On the Elliptic Distribution and Its Application to Leptokurtic Financial Data
57 Pages Posted: 3 Dec 2015
Date Written: September 26, 2015
Abstract
A novel distribution based on elliptic curves is developed to capture major leptokurtic features in the return distribution of financial assets. This distribution is based on the Weierstrass equation and depressed cubic polynomial; therefore, it is intuitive, mathematically elegant, analytically tractable, and easy to calculate. It fits well to the historical daily log-return distributions of currencies, commodities, Treasury yields, VIX, and, most difficult of all, DJIA whose kurtosis is above 20. Various asymptotic behaviors are studied that encompasses a wide range of kurtosis from 2.3 to 35. The formal expansion near O of elliptic curves also suggests a possible different perspective on the tail behavior of the financial data. The numerical methods are built into an R package. A comprehensive sqlite database stores pre-calculated statistical attributes that cover a large parameter space. This allows practitioners to quickly examine any time series of interest. The distribution serves as a viable alternative to other fat-tailed distributions used in financial and risk modeling.
Keywords: distribution, elliptic curves, timeseries, fat tail, leptokurtic, financial data, daily log-return
JEL Classification: C10, C63
Suggested Citation: Suggested Citation