A Novel Analytical Formula for the Discounted Moments of the ECIR Process and Interest Rate Swaps Pricing

This paper presents an explicit formula of conditional expectation for a product of polynomial functions and the discounted characteristic function based on the Cox–Ingersoll–Ross (CIR) process. We also propose an analytical formula as well as a very efficient and accurate approach, based on the finite integration method with shifted Chebyshev polynomial, to evaluate this expectation under the Extended CIR (ECIR) process. The formulas are derived by solving the equivalent partial differential equations obtained by utilizing the Feynman–Kac representation. In addition, we extend our results to derive an analytical formula of conditional expectation of a product of mixed polynomial functions and the discounted characteristic function. The accuracy and efficiency of the proposed scheme are also numerically shown for various modeling parameters by comparing them with those obtained from Monte Carlo simulations. In addition, to illustrate applications of the obtained formulas in finance, analytical pricing formulas for arrears and vanilla interest rate swaps under the ECIR process are derived. The pricing formulas become explicit under the CIR process. Finally, the fractional ECIR process is also studied as an extended case of our main results.