ABSTRACT The Cox–Ingersoll–Ross CIR short rate model is a mean-reverting model of the short rate which, for suitably chosen parameters, permits closed-form valuation formulae of zero-coupon bonds and options on zero-coupon bonds. This article supplies proofs of the formulae for the expected present value of payoffs under the real-world probability measure, known as actuarial valuation. Importantly, we give formulae for asymptotic levels of bond yields and volatilities for extended CIR models when suitable conditions are imposed on the model parameters.

中文翻译：

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实际测量下扩展 CIR 模型的债券收益率和波动率的渐近线

ABSTRACT The Cox–Ingersoll–Ross CIR short rate model is a mean-reverting model of the short rate which, for suitably chosen parameters, permits closed-form valuation formulae of zero-coupon bonds and options on zero-coupon bonds. 本文提供了现实世界概率测度（称为精算估值）下预期收益现值的公式的证明。重要的是，当对模型参数施加合适的条件时，我们为扩展的 CIR 模型给出了债券收益率和波动率的渐近水平的公式。