遵循一定随机过程的假设下讨论的。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不容易。（二）.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“无模型”方法进行衍生品定价和投资组合优化。在这方面还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.*****