Closed-form formulas for conditional moments of generalized cox-ingersoll-ross processes

Abstract

Cox-Ingersoll-Ross (CIR) process, introduced in 1985, is a one factor model used to describe the evolution of interest rate and pricing the financial derivatives. It was later extended to have time-dependent parameters called the extended CIR (ECIR) process, which is more widely studied and used in a variety of applications. The generalized versions of CIR process are also studied and investigated for more applications in finance. However, most of these applications rely on the knowledge and properties of conditional expectations and moments, which most of them are not yet fully developed into closed form. In this work, we propose closed-form formulas derived from applying the Feynman--Kac representation for the two generalized CIR processes: the nonlinear drift constant elasticity of variance and the Pearson diffusions which are developed from the ECIR process, as well as further study of their properties such as variance and conditional mixed moments. In addition, we also extending the result of ECIR process by applying it for valuation of interest rate swaps. The formulas derived in this work are numerically verified and validated based on the Monte-Carlo simulations.