Popescu Dan M, Lipan Ovidiu
Department of Physics, University of Richmond, Richmond, Virginia, United States of America.
PLoS One. 2015 Jan 27;10(1):e0116752. doi: 10.1371/journal.pone.0116752. eCollection 2015.
We propose the use of the Kramers-Moyal expansion in the analysis of third-order noise. In particular, we show how the approach can be applied in the theoretical study of option valuation. Despite Pawula's theorem, which states that a truncated model may exhibit poor statistical properties, we show that for a third-order Kramers-Moyal truncation model of an option's and its underlier's price, important properties emerge: (i) the option price can be written in a closed analytical form that involves the Airy function, (ii) the price is a positive function for positive skewness in the distribution, (iii) for negative skewness, the price becomes negative only for price values that are close to zero. Moreover, using third-order noise in option valuation reveals additional properties: (iv) the inconsistencies between two popular option pricing approaches (using a "delta-hedged" portfolio and using an option replicating portfolio) that are otherwise equivalent up to the second moment, (v) the ability to develop a measure R of how accurately an option can be replicated by a mixture of the underlying stocks and cash, (vi) further limitations of second-order models revealed by introducing third-order noise.
我们建议在三阶噪声分析中使用克莱默斯-莫亚尔展开。特别是，我们展示了该方法如何应用于期权估值的理论研究。尽管有帕乌拉定理，即截断模型可能表现出较差的统计特性，但我们表明，对于期权及其标的资产价格的三阶克莱默斯-莫亚尔截断模型，会出现重要特性：（i）期权价格可以写成包含艾里函数的封闭解析形式，（ii）对于分布中的正偏度，价格是正函数，（iii）对于负偏度，价格仅在接近零的价格值时变为负数。此外，在期权估值中使用三阶噪声还揭示了其他特性：（iv）两种常用期权定价方法（使用“delta套期保值”投资组合和使用期权复制投资组合）之间的不一致性，这两种方法在二阶矩之前在其他方面是等效的，（v）能够开发一种衡量指标R，以衡量期权可以被标的股票和现金的混合组合复制的准确程度，（vi）引入三阶噪声揭示的二阶模型的进一步局限性。
