GEOMETRIC
BROWNIAN MOTION: AN ALTERNATIVE TO HIGHFREQUENCY TRADING FOR SMALL INVESTORS
Eder Oliveira
Abensur
Universidade
Federal do ABC (UFABC), Brazil
Email: eder.abensur@ufabc.edu.br
Davi Franco
Moreira
Universidade
Federal do ABC (UFABC), Brazil
Email: davifm1@gmail.com
Aline Cristina
Rodrigues de Faria
Universidade
Federal do ABC (UFABC), Brazil
Email: alinecrfaria@gmail.com
Submission: 8/13/2019
Revision: 9/18/2019
Accept: 10/22/2019
ABSTRACT
Highfrequency
trading (HFT) involves shortterm, highvolume market operations to capture
profits. To a large extent, these operations take advantage of early access to
information using fast and sophisticated technological tools running on
supercomputers. However, highfrequency trading is inaccessible to small
investors because of its high cost. For this reason, price prediction models
can substitute highfrequency trading in order to anticipate stock market movements.
This study is the first to analyze the possibility of applying Geometric
Brownian Motion (GBM) to forecast prices in intraday trading of stocks
negotiated on two different stock markets: (i) the Brazilian stock market (B3),
considered as a low liquidity market and (ii) the American stock market (NYSE),
a high liquidity market. This work proposed an accessible framework that can be
used for small investors. The portfolios formed by Geometric Brownian Motion
were tested using a traditional risk measure (meanvariance). The hypothesis
tests showed evidences of promising results for financial management.
Keywords: Geometric brownian
motion, Highfrequency trading, Algorithmic trading, Financial engineering,
Statistical inference
1.
INTRODUCTION
In financial markets,
decisionmaking involves four main variables: profitability, risk, liquidity,
and income taxation. In this context, an alternative strategy used by many
investors includes buying and selling stocks on the same day, known as intraday
trading. These applications allow satisfactory results from trading highliquidity stocks on the same day. The
short interval between buying and selling allows higher assertiveness in making
inferences about their risk, which depends primarily on random processes
(COLMAN; WIENANDTS; DE PIETRO, 2013).
In addition to intraday
trading, negotiations in major stock markets have been drastically improved in
recent years by using advanced technological resources such as algorithmic
trading (AT). Hendershott et al. (2011) have shown that AT is usually defined as
the use of computational algorithms to make specific business decisions, send
orders, and manage these orders after order submission.
Highfrequency trading
(HFT) can be considered as AT. The Securities and Exchange Commission (SEC,
2014) declared that, although HFT currently represents approximately 50% of the
volume traded in the United States, the concept of HFT is still unclear.
However, according to Menkveld (2013), the characteristics of users of this
type of trading including (i) predominance of zero inventory positions at the
end of each day; (ii) frequency of trading at time intervals <5s; (iii)
profit obtained primarily by transaction spreads (sale price minus purchase
price); (iv) use of passive strategies in most cases in line with market price
opportunities; (v) thousands of transactions a day on average; (vi)
negotiations of orders including large batches of stocks; and (vii) operation
in markets with advantageous operating taxes and technological resources
compatible with market needs.
The response speed is
one of the key characteristics of this category. The use of algorithms
incorporated into powerful supercomputers allows profits in operations executed
in stock market trading and completing these operations in fractions of
seconds.
Liquidity is an
essential attribute for trading intervals of ultrahigh frequency (<5s),
high frequency (<1 min), or medium frequency (<1 h) in intraday
operations (MENKVELD, 2013). In this study, the traditional concept of
liquidity was used, i.e., speed and ease with which stock can be converted into
cash.
The analysis of HFT
characteristics raises the question of whether it is possible for small
investors (private individuals and small and mediumsized enterprises) to apply
HFT concepts to conventional computing resources.
One obstacle found in
this study was the lack of studies on intraday trading in the Brazilian stock
market. In this sense, this study offers an additional contribution showing
whether this kind of transaction is feasible in Brazil.
In recent years, the
use of AT has become common in major financial markets worldwide (NYSE,
CMEChicago, NASDAQ, Euronext, ChiX, B3). AT was first used in the United
States capital market in 1990 (CHABOUD; CHIQUOINE; HJALMARSSON; VEGA, 2014).
However, there is still controversy over the number of resources available to
professional and small investors. Pentagna (2015) has found that HFT firms take
advantage of short time intervals to outperform traditional investors and earn
a slightly higher profit margin.
Geometric Brownian
Motion (GBM) is a Markovian process, in which future prices are predicted by
considering the last observed record (LAGE, 2011). Developing a stochastic
model that efficiently represents the trading prices of assets (stocks, oil,
soy, coffee, steel, rubber) is an advantageous feature for small investors,
whose access to computing resources from highfrequency trading is limited. In
addition, mathematical models that can form lower risk portfolios mean lower chances of systemic crises for society
as a whole.
The objectives of this
study were (i) to analyze the feasibility of applying GBM to forecast prices in
intraday trading (during market hours and its variations) by small investors as
an alternative to the highfrequency trading adopted by large corporations.
Data with a time interval of 30 min and 60 min were collected from the
Bloomberg system from September 2014 to April 2015, with free registrations of
market operations made at 1min intervals in the Brazilian Stock Exchange (B3) and
New York Stock Exchange (NYSE); and (ii) to evaluate the profitability of
investment portfolios created by GBMbased price forecasts using the
meanvariance (MV) optimization model of Markowitz.
This study presents the
(i) fundamentals of the GBM and MV models; (ii) methodology used; (iii)
characterization of the samples and applied tests; (iv) results; (v) discussion
and (vi) conclusions.
2.
THEORETICAL BASIS
In this section, we present the main
references used for evaluating the theoretical basis of the study using the GBM
and MV models for portfolio optimization.
2.1.
Geometric Brownian Motion
GBM is a stochastic
model discovered by Robert Brown in 1827 by observing the continuous movement
and irregular trajectories of pollen grains in an aqueous suspension. The
stochastic process that describes GBM properties was first defined by Wiener
(1923). Ito (1944) developed the fundamentals of stochastic calculus to allow
the differential calculation of Brownian stochastic processes. Over time, GBM
was applied to several types of situations (BODINEAU; GALLAGHER; SAINTRAYMOND,
2016; SEYF; NIKAAEIN, 2012; ZHANG; ZHOU, 2015). The famous model developed by
Black and Scholes (1973), Nobel Prize in Economics, adopted GBM for forecasting
market stock prices (IWAKI; LUO, 2013; KOGAN; PAPANIKOLAOU, 2014).
GBM studies focus on
market indices to assess the efficiency of forecasting market stock prices in
short intervals (ABIDIN; JAFFAR, 2012; REBOREDO, RIVERACASTRO; MIRANDA;
GARCIARUBIO, 2013; REDDY; CLINTON, 2016; ZHOU, 2015). However, few studies
applied GBM based prediction to form portfolios in different stock markets and
in HFT conditions, especially in the Brazilian stock market. A summary of
studies on the performance of GBM in different scenarios is shown in Table 1.
Table 1: GBM applications at different time
intervals.
Reference 
Description 
Main results 
Abdin and Jaffar 
Evaluated the performance of GBM
for predicting daily closing prices of 77 small and mediumsized enterprises
in the stock market of Malaysia over a 30day period. 
GBM presented a high forecast
performance for up to 14 days. 
Reboredo et al. (2013) 
Investigated the period necessary
for prices to adjust to GBM by analyzing stock prices at 1min intervals on
two stock market indices, one exchange market, and one Spanish stock during
74 trading days. 
There was a quick adjustment to
GBM for time intervals of <1 day. GBM presented good performance
for forecasting stock prices. 
Reddy and Clinton (2016) 
Tested the efficiency of GBM to
predict stock prices at daily closing prices of 50 large Australian companies
in the year 2013. 
In the evaluated period, GBM
showed that actual prices were like those of projected prices in more than
50% of the cases. 
Zhou (2015) 
Tested the efficiency of GBM for
forecasting daily closing prices of one option during 81 consecutive trading
sessions from January to May 2014. 
GBM presented favorable results. 
GBM is also a Markovian stochastic
process in which only the last observed record is considered for forecasting
future prices. Hillier and Lieberman (2015) have reported that a Markovian
stochastic process is defined as an indexed set of random variables {X_{t}},
where the index t includes a given set T. In most cases, it is
assumed that T is the set of nonnegative integers and X_{t}
represents a measurable characteristic of interest at time t.
In the average GBM, the most common
equations used to generate a stochastic process of a random variable S
(price) assuming an initial value s_{0} in t_{0}
and a final value in t_{f} are
(ROSS, 2014; SIGMAN, 2006):
(1)
(2)
2.2.
MeanVariance Optimization Model
Markowitz (1952) proposed a
portfolio optimization model known as the meanvariance (MV) model, which is
based on the riskreturn duality and defines the optimum combination of stocks
at the lowest possible risk to surpass a rate of return. This model formed the
basis of modern economic theory and was awarded the Nobel Prize in Economics.
The adopted risk measure was variance, which is usually obtained by analyzing
historical data of the evaluated stocks.
The meanvariance model created by Markowitz is
shown below.
(3)
Subject
to:
(4)
(5)
x_{i} ≥ 0 i = 1,......,N (6)
Where:
N – Number of stocks evaluated in the
portfolio;
x_{i} – Percentage of capital to be
invested in stock i;
σ_{i
j} –
Covariance between stocks i and j;
µ_{i} – Expected rate of return of stock i;
ρ – Minimum rate of return defined by
the investor.
The objective function (3) of the
model is to minimize the risk of the portfolio, and the risk is represented by
the covariance between stocks. The first restriction (4) presented in the model
is the rate of return expected by the investor, which should be met by the portfolio
used. The second restriction (5) requires that all capital must be invested.
The latter restriction avoids negative rates of return for any of the stocks.
There are three primary data inputs:
(i) expected rates of return of the candidate stocks; (ii) correlation between
the rates of return; and (iii) covariance. The model is based on portfolio
valuation considering the expected stock price (return) and variance of the
rates of return (risk). Therefore, when choosing between two portfolios with
the same risk, investors should choose the one with the highest return.
The principle of optimal allocation
of available resources based on risk developed by Markowitz is highly
applicable to other areas, which amplified the relevance of this study. In
particular, the covariance matrix, which represents the dependency
relationships between the stocks involved, is used as a risk management
strategy in other scientific fields. Some of the main contributions of the
model are: (i) portfolio optimization (CASTELLANO; CERQUETI, 2014; CUI; GAO;
LI; LI, 2014; LIOUI ; PONCET, 2016; QIN, 2015); (ii) risk conceptualization
(MCNEIL; FREY; EMBRECHTS, 2015); (iii) risk measures (AHMADIJAVID, 2012; CHOI;
CHUI, 2012; MARKOWITZ, 2014); and (iv) stochastic calculus (KHARROUBI; LIM;
NGOUPEYOU, 2013).
Despite the development of new risk
measures (ROCKAFELLAR; URYASEV, 2000; NOYAN; RUDOLF, 2013), the MV model is
still a relevant reference for improving portfolio optimization.
3.
METHODOLOGY
This study is characterized as
applied research and quantitatively evaluated the problem of price projection
when creating investment portfolios in intraday trading. The public data
(prices) used in the study were collected in a cash market database from the
Bloomberg stock market. Bloomberg is one
of the leading providers of business market information worldwide. This
database was also chosen because it is used in several markets. The data cover
trading records made at 1min intervals from September 18, 2014, to April 2,
2015, on B3 (Brazil) and NYSE (USA). This period comprised 196 calendar days or
131 and 139 trading sessions on B3 and NYSE, respectively.
The trading floor data from the
Bloomberg system of B3 and NYSE comprised a population of 445 and 3255 traded
stocks, respectively, beginning and ending at
9 a.m. and 4 p.m., respectively,
in the analyzed period. The trading sessions were divided into twotime
intervals: (i) 14 intervals of 30 min and seven intervals of 60 min on B3 and
(ii) 12 intervals of 30 min and six intervals of 60 min on the NYSE.
3.1.
Sample Size and Assessment of Normality of Residuals Obtained in GBM
Predictions
The use of GBM for predicting prices
implies residues with normal distribution (LAGE, 2011). Therefore, assessing
this property in the predictions is essential to guarantee compliance with the
basic assumptions.
The statistical test used for this
analysis was the nonparametric AndersonDarling (AD) test, which is available
as default in software Action (integrated into Microsoft Excel) and is
considered suitable for normality tests (CARRADORI; RAMOS, 2014; SHIN; JUNG;
JEONG; HEO, 2012). A level of significance α of 5% was considered
in the tests. Therefore, rejection of the null hypothesis occurred in cases in
which the pvalue was less than 0.05.
There are no references from other
studies that could be used to estimate the population proportion of interest
(success rate of the normality test). Under these conditions, Anderson et al.
(2013) recommend a planned p* equal to 0.50 (50%). The use of p*
equal to 0.50 allows obtaining the largest possible sample size and ensures
that the sample size is enough to reach the desired error margin. In fact, the
error margin calculated after sample definition should be less than the error
margin adopted before.
A 95% confidence level and a 7.5%
error margin for the intraday data in both capital markets were assumed to
calculate the sample size. Therefore, the recommended sample size for each 60
and 30min section was 171, totaling 684 tests considering both markets.
The number of trading records in
both markets at 1min intervals in the analyzed period was 1.87 x 10^{8}.
The manipulation of these records was unfeasible. All the stocks from the two
capital markets were initially classified in descending order of trading volume
(liquidity) and were later selected by random sampling.
Based on the estimated sample size,
we chose 30 shares from B3 and 45 shares from the NYSE. In addition, the number
of days selected at random for applying the normality tests on B3 and NYSE was
30 and 45 days, respectively. This number of stocks met the statistical
requirements necessary to evaluate the time intervals, ensured total
reliability in executing the tests, and is consistent with the study objectives
regarding accessibility to small investors.
The number of stocks used did not
jeopardize the results because the inclusion of more stocks increases the
chances of creating portfolios and making profit by incorporating more liquid
stocks. The following intervals were tested: (i) 182 intervals of 60min and
364 intervals of 30min on B3 and (ii) 190 intervals of 60 min and 269
intervals of 30 min on the NYSE. In total, 1005 samples were distributed as
follows: (i) 372 intervals of 60 min and (ii) 633 intervals of 30 min.
The test was applied for the 60 and
30min samples from both capital markets. Based on the studies by Iman and
Conover (1983) and by Sheng et al. (2015), the first 60% of the price records
of each selected time interval (60 and 30 min) were used to adjust the distribution
(determination of µ and s) and the remaining 40% of the price
records were used for predicting prices. In the case of the intraday market
price predictions made by GBM, the results of the hypothesis test generated
internally by the software Action were used.
Extracts with less than eight
observations necessary for the AD normality test were discarded. The number of successful predictions
(adherence to normal distribution) in 60 and 30min time intervals was counted
and divided by the total of valid periods in each interval (e.g., 5 successful intervals / 6 valid intervals =
83.3%). This study analyzed the following hypothesis for each evaluated time
interval of each chosen stock/day:
· H1: The residuals obtained by
predicting prices using GBM follow a normal distribution.
3.2.
Evaluation of the Success Rates of GBM Predictions
The stocks chosen at random were
evaluated for six 60min intervals and twelve 30min intervals on the NYSE and
seven 60min intervals and fourteen 30min intervals on the B3. The quoted
prices exclude the cases with fewer than eight price observations, the minimum
value required for the AD test, or stocks without results in one of the time
intervals.
After calculating the rate of
success of price prediction using GBM for each selected interval, the time
interval (60 or 30 min) that achieved the best result was determined. A ttest
on the difference in the means of related (dependent) samples in the 60 and
30min time intervals was conducted using the Microsoft Excel® data analysis
functions. The following hypothesis was
evaluated:
· H2:
The mean success rate in the normality test in 30min intervals is higher than
that in 60min intervals.
3.3.
Evaluation of the Profitability of the Portfolios Created using the MV
Model
The profitability of the portfolios
formed using the MV model was tested as follows:
· 30min time intervals were selected
(according to the result obtained in the previous phase) to assess the history
of the stock price;
· For each 30min interval, the stock
price history from the first to the 18^{th} minute was determined
(phase of estimation of GBM parameters);
· Portfolios were formed by predicting
prices using GBM for the remaining 12 min of each time interval;
· For all time intervals, a single
strategy was defined as the assembly (purchase) of the portfolio in the 18^{th}
minute and its disassembly (sale) in the 30^{th} minute;
· The prices effectively practiced by
the market in the assembly (18 minutes) and disassembly (30 minutes) were
recorded to determine the profit or loss of the formed portfolio;
· The Markowitz MV model was used to
select the stocks in the portfolios (ABENSUR, 2014).
In this phase of testing, time
intervals were chosen at random for applying the MV optimization model. The
independent effect of stock prices was ensured by using intervals of at least 2
days between the trading sessions, and Mondays and Fridays were avoided
whenever possible because these days were used by market managers to adjust
their investment allocation strategies (KEIM; STAMBAUGH, 1984). A total of 88
trading sessions were selected for testing the formation of the portfolios.
Optimization simulations were made
using the Solver optimization application available in Microsoft Excel®. The
results of the minimumrisk portfolios formed were subsequently subjected to a Ztest
of the mean, according to the following hypothesis:
· H3: The mean rate of return was
positive (profit).
4.
RESULTS
This section includes (i) evaluation
of normality of the residuals and comparison of price forecasts; and (ii)
evaluation of the profitability of the investment portfolios formed by GBM
forecasts. A summary of the statistical treatments applied in this study is
presented.
4.1.
Evaluation of the Normality of Residuals and Comparison of Price
Predictions
The AD tests were executed on the
GBMbased price projections obtained from each of the 372 60min time intervals
and 633 30min time intervals evaluated. The results generated by software
Action for the 30min prediction of one share are shown in Figure 1. The top
tables represent (i) the fitting phase of the coefficients (µ, s) for the initial 60% of the data in that time
interval and (ii) GBMbased price predictions (testing) for the remaining 40%
of the data.
The graphical analysis is divided
into (i) visualization of the probability paper (PP) graph for the conducted AD
test, in which the adequacy of the analyzed statistical model to the data is
considered useful in cases in which the distribution of the points is a
straight line and (ii) the QQPlot graph, which considers that the two analyzed
probability distributions (actual versus Gaussian) are similar in cases in
which the points lie on a straight line.
The table with the pvalue of
the performed AD test is presented. The obtained pvalue was 0.6895,
indicating that, at a significance level of 5%, the hypothesis that the
residuals obtained by GBM projection follow a normal distribution is accepted
(Figure 1) The QQPlots of three other stocks (PCAR4, GOAU4, LIGT3) derived from
the tests in the 30min time interval are shown in Figure 2. The first two
graphs are examples of approval, and the last graph is an example of rejection
of hypothesis H1.
As a strategy to organize and
facilitate the interpretation of the results, Tables 2 and 3 show the obtained
results in the two capital markets (B3 and NYSE). The average success rate in
the 30min time interval was 87.0% and 70.8% for B3 and NYSE, respectively. The
mean rate of success in the 60min interval was 70.8% and 62.6% for B3 and
NYSE, respectively. The overall mean
rate of success was 78.8% (30 min) and 66.6% (60 min).
The rate of success of adherence to
normal distribution in the 60 and 30min time intervals was compared for
determining the best time interval for the GBMbased forecast. H2 was validated
at a significance level of 1%, i.e., the mean success rate in the normality
test in the 30min interval was higher than that in the 60min interval in B3.
However, the same result was not observed in the NYSE.
Figure 1:
GBMbased Prediction and AD Test in the 30min time interval.
Figure 2:
QQPlots (PCAR4, GOAU4, LIGT3)
Table 2: Success rate of the normality test for
B3 stocks in 60min and 30min time intervals
Date 
Stock 
Success rate (60 min) 
Success rate (30 min) 
2014/09/18 
66.7 
92.3 

2015/03/06 
100.0 
100.0 

2014/09/19 
100.0 
100.0 

2014/09/22 
50.0 
84.6 

2015/01/20 
20.0 
66.7 

2014/12/04 
33.3 
69.2 

2015/03/05 
JBSS3 
33.3 
61.5 
2015/03/18 
33.3 
100.0 

2014/11/13 
66.7 
84.6 

2014/11/07 
100.0 
100.0 

2015/02/05 
33.3 
53.9 

2015/01/12 
100.0 
100.0 

2014/10/27 
50.0 
76.9 

2014/10/15 
100.0 
100.0 

2015/02/20 
100.0 
100.0 

2015/01/06 
83.3 
84.6 

2015/01/23 
40.0 
66.7 

2014/10/09 
83.3 
84.6 

2014/12/03 
83.3 
69.2 

2014/12/10 
50.0 
92.3 

2014/10/29 
100.0 
100.0 

2014/11/19 
TUPY3 
100.0 
100.0 
2015/01/30 
100.0 
100.0 

2014/11/05 
33.3 
76.9 

2015/03/11 
100.0 
100.0 

2014/10/14 
66.7 
84.6 

2015/01/09 
80.0 
100.0 

2014/10/08 
50.0 
69.2 

2014/10/06 
66.7 
92.3 

2015/03/30 
100.0 
100.0 


Average 
70.8 
87.0 
Table
3: Success rate of the normality test for NYSE stocks in 60min and 30min time
Date 
Stock 
Success rate (60 min) 
Success rate (30 min) 
2015/03/06 
RAX UN 
83.3 
72.7 
2014/10/30 
PHM UN 
50.0 
83.3 
2015/02/11 
CLX UN 
50.0 
87.5 
2014/10/29 
PCP UN 
100.0 
75.0 
2015/03/03 
ABB UN 
50.0 
60.0 
2015/04/02 
SLB UN 
33.3 
66.7 
2015/02/20 
DAL UN 
66.7 
75.0 
2015/03/30 
SUNE UN 
50.0 
75.0 
2015/02/12 
PX UN 
66.7 
90.0 
2014/11/10 
UA UN 
66.7 
81.8 
2014/10/31 
SWFT UN 
83.3 
81.8 
2015/02/23 
RCL UN 
50.0 
37.5 
2014/11/11 
EMN UN 
66.7 
88.9 
2015/01/27 
OIS UN 
0.0 
40.0 
2014/09/30 
PAY UN 
33.3 
60.0 
2015/01/20 
BURL UN 
83.3 
66.7 
2014/10/07 
DD UN 
33.3 
80.0 
2014/11/20 
GIS UN 
100.0 
33.3 
2014/09/30 
RAI UN 
66.7 
0.0 
2015/01/15 
OFC UN 
80.0 
80.0 
2014/12/30 
FRO UN 
100.0 
100.0 
2015/01/05 
HME UN 
33.3 
100.0 
2014/12/29 
KIM UN 
80.0 
100.0 
2015/02/26 
HP UN 
16.7 
58.3 
2015/03/10 
UHS UN 
33.3 
0.0 
2015/01/02 
BBL UN 
83.3 
66.7 
2015/01/13 
TNK UN 
100.0 
100.0 
2014/10/13 
PH UN 
83.3 
88.9 
2014/12/09 
LHO UN 
80.0 
88.9 
2014/12/18 
CE UN 
100.0 
100.0 
2014/11/24 
FLR UN 
16.7 
58.3 

Average 
62.6 
70.8 
4.2.
Assessment of the Profitability of the Portfolios using GBMbased
Prediction
The decisive test for assessing the
efficiency of the GBM model is its ability to form profitable portfolios.
Because of the restrictions explained in the methodology, it was possible to
form 85 distinct minimumrisk portfolios with a goal of achieving a positive
rate of return (≥ 0). A minimum positive rate of return was chosen
because the optimization model could freely identify all possible profitable
portfolios. The configuration of the MV model used in the test was as follows:
(7)
Subject to:
(8)
(9)
(10)
A summary of the statistical
procedures and the rates of returns of the formed portfolios are shown in
Tables 4 and 5, respectively. In addition to the stocks shown in Tables 2 and
3, the following stocks were evaluated: ACCO UN, AMG UN, APD UN, ARES UN, FN
UN, GLOB UN, HQL UN, IGT UN, LAD UN, MXF UN, MY UN, OMAM UN, SGF UN, WES UM.
Table 4: Statistical summary
Statistical procedure 
Objective 
Level of significance 
Result 
Minimum
sample size 
Determine
the minimum sample size for the AD tests 
5% 
n_{30}=n_{60}=171 n_{total }= 684 
Nonparametric
AD test (H1) 
Evaluate
the normality of GBM residuals 
5% 
Adherence to normal distribution in 78.8% of
cases for n = 633 (30min interval) and 66.6% for n = 372 (60min
interval) 
Hypothesis
test 
Comparison
of the normality rate in 30min and 60min intervals on the B3 
1% 
There was evidence of 
Hypothesis
test 
Comparison
of the normality rate in 30min and 60min intervals on the NYSE 
1% 
There was no evidence of 
Hypothesis
test 
Evaluation
of the profitability of the portfolios formed on the B3 
1% 
There
was evidence of 
Hypothesis
test 
Evaluation
of the profitability of the portfolios formed on the NYSE 
1% 
There
was evidence of 
Table 5: Rate of return of the formed
portfolios (%)
Date 
Day 
Time 
B3 Rate of return

NYSE Rate of return


09/18/2014 
Thursday 
12:00 
0.0410 
0.0503 

09/23/2014 
Tuesday 
15:30 
0.0146 
0.0309 

09/25/2014 
Thursday 
14:00 
0.0357 
0.1139 

09/30/2014 
Tuesday 
11:00 
0.0168 
0.0760 

10/01/2014 
Wednesday 
13:30 
0.0002 
0.0127 

10/07/2014 
Tuesday 
11:30 
0.0387 
0.0782 

10/09/2014 
Thursday 
14:30 
0.2585 
0.0008 

10/14/2014 
Tuesday 
12:00 
0.0332 
0.0456 

10/16/2014 
Thursday 
10:30 
0.2418 
0.0195 

10/21/2014 
Tuesday 
14:30 
0.1463 
0.0682 

10/23/2014 
Thursday 
09:30 
0.0523 
0.0469 

10/29/2014 
Wednesday 
10:00 
0.0007 
0.0494 

11/05/2014 
Wednesday 
15:00 
0.0559 
0.0083 

11/11/2014 
Tuesday 
11:30 
0.0609 
0.0346 

11/13/2014 
Thursday 
13:00 
0.0659 
0.0229 

11/20/2014 
Thursday 
12:30 
(Holiday) 
0.0501 

11/25/2014 
Tuesday 
13:00 
0.4736 
0.0635 

11/27/2014 
Thursday 
10:30 
0.0524 
(Holiday) 

12/03/2014 
Wednesday 
09:30 
0.0405 
0.3983 

12/16/2014 
Tuesday 
10:00 
0.0570 
0.2110 

12/18/2014 
Thursday 
11:30 
0.0588 
0.1223 

12/23/2014 
Tuesday 
11:00 
0.1246 
0.0044 

01/06/2015 
Tuesday 
14:00 
0.0931 
0.0293 

01/08/2015 
Thursday 
15:30 
0.2147 
0.0608 

01/13/2015 
Tuesday 
13:00 
0.0096 
0.3452 

01/15/2015 
Thursday 
10:00 
0.0485 
0.1552 

01/20/2015 
Tuesday 
10:30 
0.0207 
0.0407 

01/27/2015 
Tuesday 
09:30 
0.0884 
0.0252 

01/29/2015 
Thursday 
15:30 
0.0741 
0.0064 

02/03/2015 
Tuesday 
13:30 
0.1565 
0.0713 

02/05/2015 
Thursday 
12:00 
0.0148 
0.0877 

02/11/2015 
Wednesday 
15:30 
0.0376 
0.0713 

02/17/2015 
Tuesday 
15:00 
(Holiday) 
0.0301 

02/19/2015 
Thursday 
09:30 
0.0210 
0.0891 

02/24/2015 
Tuesday 
11:30 
0.4289 
0.0101 

02/26/2015 
Thursday 
10:00 
0.0080 
0.1313 

03/4/2015 
Wednesday 
13:00 
0.1229 
0.0469 

03/10/2015 
Tuesday 
11:00 
0.0486 
0.0330 

03/12/2015 
Thursday 
13:30 
0.2579 
0.0899 

03/18/2015 
Wednesday 
14:00 
0.3486 
0.1477 

03/24/2015 
Tuesday 
15:30 
0.0717 
0.0030 

03/26/2015 
Thursday 
14:00 
0.2193 
0.0121 

03/31/2015 
Tuesday 
10:30 
0.0652 
0.0812 

04/02/2015 
Thursday 
14:00 
0.2333 
0.0537 



Annual Return 
14.61 
10.63 



CI (α=0.05) 
14.61 ± 4.21 
10.63 ± 3.07 

5.
DISCUSSION
The results confirmed that GBM and
HFT can be used by small investors. The sequence of the performed tests was
logical. We first evaluated whether GBM could provide satisfactory results for
forecasting prices at 1min intervals. After that, the best time interval of
data accumulation (30 or 60min) for trading was analyzed. Finally, we
assessed whether the portfolios formed by GBMbased price prediction were
profitable.
It should be noted that ordinary
investors have public access only to the negotiated prices of the stocks in
each time interval. The decision structure used in this study considered that
small investors had resources and had access only to the negotiated prices of
the stocks. Professional investors (e.g., brokers and pension funds) have privileged
access to other data (e.g., stock lots traded per price) that allow better
price forecasts.
Following the tests, each stage
supported the execution and results of the following phases. The results of the
applied tests indicated that GBM might be used in decisionmaking by small
investors. Similar to other studies that evaluated different markets and time
intervals (ABIDIN; JAFFAR, 2012; REBOREDO ET AL., 2013; REDDY; CLINTON, 2016;
ZHOU, 2015), our results confirmed the quality of GBMbased forecasts in the B3
and NYSE (H1 hypothesis test), particularly in the 30min interval. For a
sample size based on the conservative premise of p* = 0.50, the obtained
error margins were lower than the margin of 7.5%. As an example, the error
margin of the success rate of B3 in the 30min interval was 5% after sampling.
Decisionmaking based on data
collected in short intervals is one of the foundations of HFT (MENKVELD, 2013).
HFT made at short intervals reduces the volatility of trading returns (CHABOUD
ET AL., 2014; HASBROUCK; SAAR, 2013). Therefore, the shorter is the time
interval, the lower is the exposure to market volatility and, consequently, the
better are the conditions of predicting prices. It is not surprising that this
type of negotiation was increased and is ideal for markets that present
technological infrastructure capable of offering data in milliseconds, such as
the NYSE and ChiX.
Therefore, the results of the H2
test for the 30min success rates are consistent, especially for the Brazilian
market. There was no significant difference between the 30 and 60min time
intervals probably because of the higher predictability and the lower
volatility of the American market, and therefore any of these intervals might
be used. Nonetheless, the success rate was higher in the 30min interval.
One of the assumptions for
statistical inference analysis, such as hypothesis testing, is the independence
between the evaluated events. An interval of at least two business days was
used between the dates and times selected for portfolio formation. This strategy
decreased the number of formed portfolios but increased the reliability of the
confirmation of profitable portfolios evaluated by hypothesis H3.
The rate of return depends on market
characteristics, including market size (e.g., number of participants and traded
volume), volatility, liquidity, trading costs, and opportunity costs. Although the mean return of GBM portfolios on
B3 was higher than that on the NYSE and is consistent with the observed success
rates, these results should be viewed individually.
The annual rates of return (250
business days) were high even when discounting the inflation of the respective
periods and markets. In the Brazilian capital market, the estimated mean annual
rate of return was 14.61% for an inflation of 11.09% with an actual gain of
3.17%. In the United States, the actual gain of 9.81% can be considered
exceptional and was obtained from an estimated mean annual return of 10.63% for
an inflation of 0.75%.
6.
CONCLUSIONS
This study is the first to
demonstrate that HFT characteristics (intraday trading, short time forecasts,
zero inventory) can be used by small investors for investment allocation. GBM
was feasible for predicting stock prices in two capital markets (Brazil and the
United States) with different rates of liquidity and volatility.
The adherence of the residuals to
normal distribution, evaluated by the AD test, was satisfactory and consistent
because the percentage of adherence in 30min intervals (78.8%) was higher than
that in 60min intervals (66.6%). With respect to the profitability of the
portfolios, the use of the Markowitz meanvariance model indicated higher
assertiveness in gaining profit, and therefore the measurements of the risk and
yield were satisfactory. High actual rates were obtained by annualizing the
mean percentage returns.
GBM was assessed in free 1min
intervals obtained on the B3 and NYSE. A promising line of research is the
incorporation of this model into other portfolio optimization models (CVaR,
Downside risk) to compare the efficiency of portfolio formation under different
risk conditions. Furthermore, this study opens the way for exploring the
integration of the riskreturn duality into the concept of liquidity because,
in HFT negotiations, it is essential to guarantee the sale of formed
portfolios.
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