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.

大多数实证研究表明，三个因素足以解释期权价格固有的所有相关不确定性。在本文中，我们考虑了一个表现出非跨度随机波动率（USV）的三因素 CIR 模型，这意味着不可能用债券或掉期投资组合完全对冲波动风险。债券市场的不完备性是USV存在的必要条件。由于不完整性，需要对模型参数进行限制。我们为产生不完全债券市场的三因素 CIR 模型提供了充分必要条件。债券价格是仅两个期限结构因素的指数仿射函数，与非跨度因素无关。通过我们展示 USV 的三因素 CIR 模型，我们推导出债券期货价格的动态形式。通过引入变换的指数解并使用傅立叶反演定理，我们获得了欧洲零息期权价格的闭式解。该定价方法对于考虑非跨度随机波动的存在是有效的。