Early Exercise of Options in Fixed Income Markets: An Eigenfunction Expansion Approach.

摘要

This dissertation develops the eigenfunction expansion approach to evaluating derivatives with embedded early exercise features in the fixed income markets. The first chapter gives an overview of the dissertation. In the second chapter we propose an efficient method to evaluate callable and putable bonds under a wide class of interest rate models, including the popular short rate diffusion models, as well as their time changed versions with jumps. The method is based on the eigenfunction expansion of the pricing operator. Under some technical conditions, the callable and putable bond pricing function is shown to have an eigenfunction expansion in eigenfunctions of the pricing operator with the expansion coefficients determined through a backward recursion. For popular short rate diffusion models, such as CIR and Vasicek, the method is orders of magnitude faster than the alternative approaches in the literature. In the third chapter we consider the SubCIR++ model for applications in interest rate and credit risk modeling. The SubCIR++ model is constructed from the CIR diffusion model by a time-change and by an extension with a deterministic function of time. The SubCIR++ model provides for a flexible model in which the interest rate or the intensity process has jumps and matches the initial term structure of interest rates or the initial market implied hazard function in the interest rate or credit risk model, respectively. The price of Bermudan swaptions in the SubCIR++ interest rate model and Bermudan credit default swap options in the SubCIR++ default intensity model can be evaluated efficiently with the eigenfunction expansion method. In the fourth chapter we propose the convertible bond model based on the jump to default extended CEV (JDCEV) process of Carr and Linetsky (2006). We present the eigenfunction expansion approach to price convertible bonds which have conversion, call, and put options that can be exercised early. We also study the impact of stochastic interest rates in pricing convertible bonds by directly comparing the valuation obtained from a model with stochastic interest rates to that obtained from a model with the corresponding deterministic interest rates.