An investigation of higher order moments of empirical financial data and their implications to risk

Luke De Clerk,Sergey Savel'ev

doi:10.1016/j.heliyon.2022.e08833

Abstract

Here, we analyse the behaviour of the higher order standardised moments of financial time series when we truncate a large data set into smaller and smaller subsets, referred to below as time windows. We look at the effect of the economic environment on the behaviour of higher order moments in these time windows. We observe two different scaling relations of higher order moments when the data sub sets' length decreases; one for longer time windows and another for the shorter time windows. These scaling relations drastically change when the time window encompasses a financial crisis. We also observe a qualitative change of higher order standardised moments compared to the gaussian values in response to a shrinking time window. Moreover, we model the observed scaling laws by analysing the hierarchy of rare events on higher order moments. We extend the analysis of the scaling relations to incorporate the effects these scaling relations have upon risk. We decompose the return series within these time windows and carry out a Value-at-Risk calculation. In doing so, we observe the manifestation of the scaling relations through the change in the Value-at-Risk level.

Highlights

In many financial settings, the behaviour of market data is analysed to better understand: the logarithmic price change, [1, 2, 3, 4], the historic or implied volatility, [5, 6] or the actual price behaviour [7, 8, 9]
In [10], the higher order moments were used to study the applicability of certain Generalised AutoRegressive Conditional Heteroskedasticity (GARCH) models for mimicking price dynamics
By investigating the higher order moments we can get an insight to the distribution of price change and how it varies over time

Summary

Introduction

The behaviour of market data is analysed to better understand: the logarithmic price change, [1, 2, 3, 4], the historic or implied volatility, [5, 6] or the actual price behaviour [7, 8, 9]. In [10], the higher order moments were used to study the applicability of certain Generalised AutoRegressive Conditional Heteroskedasticity (GARCH) models for mimicking price dynamics. The use of higher order moments within financial modelling is well established, [11]. By investigating the higher order moments we can get an insight to the distribution of price change and how it varies over time.

INTRODUCTION

Estimation of higher order moments from empirical data

The Impact of Economic Environments upon Scaling Relations

Higher Order Standardised Moments in Empirical Data

GARCH Simulations

Hierarchical Analysis of Rare-Events

Implications on Value-at-Risk

Findings

Conclusion