New Integrability Conditions to the Riccati Equation and Application to Finance

The relationship between the Riccati equation and Finance has long been discussed. The Cox-Ingersol-Ross model for the short interest rate combined with the affine class of term-structure models for a zero-coupon price yields to the Riccati equation in which one of the coefficients is a constant. Unfortunately, there is no general method to find an analytic solution for the Riccati equation. Only special cases can be considered. This paper introduces two new methods to solve the Ricatti equation analytically, emphasizing the use of integrating factors, the connections to the Bernouilli equation, and the fact that a solution only de-pends on at most two coefficients of the Riccati equation. One of the methods was applied to the Cox-Ingersoll-Ross (CIR) interest rate model, demonstrating the derivation of bond prices in terms of power series along with several graphs. Those graphs may give a sense of when to acquire the zero-coupon bond with low risk.