Applications of the eigenfunction expansion method in interest rate modeling

The Eigenfunction Expansion method (EEM) is a powerful analytical tool to derive analytical solutions to derivatives pricing problems in finance. The methodology is based on unbundling all contingent claims (derivative securities) written on an underlying state variable assumed to follow a Markov process into portfolios of primitive securities called eigensecurities . Eigensecurities (eigenfunctions) are the eigenvectors of the pricing (present value) operator mapping future cash flows into present values. Once the analytical and computational work of determining eigenvalues eigenfunctions is done, the pricing of the contingent claim is immediate by the linearity of the pricing operator and eigenvector property of eigensecurities. The result is the eigenfunction expansion of the value function for the derivative security.; Because this method provides fast convergence for instruments with long maturities, fixed income markets and interest rate derivatives is a natural application area for this method. This dissertation applies the general eigenfunction expansion method for the purpose of pricing and hedging interest rate instruments when the underlying state variable follows a scalar diffusion process and, in particular, develops two applications of the EEM interest rate modeling. The first application is the development an interest rate model to describe the dynamics of interest rates in low interest rate regimes, with applications to Japanese interest rates and the pricing of Japanese Government Bonds (JGB). The second application is the development of the intensity-based prepayment approach for modeling residential mortgages.