SOLUTION OF THE EXTENDED CIR TERM STRUCTURE AND BOND OPTION VALUATION

Abstract

The extended Cox‐Ingersoll‐Ross (ECIR) models of interest rates allow for time‐dependent parameters in the CIR square‐root model. This article presents closed‐form pathwise unique solutions of these unsolved stochastic differential equations (s.d.e.s) in terms of functionals of their driving Brownian motion and parameters. It is shown that quadratics in solution of linear s.d.e.s solve the ECIR model if and only if the dimension of the model is a positive integer and that this solution can be achieved by construction of a pathwise unique generalized Ornstein‐Uhlenbeck process from the ECIR Brownian motion. For real valued dimensions an extension of the time‐change theorem of Dubins and Schwarz (1965) is presented and applied to show that a lognormal process solves the model through a stochastic time change. Pathwise equivalence to a rescaled time‐changed Bessel square process is also established. These novel results are applied to characterize zero‐hitting time and to produce transition density and zero‐hitting conditions for the ECIR spot rate. the CIR term structure is then extended to ECIR under no arbitrage, and its solutions and the transition density are represented under a new ECIR martingale measure. the findings are employed to derive a closed‐form ECIR bond option valuation formula which generalizes that obtained by CIR (1985).