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

衍生品定价、套期保值、投资组合优化和风险管理是现代金融学中的重要问题。Black-Scholes-Merton模型是期权定价的著名模型。然而，利用世界范围内真实的金融数据的实证研究表明，基于某些特殊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模型更精确。在更现实、更复杂的Levy过程模型下，存在着许多新的、具有挑战性的理论和计算问题。有效的数值方法来处理这些问题的Levy模型下的需求很高，因为缺乏封闭的公式在Levy模型。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形的不可或缺的工具。该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微积分等，和有效的数值方法来解决某些问题的投资组合优化，金融衍生品定价，对冲和风险管理下更现实的Levy过程模型。我计划解决的问题包括以下几个方面：**1。投资组合优化问题的最优投资组合和有效边界的推导。** 2.构造有效的蒙特卡罗和拟蒙特卡罗方法，并将其应用于期权定价、投资组合优化等。3.通过有效的蒙特卡罗和准蒙特卡罗方法模拟多资产衍生品。** 4.多资产期权希腊公式的推导和这些希腊的模拟。** 5.美式期权的定价和对冲。**上述问题无论在学术研究还是在金融产业应用上都是新的、重要的，其研究成果对加拿大金融相关产业的发展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.