Some Problems on Derivative Pricing and Portfolio Optimization under*More Realistic Asset Price Models

衍生产品定价、套期保值、投资组合优化和风险管理是现代金融中的重要问题。布莱克-斯科尔斯-默顿模型以期权定价而闻名。然而，使用全球真实金融数据的实证研究表明，基于一些特殊的Levy过程的模型，称为从属布朗运动或时变布朗运动，比Black Scholes-Merton模型更准确。在更现实和更复杂的Levy过程模型下，有许多新的和具有挑战性的理论和计算问题。由于在Levy模型下缺乏闭合公式，因此在Levy模型下处理这些问题的有效数值方法的需求量很大。蒙特卡罗(MC)/准蒙特卡罗(QMC)模拟方法已成为金融工程中处理高维情况不可缺少的工具。**本研究计划将应用先进的数学工具，如随机分析、随机最优控制、(随机)微分方程、Malliavin演算等，以及高效的数值方法，在更现实的Levy过程模型下解决投资组合优化、金融衍生品定价、套期保值和风险管理等问题。我计划解决的问题包括：**1.最优投资组合的推导和投资组合优化问题的有效前沿。**2.构造有效的蒙特卡罗和准蒙特卡罗方法，并将其应用于期权定价、投资组合优化等。**3.用有效的蒙特卡罗和准蒙特卡罗方法模拟多资产衍生品。**4.多资产期权希腊人公式的推导和这些希腊人的模拟。**5.美式期权的定价和对冲。**上述问题对于学术研究和金融工业应用都是新的和重要的。研究结果对加拿大金融相关行业具有一定的借鉴意义。

Derivative pricing, hedging, portfolio optimization and risk management are important problems in modern finance. The Black-Scholes-Merton's model is well known for option pricing. However, empirical studies using worldwide real financial data show that models based on some special Levy processes, called subordinated Brownian motions or time-changed Brownian motions, are more accurate than the Black Scholes-Merton's model. There are many new and challenging problems, both theoretical and computational, under the more realistic and more complex Levy process models. Efficient numerical methods dealing with these problems under Levy models are in high demand since lack of closed formulas under Levy models. The Monte Carlo (MC)/quasi-Monte Carlo (QMC) simulation methods have become indispensable tools in financial engineering in handling high dimensional situations.**This research program will apply advanced mathematical tools, such as stochastic analysis, stochastic optimal control, (stochastic) differential equations, Malliavin calculus, etc., and efficient numerical methods to solve certain problems in portfolio optimization, financial derivative pricing, hedging and risk management under the more realistic Levy process models. The problems I plan to tackle include the following:**1. Derivation of optimal portfolios and efficient frontiers for portfolio optimization problems.**2. Construction of efficient Monte Carlo and quasi-Monte Carlo methods with applications to options pricing, portfolio optimization, etc.**3. Simulation of multi-asset derivatives by efficient Monte Carlo and quasi-Monte Carlo methods.**4. Derivation of formulas for Multi-asset option Greeks and simulation of these Greeks.**5. Pricing and hedging American style options.**The above problems are new and important both for academic research and financial industrial applications, and the results can be beneficial to the Canadian financial relevant industries.