Flexible two-point selection approach for characteristic function-based parameter estimation of stable laws

Abstract

Stable distribution is one of the attractive models that well describes fat-tail behaviors and scaling phenomena in various scientific fields. The approach based upon the method of moments yields a simple procedure for estimating stable law parameters with the requirement of using momental points for the characteristic function, but the selection of points is only poorly explained and has not been elaborated. We propose a new characteristic function-based approach by introducing a technique of selecting plausible points, which could bring the method of moments available for practical use. Our method outperforms other state-of-art methods that exhibit a closed-form expression of all four parameters of stable laws. Finally, the applicability of the method is illustrated by using several data of financial assets. Numerical results reveal that our approach is advantageous when modeling empirical data with stable distributions. Stable distribution is a class of probability distributions including the well-known Gaussian distribution. Besides its rich theoretical properties, it can effectively describe heavy-tails and skewness in financial markets and other various science fields. One of the primary and challenging issues when modeling financial behaviors with stable laws is to estimate all four parameters of stable distribution, due to the lack of stable densities and cumulative distribution functions (CDFs). We tackle this issue by proposing a new technique that allows us to benefit from the interrelations between the scaling exponent parameter and the characteristic function. Differently from the existing literature, our approach enables us to flexibly choose the proper points at which the characteristic function should be evaluated. Therefore, we can detect stable laws in financial data without any inconvenient restrictions on parameter ranges. This makes the estimation significantly practical. We explore price behaviors in crude oil futures and US dollar-Japanese Yen (USDJPY) exchange rate and show numerical evidence that our approach provides the most accurate detection of stable laws.