Essays in Financial Engineering

This thesis consists of three essays in financial engineering. In particular we study problems in option pricing, stochastic control and risk management. In the first essay, we develop an accurate and efficient pricing approach for options on leveraged ETFs (LETFs). Our approach allows us to price these options quickly and in a manner that is consistent with the underlying ETF price dynamics. The numerical results also demonstrate that LETF option prices have model-dependency particularly in high-volatility environments. In the second essay, we extend a linear programming (LP) technique for approximately solving high-dimensional control problems in a diffusion setting. The original LP technique applies to finite horizon problems with an exponentially-distributed horizon, T. We extend the approach to fixed horizon problems. We then apply these techniques to dynamic portfolio optimization problems and evaluate their performance using convex duality methods. The numerical results suggest that the LP approach is a very promising one for tackling high-dimensional control problems. In the final essay, we propose a factor model-based approach for performing scenario analysis in a risk management context. We argue that our approach addresses some important drawbacks to a standard scenario analysis and, in a preliminary numerical investigation with option portfolios, we show that it produces superior results as well.