Extreme value of intraday returns

• Author: Aranit Muja
• Position: Dept of Mathematics, Statistics and applied Informatics, Faculty of Economics, Tirana University
• Pages: 284-294

ISSN 2410-759X Balkan Journal of Interdisciplinary Research Vol 1 No 1

Acces online at www.iipccl.org IIPCCL Publishing, Tirana-Albania May 2015

284

Extreme value of intraday returns

Aranit Muja

Dept of Mathematics, Statistics and applied Informatics

Faculty of Economics

Tirana University

Abstract

e aim of this research paper is to studythe properties of intraday returns, in a time range

from one to een minutes. In order to perform this analysis, we consider four sets of historical

intraday returns for FTSE-MIB1index. e rst series consist of intraday returns with one-minute

frequency, represented in log scale, which includes the period from 01.04.2011 till 30.09.2011. e

consideration period for the other series does not vary, but the frequencies which we calculate the

returns with, do. In detail, we took in consideration returns generated in 1, 5, 10 and 15 minutes.

First, the study analyses the distribution of intraday returns by employing both graphical methods

and moments calculation on dierent time scales. Secondly, the study analyses the returns maximum

distribution on dierent time scales, checking the type GEV (Generalized Extreme value) returns

distribution goodness of t. e GEV parameters estimation was made by maximum likelihood

using EVIM2 toolbox in Matlab.

Key Words: Intraday returns, block maxima, extreme value theory, GEV, stylized facts.

Introduction

e assessment and modelling of extreme event is recently gaining momentum in

actuarial and nancial disciplines. In many cases, we are more interested in the probability

of extreme events (events distribution minimum or maximum) rather than normal ones.

e extreme event in nancial applications may bring down the company, an equity price

or a portfolio and, thus, give rise tothe claim amountby the reinsurance company. It is

helpful to use the principles of extreme value theory to obtain a good estimate of these

events. e theory deals with the study of the asymptotic distribution of extreme events,

i.e. events that occur with a low frequency and are relatively large compared to most of

the observations in a given sample. e statistical methods derived from this theory have

had an increasing use in nance, especially in the risk assessment. e EVT application in

nance, estimates, particularly, the tail shape of a return distributions and uses, only for

analysis, past performance time-series“extreme” data.

e rest of the paper is organized in the following fashion. e rst section describes the

extreme value theory. e goal, instead, is to nd a limit theorem that gives a non-trivial

result on the distribution of the maximum for suciently large samples. We have followed

the analogy with the central limit theorem, in process.

In literature, the most important result in the extreme value theory is the “ree types

theorem”(Fisher, Tippett 1928; Gnedenko 1943). e theory deals with the convergence

of maxima.

It was considered that, if the maximum normalized sample admits a non-degenerate

1e FTSE MIB is the benchmark stock market index for Borsa Italiana, the Italian national stock

exchange. e index consists of 40 most-traded stock classes on the exchange.

2EVIM is a free soware package for extreme value analysis in Matlab.