CIR过程与Heston模型的卡方模拟
Chi-square simulation of the CIR process and the Heston model
AI总结：
提出基于广义高斯变量幂和的中心卡方密度新表示，结合Marsaglia极坐标法扩展与Beasley-Springer-Moro直接反演法，实现高精度、鲁棒且高效的卡方采样，并应用于Heston模型的非中心卡方方差模拟。
摘要：
Cox-Ingersoll-Ross过程的转移概率可由非中心卡方密度表示。首先，我们基于广义高斯随机变量的幂和证明了中心卡方密度的新表示。其次，我们证明了Marsaglia极坐标法可推广到该分布，为广义高斯采样（从而为中心卡方采样）提供了一种简单、精确、稳健且高效的接受-拒绝方法。第三，我们基于Beasley-Springer-Moro方法推导了一种简单、高精度、稳健且高效的广义高斯采样直接反演法，其对逆累积分布函数的逼近精度可达小数点后第十位。随后，我们将方法应用于Heston模型中的非中心卡方方差采样。我们重点关注自由度较小且零边界具有吸引性且可达的情况，这在外汇市场中很典型。利用卡方分布的可加性，我们的方法适用于所有参数区域。
英文摘要：
The transition probability of a Cox-Ingersoll-Ross process can be represented by a non-central chi-square density. First we prove a new representation for the central chi-square density based on sums of powers of generalized Gaussian random variables. Second we prove Marsaglia's polar method extends to this distribution, providing a simple, exact, robust and efficient acceptance-rejection method for generalized Gaussian sampling and thus central chi-square sampling. Third we derive a simple, high-accuracy, robust and efficient direct inversion method for generalized Gaussian sampling based on the Beasley-Springer-Moro method. Indeed the accuracy of the approximation to the inverse cumulative distribution function is to the tenth decimal place. We then apply our methods to non-central chi-square variance sampling in the Heston model. We focus on the case when the number of degrees of freedom is small and the zero boundary is attracting and attainable, typical in foreign exchange markets. Using the additivity property of the chi-square distribution, our methods apply in all parameter regimes.
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