四阶Edgeworth密度函数上的有效域与隐含波动率应用

Valid Region of Fourth-order Edgeworth Density Function and Its Application on Implied Volatility Smirk

以正态分布为基底的Edgeworth级数展开是一个渐进展开序列,其截断形式常用以逼近未知的概率密度函数.若截断的Edgeworth级数能成为一个有效的(非负的)概率密度,前提条件是对参数(累积量)的取值做一些限制.文章介绍了在数值上求解四阶Edgeworth展开中参数的约束区域的算法,从而保证参数限制在有效区域内的四阶Edgeworth展开序列可以被认为是有效的概率密度.此外,给出了基于Black-Scholes公式的四阶Edgeworth密度函数的期权定价公式,并建立了隐含波动率微笑的水平、斜率和曲率与风险中性标准差、偏度和超值峰度之间的联系.

The Edgeworth series expansion based on a normal distribution is a sequence of asymptotic expansion,whose truncation form is commonly used in approximately unknown probability density function.The Edgeworth expansion is widely used,and its truncation form is defined as an effective(non-negative)probability density but with some given restrictions on the value of the parameters(cumulants).In this paper,the algorithm for numerically solving the constrained region of the parameters in the fourth-order Edgeworth expansion was introduced,thus ensuring that the fourth-order Edgeworth expansion sequence with parameters restricted within the effective region could be considered as an effective probability density.Furthermore,an option pricing formula for the fourth-order Edgeworth density function based on the Black-Scholes formula was proposed.The relationships among the level,slope,and curvature of the implied volatility smirk and the risk-neutral standard deviation,skewness,and overvalue kurtosis were established.