A Bond Option Pricing Formula in the Extended CIR Model, with an Application to Stochastic Volatility

摘要

We provide a complete representation of the interest rate in the extended CIRmodel. Since it was proved in Maghsoodi (1996) that the representation of theCIR process as a sum of squares of independent Ornstein-Uhlenbeck processes ispossible only when the dimension of the interest rate process is integer, weuse a slightly different representation, valid when the dimension is notinteger. Our representation consists in an infinite sum of squares of basicprocesses. Each basic process can be described as an Ornstein-Uhlenbeck processwith jumps at fixed times. In this case, the price of a bond option resemblesthe Black-Scholes formula, where the normal distribution is replaced by thegeneralized chi-square distribution. The formula is in closed form, up to thesolution of a Riccati equation for the bond price of the option. We thenprovide a generalization of our representation to an extended CIR model withstochastic volatility. We present a closed form approximation of the price of abond option, valid when the expiration of the option is small and the speed ofmean-reversion of volatility is high. The approximation is in "full" closedform, i.e., it does not require to solve an ordinary differential equation.