Applications of certain non-Gaussian processes in financial mathematics

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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.