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Abstract—After the widespread release of electronic trading,
automated trading systems have become a significant part of the
business intelligence system of any modern financial investment
company. An important part of the trades is made completely
automatically today by computers using mathematical algorithms.
The trading decisions are taken almost instantly by logical models
and the orders are sent by low-latency automatic systems. This paper
will present a real-time price prediction methodology designed
especially for algorithmic trading. Based on the price cyclicality
function, the methodology revealed will generate price cyclicality
bands to predict the optimal levels for the entries and exits. In order
to automate the trading decisions, the cyclicality bands will generate
automated trading signals. We have found that the model can be used
with good results to predict the changes in market behavior. Using
these predictions, the model can automatically adapt the trading
signals in real-time to maximize the trading results. The paper will
reveal the methodology to optimize and implement this model in
automated trading systems. After tests, it is proved that this
methodology can be applied with good efficiency in different
timeframes. Real trading results will be also displayed and analyzed
in order to qualify the methodology and to compare it with other
models. As a conclusion, it was found that the price prediction model
using the price cyclicality function is a reliable trading methodology
for algorithmic trading in the financial market.
Keywords—Algorithmic trading, automated trading systems,
financial markets, high-frequency trading, price prediction.
I. INTRODUCTION
RADING and investing in financial markets is a common
activity today. An important number of market
participants are buying and selling every day in the free
markets. Private and public investors, different types of
companies and traders are continuously speculating the
markets in order to make a profit. The main objective is to
catch the price difference in time, to buy cheap and to sell
more expensive using different trading and investing
strategies.
“Nowadays, in the current challenging economic
environment, businesses have changed their models in order to
be more service oriented and serve a broader and global
audience.” [1] The trading and investment domain is one on
the top in this long list. “The increases in complexity of the
C. Păuna is with the Economic Informatics Doctoral School, Bucharest
University of Economic Studies, 11th Tache Ionescu str. 010352 Bucharest
Romania (phone: +407.4003.0000; e-mail: cristian.pauna@ie.ase.ro).
This paper was co-financed by the Bucharest University of Economic
Studies during the Ph.D. program and Algorithm Invest company
(https://algoinvest.biz).
phenomena that characterize a firm’s activity in general and of
financial aspects in particular, have led to an exponential
increase in the volume of data and information operated from
any field of financial activity.” [2] All of these issues involve
new aspects to organize the activity. All are competing to new
directions for the business environment in financial trading.
After the widespread release of electronic trading, “the role
of the automated trading software in the business intelligence
systems of any financial or investment company became
significant.” [3] An important part of the trades is set up
completely automatically today by computers using advanced
mathematical algorithms. The low-latency real-time
automated systems are used today to build and send trading
orders without any human intervention. Designing, testing and
developing automated software for trading decisions has
become a sustained activity nowadays and this is the field this
paper is addressed for. In this article, a computational model
will be revealed to predict the price evolution using the real-
time price series. Being exclusively a mathematical algorithm,
the methodology presented can be applied in any algorithmic
trading system to automate the trading decisions. In the first
part of the paper, the prediction model will be described and
explained. Computing a price prediction line is the core of the
model based on the price cyclicality function [4]. The
prediction model is presented in the form of two bands, one
for the stop loss and one for the take profit price levels. Some
clear trading strategies will be developed based on this model
together with the implementation steps into an automated
trading system.
In the second part of this paper, the main direction will be
revealed to integrate the developed model into an algorithmic
trading system. The general logical scheme for automated
trading software will be presented together with the steps to
compute the Price Cyclicality Function. The automated
trading signals are built based on the Price Prediction Line.
The article will also include code examples to compute these
functions. In order to present the reliability of the developed
model, in the last part, real trading results obtained with this
methodology will be included. A comparative study is also
presented in order to compare the developed trading model
with other known trading strategies. The study will permit to
highlight the advantages of the price prediction bands
methodology and its place in the algorithmic trading domain.
In the last chapter, different practical conclusions will be
presented. It was found that the methodology presented in this
paper can obtain good results to predict the changes in price
Cristian Păuna
A Prediction Model Using the Price Cyclicality
Function Optimized for Algorithmic Trading in
Financial Market
T
behavior. This article will conclude that the price prediction
model built with the price cyclicality bands is a reliable
method which can be applied for algorithmic trading in a wide
range of capital markets.
II. THE PRICE PREDICTION MODEL
In order to answer the questions “when to buy?” and “when
to sell?” on financial markets to make profits, a mathematical
model will be developed in this chapter. Based only on the
price action, this model will permit to set up the buy and sell
orders by specialized software in order to automate the trading
process. The concepts used in this chapter are not new. The
Price Cyclicality (PCY) function was presented for the first
time in [4] and represents a function describing the cyclical
behavior of the price movement and the intervals when the
price is approaching to change its direction. The PCY function
can be used in order to set limit conditions to entry and to exit
the markets. The Price Prediction Line (PPL) was developed
and explained in [5]. It represents an accurate trend line that
describes and predicts the price evolution with good results.
What is new in this paper is assembling all of these tools into a
model building price bands that will give a clear indication for
the entry and exit points, a reliable mathematical that can be
adapted into automated software.
A. PCY Function
Being given a data price series of i intervals, the Price
Cyclicality Function (PCY) is defined by:
11 iiii PCYPCYPCY
with
0
0PCY
(1)
where
ii
ii
iminmax
max
with
iii maMA
(2)
and
kk
Niik
imaMA ,
maxmax
(3)
kk
Niik
imaMA ,
minmin
(4)
in which MA and ma represent the moving averages [6] with
two different periods. In (3) and (4), the term N is the period
of the cyclicality function and represents the number of the
time intervals taken into account to build the PCY function; α
is a functional parameter that can be optimized for each
financial market in order to obtain the best results.
More details about how PCY function was created,
developed and optimized are presented in [4] together with the
influence of the functional parameter α. It is also presented a
study which reveals how the PCY function can be used in
order to build a trading and limit conditions for financial
markets. The PCY function can be applied in any timeframe
and is a part of the trading model developed later.
B. Price Prediction Line
The Price Prediction Line (PPL) is a trend line obtained
using a transformation function of the PCY function back into
the price space using the formula:
iiiii PPPPCYPPL min100/minmax
(5)
where
i
Pmax
and
i
Pmin
represent the maximum and
minimum price values in the current monotony interval of the
PCY function given by (1). The PCY function is limited in the
interval [0; 100]. Meanwhile, the PPL function is defined into
the price interval and predicts the price evolution with good
accuracy, as proved in [5]. For some financial markets with
high price volatility, the PPL function defined by (5) needs an
attenuation process in order to have a smooth evolution. For
this, any known methods can be applied as smoothing with
Spline line interpolations [7], polynomial or trigonometric
interpolations [8] or just a simple, exponential or weighted
moving averages [6] with a small period.
PCY and PPL functions for a daily price series of Frankfurt
Stock Exchange Deutscher Aktienindex DAX30 financial
market [9] are presented in Fig. 1. The functions can be used
in order to trade the markets. When PCY and PPL functions
are starting to increase, a buy condition is met because the
price will increase in the next time intervals. In [4] and [5] are
presented different types of trading signals developed with this
idea. As we can see in Fig. 1, the information about the entry
point in the market can be easily found. For long term or
investment trading systems, this point is a good trading
opportunity. Even so, the exit point is not so obvious, once the
market is bouncing up and down many times until the
direction changes. For high-frequency trading systems, with
small profit targets, to know the trend direction and the
starting point of the trend is not enough. For this kind of
algorithms, a take profit value and a stop loss level are
required in order to have a complete trading strategy. For this
purpose a methodology will be developed in the next sections,
using the Parabolic Stop and Reverse (PSAR) indicator
developed by Wilder [11].
Fig. 1 PCY Function and PPL
C. Price Prediction Bands
In this section we will define the Price Prediction Bands
(PPB) using for ascending trend (current price higher than
PSAR):
ii PSARndStopLossBa
(6)
iiii PSARPPLPPLOneTakeProfit
(7)
iiii PSARPPLPPLTwoTakeProfit
(8)
and for the descending part of the trend (current price lower
than PSAR) we have similarly:
ii PSARndStopLossBa
(9)
iiii PPLPSARPPLOneTakeProfit
(10)
iiii PPLPSARPPLTwoTakeProfit
(11)
Fig. 2 PPB defined with PPL and PSAR
D. How to Use the PPB
As we can see in Fig. 2, once the price touched the PSAR
level above PPL, the new PASR level will be calculated under
the PPL and the current price. This is the moment when the
uptrend begins. There are many trading strategies using this
point as an entry point, but the PSAR methodology [11] is
limited when it is about the exit point. The usual exit point is
located when the price level is equal with PSAR when a new
downtrend is starting. This idea is not productive at all,
because many times the profit level is too low. It was found
that exit the trade will produce a higher profit when the price
level is equal with PPB, once the profit is taken near a local
maximum point.
The first take profit level predicted by the PPB is calculated
in direct relation with the distance between PPL and PSAR.
For each time interval the values for PPL and PSAR are
different, consequently, the level for the take profit band
(TPB) is different. Because the values of the PPL depend on
the price evolution and PSAR is updated in strong correlating
with the Average True Range (ATR) developed also by
Wilder in [11], the take profit level given by PPB is in direct
correlation with the price behavior and ATR evolution. The
price band formed with the PSAR values levels will be noted
here as SLB (stop-loss band). These values are used as stop-
loss level in our model.
As we can see in Fig. 2, the distance between TPB and SLB
is variable. We will call this distance as to be a safe trade
range (STR), because under SLB a stop loss is touched and for
values higher than TPB the price is too high for a new entry.
There are intervals where the STR distance is increasing. In
this case, we will say that we have a price expansion, these are
the cases associated with a strong trend when the price is
making new highs. If the STR is decreasing, we will say we
have a price contraction. If we have an uptrend, this is the case
when the trend is preparing to reverse or to slow down the
price motion. Starting from the analysis of STR we will
develop a trading strategy as it is presented in the next section.
As it can be seen in Fig. 2, sometimes the price goes higher
the TPB levels. For these cases, the second TPB is included
using (8) and (11). We will note the second TPB with TP2B.
In Fig. 2, the TP2B was plotted using the gold ratio (δ=1.618).
This band is used in case of powerful trends to know where to
close the trade with good profit, before the price turning point.
In the majority of the financial market, after the price
touched the TPB or TP2B, a down movement is present before
the next up movement. This is the interval when a new entry
in the trade is possible in order to maximize the profit. Once
the PPL is increasing and the PCY function is increasing too,
the uptrend is still present and a new entry neat the PPL is a
good opportunity. The PCY values give us good information
about the proximity of the price turning point when the trend
is reversing. In addition, large values of the STR involve good
profit expectation meanwhile lower values for the STR can be
a good indication to stay away from the market risk.
The PPB can be used for manual trading on daily timeframe
or 4 hours (4H) timeframe. The information included in the
PPB is good enough for short and medium time trades. Better
results are expected using algorithmic trading and even high-
frequency trading as we will see in the next section. An
automated algorithm will use the PPB values in order to build
automated trading signals. For high-frequency trading small
profit target range will be used, the TPB and TP2B being used
in order to limit the entry into a new position. For these cases,
SLB is also used as a stop-loss level.
The PPB can be also used in order to find some cases when
the price is oversold. Sometimes the price exceeds TPB and
TP2B into downtrends with the STR is in contraction. In these
cases, the price can be considered oversold and a buy trade
can be a good opportunity until the price is bouncing again in
the STR area. A case like this is plotted in Fig. 3. The oversold
cases found with PPB will be a subject for additional trading
signals included in the next section.
Fig. 3 Oversold price area detected with PPB
E. Trading Signals Based on the PPB
To automate the trading decisions we have to include the
significance of the PPB levels into some Boolean variables
called trading signals. These will be the core of the automated
trading software presented in the next chapter. The first type
of trading signal based on PPB is related to the point when the
uptrend is starting. The signals for each time interval defined
by (i) index are given by:
iii
ii
iiiii
PCYPCYPCY
TPBAskSLBAsk
SLBPPLPPLPPLBuy
1
1
(12)
where Ask is the current ask price for the equity traded, θ is the
minimal take profit level and ρ is the maximum PCY value as
protection for the trend reversal. The functional parameters θ
and ρ can be optimized for each financial market traded in
order to maximize the profit. These trading signals can be used
with good results for daily and four hours timeframes.
For the rest part of the uptrend, on the expansion periods of
the STR, the trading signals can be given by:
iii
ii
iiii
PCYPCYPCY
TPBAskSLBAsk
SLBPPLExpansionBuy
1
(13)
where
11 iiiii SLBTPBSLBTPBExpansion
(14)
defines the price expansion intervals, where STR is increasing.
These types of signals are used with good results for daily and
four hours timeframes.
The signal given by (13) gives us good results for the price
expansion intervals as we will see in the last chapter. In the
rest of the intervals, when we have not an expansion for the
STR, there are also good cases for trading opportunities. We
have found that cases are filtered by an additional condition
imposed for the TPB. When the TPB is increasing, even STR is
decreasing, good trading opportunities can be found. All of these are
given by the trading signals assembled with the formula:
iii
ii
iiiii
PCYPCYPCY
TPBAskSLBAsk
SLBPPLTPBTPBBuy
1
1
(15)
It was found that the trading signals given by the formula (15) give us
good trading results for the daily and four hours timeframes. To
automate the cases for the oversold price intervals detected with PPB,
the trading signals can be given by:
ii
iiii
PCYTPBAsk
TPBPPLExpansionBuy !
(16)
where μ is a minimal value for the PCY function for which long trade
is accepted in the oversold area. This functional parameter can be
optimized for each financial market in order to improve the results.
These signals give us good results for four hours timeframe. For the
daily timeframe, the trading signals (16) need to be filtered with an
additional condition in order to set only that trades in presence of a
strong trend. For this purpose, a simple limit condition for the ATR
values is strong enough. In this section, we presented the buy side
signals developed with PPB. These are the most used trading signals
for the majority of the markets. For those markets where sell trades
can be considered, the sell trading signals can be assembled similarly.
3. Informatics for automatic trading
The place of the automated trading software in the business
intelligence system of a modern investment company is well defined
in [3]. “An automated trading system is a software which is receiving
the real-time and historical price data of an equity, generates the
signals for buying and selling of the equity based on well-determined
algorithms, sets the volume of trading based on the capital liquidity
and the capital a defined risk level, builds the trading orders and send
them to the brokerage account without any human intervention” [3].
A logical scheme for automated trading software is presented in
figure 4.
There are three data inputs for automated trading software. First is the
low-latency real-time price data from the stock exchange. The second
is the historical price data coming from a data warehouse. These two
data fluxes are managed by two modules for real-time and historical
data-mining processes. A low latency data management module will
assure the speed for the data processing. The price data series are
stored in memory and set up to be ready for the mathematical model.
The trading algorithms use low-latency price data and build trading
signals. The third data flux includes real-time capital and liquidity
data from the brokerage account. These data are the core of the risk
management module. In this module, depending on the liquidity and
the risk level established, the volume for the trading orders is set up.
A reliable capital and risk management method is presented in [12].
With the signals and trading volume, the orders can be assembled and
automatically sent to the brokerage account.
Figure 4. Data logical scheme of automated trading system
In this paper details about the integration of the trading model
developed will be presented. All technical aspects regarding data
acquisition are already solved. There are many trading platforms
available that integrate all of these features. One of them is Meta
Trader 4 [13] which permits algorithmic trading using a Meta Quotes
programming language [14]. This language will be used in the next
sections in order to exemplify the codes for different procedures.
3.1. Integration of PCY Function
The PCY function is the core of the presented trading model. The
formulas presented in section 2.1. must be computed with low time
consumption for a prompt response, to permit assemblage and
sending the trading orders as fast as possible. In figure 5. is presented
a code sample to compute the PCY function in real time.
Figure 5. Code sample to compute the PCY Function. (source [4])
3.2. Integration of PPL
Once the PCY functions values are computed, the PPL values can be
given by a procedure like in figure 6.
Figure 6. Code sample to compute the PPL.
With PCY and PPL values ready, the trading signals variables can be
easily computed. When the signal is confirmed, a trading order is
assembled and sent instantly by the trading platform to the broker.
4. Trading results
In this chapter, it will be presented trading results obtained with all
different signals presented in this paper using a high-frequency
trading methodology programmed into an automated trading system.
4.1. Trading results using PPB methodology
The results presented below were obtained using TheDaxTrader [15],
an automated trading system that uses the PPB trading signals in
order to generate buy side trades for DAX30 [10].
The results presented in table 1. were obtained with a high-frequency
trading algorithm applied for DAX30 the period 01.06.2015 –
30.09.2018 using a fixed target of 10 points for each trade. The
DAX30 index market was traded as a contract for difference (CFD)
with a spread of 1 point. The exposed capital involved and the risk
management for the high-frequency trading procedures was made
using the “Global Slot Loss Method” presented in [12].
The PPB trading signals were built for daily and four hours
timeframe interval. An additional condition was imposed regarding
the hourly intervals of the executed trades between 8:00 and 20:00
coordinated universal time (UTC) in order to ensure the liquidity on
the market. In table 1. are presented the trading results obtained for
the PCY limit ρ=99.95, for the minimal take profit distance θ=10 and
for α=0.33, the functional parameter of the PCY function. For the
signals given by formula (16) it was used the value μ=1 for the
maximal level of the PCY into the oversold intervals. The moving
averages used to build the PCY function were computed for 20 and
50-time intervals. The signals were computed by the trading software
one time per minute.
Trading signals
and timeframe
Number
of trades
Profit
Draw-
down
RRR
(12) 4 Hours
53
7,410
5,350
1:1.39
(12) Daily
117
16,224
5,353
1:3.03
(13) 4 Hours
72
10,292
5,123
1:2.01
(13) Daily
119
16,069
5,688
1:2.83
(15) 4 Hours
136
19,570
5,131
1:3.81
(15) Daily
180
25,701
5,687
1:4.52
All above together
330
46,970
5,716
1:8.22
(16) 4 Hours
179
25,274
9,337
1:2.71
All signals together
430
60,969
9,337
1:6.53
Table 1. Trading results obtained with PPB signals
All trading signals assembled with the PPB values generate a
significant number of trades and good values for the risk and reward
ratio (RRR). For the signals (12), (13) and (15) traded together; the
RRR obtained is 1:8.22, a very good value compared with other
signals as we will see in the next section. The lowest capital exposure
is obtained for these signals made in direction of the main trend. For
all PPB signals traded together, the RRR is 1:6.53, also a good value.
The longest trade period for these signals obtained n the study above
was 218 hours. The Quality Trading Coefficient (QTC) [16] obtained
was between 0.52 and 0.87. All these values indicate that PPB trading
methodology is a reliable one. Even the oversold PPB trading signals
used a double capital exposure, the returns from this type of trades is
significant. The capital evolution in the time interval for all signals
assembled together is presented in the next figure.
Figure 7. Capital evolution due to the trades made by PPB signals
4.2. Comparative trading results
In order to have a clear image of the PPB trading methodology, in
this section will be presented comparative trading results made in the
same market conditions with different and known trading
methodologies. In order to compare the same type of trading
methodologies, we will compare the results obtained with the
formulas (12), (13) and (15) with results made with other trading
methodologies in the direction of the main trend. To obtain the results
in table 2. and table 3. It was used TheDaxTrader [15] automated
trading system for DAX30 Index [10] for the period 01.06.2015 –
30.09.2018 using a high-frequency trading methodology with a fixed
profit target of 10 points for all signals presented.
Trading signals and
timeframe
Number
of trades
Profit
Draw-
down
RRR
Perfect Order [17]
107
20,107
8,712
1:2.31
Fischer Signals [18]
177
47,212
6,321
1:7.46
Turtle Signals [19]
162
24,160
3,693
1:6.54
PPB (12)+(13)+(16)
330
46,970
5,716
1:8.22
Table 2. Comparative trading results obtained in long of the main trend
As we can see in table 2, the results made with PPB trading
methodology have the highest RRR value. The PPN trading signals
made almost double trades than other trading strategies in the same
trading conditions. These results are an additional confirmation that
the PPB trading methodology is a reliable one. The results made with
formula (16) will be compared with other results made also with an
oversold price trading methodology in table 3.
Trading signals and
timeframe
Number
of trades
Profit
Draw-
down
RRR
RSI Oversold [20]
76
18,243
5,901
1:3.09
PPB (16)
179
25,274
9,337
1:2.71
Table 3. Comparative trading results obtained with price oversold methods
As we can see, the PPB trading signals made a significant number of
trades with a comfortable RRR value. The PPB results are perfect
comparable with the RSI oversold trading method presented in [20].
Both methods included in table 3. Trade the cases when the rice is
oversold but they rarely intersect. Using both methods into a trading
system will generate additives profits with the same capital exposure.
5. Conclusions
The PPB presented in this paper can be used to develop a reliable
trading methodology. Based only on the price action, the trading
method presented here can be easily applied for algorithmic trading
and high-frequency trading in automated trading systems.
To build this model it was considered the PCY Function [4], a
transformation of the time price series into a subspace defined on the
[0; 100] interval. Based on the minimal and maximal price values on
a time interval, the PCY function has an asymptotic behavior and can
be used in order to impose limit conditions in order to avoid initiating
trades in the interval when the main trend is preparing to reverse.
The PPL [5] was considered as the core for this trading model. It is a
reversed transformation of the PCY Function into the price space.
This trend line is considered as the main price for the prediction
bands. In order to find the main trend, the method uses the PSAR
[11] function which define the current SLB. To have a complete
trading model, TPBs are defined using formulas (7), (8), (10) and
(11). All of these together define the PPB.
The methodology presented in this paper can be easily used for
manual trading and investment. The PPB levels can be followed in
order to set up the trades. The best results are obtained with daily and
four hours timeframes. This methodology was tested with good
results for the next financial markets: Deutscher Aktienindex
(DAX30), Dow Jones Industrial Average (DJIA30), Financial Times
London Stock Exchange (FTSE100), Cotation Assistée en Continue
Paris (CAC40), Swiss Stock Exchange Market Index (SMI20),
Australian Securities Exchange Sydney Index (ASX200), Tokyo
stock Exchange Nikkei Index (Nikkei225), NASDAQ100 Index,
Standard & Poor’s Index (S&P500) and Small Capitalization US
Index (Russell2000). Also with good and stable results, the PPB
methodology presented in this paper was applied for Gold and Bent
Crude Oil financial markets. For the currency markets, the method
can be also applied using additional conditions regarding the price
volatility level and the power of the trend. The cases when the market
is not in a major trend must be avoided.
Being exclusively a mathematical model based on the price action,
the PPB method can be adapted for algorithmic trading and can be
easily included in an automated trading system. Sample codes about
how the PCY Function, the PPL and the PPB can be automated are
also included in this paper. The simplicity of this method and the
reduced number of functional parameters made this methodology to
be one of the easiest integrated into a trading system. All parameters
for the PCY Function can be set up and used with the same valued
for all financial markets. The parameters for the trading signals
presented can be optimized for each financial market but the values
do not differ in time; once optimized for a long period of time, they
can be used with good results for the next period.
Looking at the results presented in the last chapter, the significant
number of the trades set up by the presented method and the good
values for the risk and reward ratio recommend the PPB as to be a
reliable trading methodology.
As it was presented, the method shows when a new trend begins. For
these particular moments, trading signals can be built with the
formula (12). The PPB also reveals the time intervals when the price
is in an expansion. For these cases when the STR is increasing, good
trading opportunities can be found using formula (13). For the
intervals when the price is contracting, sustained trading signals are
also built with formula (15). All of these signals traded together
obtained a risk to reward ratio value equal with 1:8.22 for the study
case took as an example. This value is a very good one for a single
strategy computed with algorithmic trading into an automated trading
system.
As it was found in this paper, using the price level related to the PPB
values, good opportunities for oversold buy trades can be found using
formula (16). These additional trading signals offer us a significant
number of trades even the risk and reward ratio is higher. The profit
made by these types of signals is a significant one and the
methodology is preferred by many investors. The signals build with
the PPB can be also included in automated investment systems with
very good efficiency.
Taking in consideration the results presented, the simplicity of the
method, all the advantages regarding the trend detection, stop loss
and take profit levels, detection of the price expansion and
contraction intervals, defining the STR in different timeframe, all of
these aspects make the PPB be a reliable and sustained trading
methodology.
The reduced number of the functional parameters and the simple
integration into any trading software recommend the method
presented to be considered for any automated trading and investment
system.
The PPB trading methodology improve the spectrum of the trading
strategies and can also be considered as a data-mining filter in
addition to any other trading methodology in order to improve the
trading efficiency.
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