Some Problems on Derivative Pricing and Portfolio Optimization underMore Realistic Asset Price Models

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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.