Valid Region of Fourth-order Edgeworth Density Function and Its Application on Implied Volatility Smirk
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.
Edgeworth expansionEdgeworth density functionimplied volatility smirkskewnesskurtosis
万方数据
维普
