| This dissertation investigates two topics in financial engineering: credit risk and American option pricing.; In the part of credit risk, we propose a two-sided jump diffusion model for credit risk. Standard models based on geometric Brownian motion cannot provide a satisfactory answer to empirical observation, such as a variety of shapes of credit yield spreads, very low leverage levels of high tech firms, and related impact on equity options. Our model leads to more realistic credit spread curves, provides a link between credit risk and equity options, and yields an explanation on why high tech firms tend to have very little debt.; In the part of American option pricing, we study and compare two dual formulations in the Monte-Carlo simulation solution for American option pricing. The conventional way to do this problem relies Dynamic Programming in simulation, which is time consuming because of the considerable computation effort in evaluation of conditional expectations. An alternative method is to use duality in simulation. Two main duals, additive and multiplicative, are proposed in the literature. An immediate question is which dual is better, and whether is beneficial to combine two duals. In this part of the dissertation, we point out that repeated use of two duals will lead to an iterative algorithm that improves the estimation of true value functions of options progressively. Meanwhile we get upper bounds of biases for both duals, and find that multiplicative dual could incur very huge variance in some cases compared to the additive method. Finally both duals lead to the same linear programming duality, if we discretize the original problem. |
