Pricing fixed income derivatives under a three-factor CIR model with unspanned stochastic volatility

Most empirical studies show that three factors are sufficient to explain all the relevant uncertainties inherent in option prices. In this paper, we consider a three-factor CIR model exhibiting unspanned stochastic volatility (USV), which means that it is impossible to fully hedge volatility risk with portfolios of bonds or swaps. The incompleteness of bond markets is necessary for the existence of USV. Restrictions on the model parameters are needed for incompleteness. We provide necessary and sufficient conditions for a three-factor CIR model that generates incomplete bond markets. Bond prices are exponential affine functions of only the two term-structure factors, independent of the unspanned factor. With our three-factor CIR model exhibiting USV, we derive the dynamic form of bond futures prices. By introducing the exponential solution of a transform and using the Fourier inversion theorem, we obtain a closed-form solution for the European zero-coupon option prices. The pricing method is efficient for taking into account the existence of unspanned stochastic volatility.