data). Generally, tick-by-tick data are not regularly spaced in time,
which leads to additional challenges for high-frequency data analysis
[1, 2]. Current research of financial data is overwhelmingly conducted
on the homogeneous grids that are defined with filtering and aver-
aging tick-by-tick data.
Another problem that complicates analysis of long financial time
series is seasonal patterns. Business hours, holidays, and even daylight
saving time shifts affect market activity. Introducing the dummy
variables into time series models is a general method to account for
seasonal effects (see Section 5.2). In another approach, ‘‘operational
time’’ is employed to describe the non-homogeneity of business activ-
ity . Non-trading hours, including weekends and holidays, may be
cut off from operational time grids.
2.2 RETURNS AND DIVIDENDS
IMPLE AND COMPOUNDED RETURNS
While price P is the major financial variable, its logarithm,
p ¼ log (P) is often used in quantitative analysis. The primary reason
for using log prices is that simulation of a random price innovation
can move price into the negative region, which does not make sense.
In the mean time, negative logarithm of price is perfectly acceptable.
Another important financial variable is the single-period return (or
simple return) R(t) that defines the return between two subsequent
moments t and t1. If no dividends are paid,
R(t) ¼ P(t)=P(t 1) 1(2:2:1)
Return is used as a measure of investment efficiency.
Its advantage is
that some statistical properties, such as stationarity, may be more
applicable to returns rather than to prices . The simple return of a
(t), equals the weighed sum of returns of the portfolio
¼ 1, (2:2:2)
are return and weight of the i-th portfolio asset,
respectively; i ¼ 1, ...,N.
Financial Markets 7