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Abstract—In the first decades of the 21st century, in the
electronic trading environment, algorithmic capital investments
became the primary tool to make a profit by speculations in financial
markets. A significant number of traders, private or institutional
investors are participating in the capital markets every day using
automated algorithms. The autonomous trading software is today a
considerable part in the business intelligence system of any modern
financial activity. The trading decisions and orders are made
automatically by computers using different mathematical models.
This paper will present one of these models called Price Prediction
Line. A mathematical algorithm will be revealed to build a reliable
trend line, which is the base for limit conditions and automated
investment signals, the core for a computerized investment system.
The paper will guide how to apply these tools to generate entry and
exit investment signals, limit conditions to build a mathematical filter
for the investment opportunities, and the methodology to integrate all
of these in automated investment software. The paper will also
present trading results obtained for the leading German financial
market index with the presented methods to analyze and to compare
different automated investment algorithms. It was found that a
specific mathematical algorithm can be optimized and integrated into
an automated trading system with good and sustained results for the
leading German Market. Investment results will be compared in order
to qualify the presented model. In conclusion, a 1:6.12 risk was
obtained to reward ratio applying the trigonometric method to the
DAX Deutscher Aktienindex on 24 months investment. These results
are superior to those obtained with other similar models as this paper
reveal. The general idea sustained by this paper is that the Price
Prediction Line model presented is a reliable capital investment
methodology that can be successfully applied to build an automated
investment system with excellent results.
Keywords—Algorithmic trading, automated investment system,
DAX Deutscher Aktienindex.
I. INTRODUCTION
APITAL investment is a sustained activity today. An
impressive number of market participants are buying and
selling every day on thousands of free markets. Traders,
investors, different types of companies, private and public
funds are participating in financial markets in order to make a
profit from the difference between the buy and the selling
price. Once electronic trading was widespread released,
especially after 2011 when ''investment firms on both the buy-
side and sell-side were increasing their spending on
technology for electronic trading'' [1], a significant part of the
capital investments is made completely automatically. The
Cristian Păuna is with the Economic Informatics Doctoral School,
Academy of Economic Studies, 11th Tache Ionescu str. 010352 Bucharest,
Romania (phone: +407.4003.0000; e-mail: cristian.pauna@ie.ase.ro).
This paper was cofinanced by Algorithm Invest (https://algoinvest.biz).
trading orders are built and sent almost instantly by servers
running automated trading systems.
Algorithmic trading is the key in financial trading
nowadays, as long as ''large general financial volatility may
increase uncertainty about the economic environment, with
long-lasting effects as investors demand a higher risk'' [2].
With increased price volatility on markets where ''there is an
exponential over-reaction to an action'' [3], the trading
decisions must be fast indeed. Special algorithms and
methodologies are used in order to send the trading orders
before a significant change in the price level. This paper will
present one of these mathematical models specially designed
for algorithmic trading.
Thousands of trading strategies compete to find the best
way to buy and sell in order to make a profit. Related with the
capital investment strategies for the stock markets, advanced
studies and working models that can be used in algorithmic
trading can be found in [4]-[7]. For the currency and
commodities markets, reliable models and trading strategies
can be found in [8]-[10]. Original investment strategies
optimized for automated capital investment software can be
found in [11]-[15].
The theory is simple: buy cheap and sell more expensive.
To put it in practice is not easy nowadays, because of the price
volatility, unexpected news released, and unexpected behavior
of the significant market participants.
Dictionaries of trading rules and psychological practice to
enhance the capital investment performance are since a long
time on libraries. One of the essential books is [16], including
several practical strategies that can be included in algorithmic
trading in order to improve the mathematical models.
Recommendation lines from notable authors can be found as:
''buy the market after it’s dropped; not after it’s risen.'' [4] But
when the market it's dropped enough? Also, when is the price
too much higher, and it is preparing to change its direction?
How to determine all of these in order to have a supportable
risk? Moreover, how to automate the trading decisions,
including all of these considerations? The method presented in
this paper will offer answers to all these questions.
The algorithm presented in this paper is exclusively a
mathematical model based on the price time series. It can be
easily implemented in any automated trading system in order
to automate the capital investment process. The model uses
the cyclicality behavior of the time price variations to build a
price prediction trend line called “Price Prediction Line.”
Trading signals will be built using the particularities of this
function, in order to buy or to sell on the capital market
depending on the predicted price tendency. Realistic price
Cristian Păuna
Price Prediction Line, Investment Signals and Limit
Conditions Applied for the German Financial Market
C
World Academy of Science, Engineering and Technology
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level predictions can also be made together with limit
conditions in order to exit the investment positions before the
trend changes its direction. The presented method will be
compared with other models using trading results presented in
the last part of the paper.
II. PRICE CYCLICALITY
The model presented in this paper will use the "Price
Cyclicality function" noted as PCY [17]. Starting with the
assumption that the price has a wave behavior with variable
wavelengths, the model can be successfully applied for any
volatile market. This hypothesis is in line with the real
behavior of the investors in financial markets. After the price
starts to increase, more market participants will open more
buy positions that will grow the price continuously by
increasing demand. After a while, some of them will start to
close those trades in order to mark the profit. In that time
interval, the price evolution slows down. At a certain time
moment, more traders and investors will close their long
positions detecting the slowdown in the price evolution. More
sell orders than the buy orders will arrive on the market. A
reversed tendency will be present sooner or later in the price
behavior. It is a cyclical phenomenon which will be seen in
the price evolution. Increasing and decreasing demand
intervals are present in each market, similar to wave
propagation in time. Because the decisions do not have
constant repeatability, the price wave will have a variable
wavelength. The model presented in this paper can be adapted
automatically to this cyclical behavior. It will use the price
time series evolution and different functional parameters
which are optimized for each capital market.
The Price Cyclicality Function (PCYi) is mathematically
defined for each time price series interval (i) by the next
recurrent formula, starting from [17]:
11
iiii PCYPCYPCY
where 0
0
PCY (1)
where
ii
ii
iminmax
max
where iii maMa
(2)
and
kk
ni
ik
imaMamin
min
(3)
kk
ni
ik
imaMamax
max
(4)
i
Ma and i
ma are two moving averages [18] with different
periods (Pma and Pma, where PMa < Pma), and (n) is the
number of the time intervals considered in the time price
series, associated with the period of the presented model.
These three parameters (PMa, Pma, and n) are functional
parameters that will be optimized for each capital market and
for each timeframe used in order to obtain the maximal capital
efficiency for a minimal capital exposure, as we will see in a
further chapter. The PCY is represented in Fig. 1 for a daily
evolution of the Deutscher Aktienindex DAX30, the main
index of the Frankfurt Stock Exchange [19].
Fig. 1 PCY and Price Prediction Line
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The PCY has some important properties. The (PCYi) values
are limited into the interval [0;100]. The function presents
well defined ascending and descending intervals depending on
the price tendency and most important, the function has
asymptotic behavior on the overbought and oversold price
intervals. What is missing from the PCY model is a way to
predict the price level. The PCY function predicts with enough
accuracy, only the change in the price. The model developed
and presented in this paper proposes to predict the price
behavior for the next time intervals together with a predicted
price value level. To do this we will transform the PCY back
into the price space using the next transformation function:
iiiii PminPminPmaxPCYPPL 100/
(5)
The new function obtained (PPLi) is the Price Prediction
Line, a very useful function which will be used to obtain a
price prediction level, as we will see in this paper. Once the
terms (Pmini) and (Pmaxi) represent the maximum and
minimum price values on the current monotony interval of the
Price Cyclicality Function (PCY) is given by (1), after simple
mathematical considerations we can see that the PPLi values
are defined in the price space.
In very volatile price movements, the Price Prediction Line
given by (5) needs an attenuation process in order to have a
smoothed evolution. For this step we can apply mathematical
attenuation methods as the smoothing with Spline line
interpolations [20], polynomial or trigonometric interpolations
[21] or simple, exponential or weighted moving averages [18]
with a small period. The result using trigonometric
interpolation to attenuate the PPLi values for DAX30 index
are presented n Fig. 1.
III. PRICE PREDICTION LINE
The Price Prediction Line obtained after smoothing the
transformation function (5) has ascending and descending
intervals depending on the price behavior. The monotony
intervals are determined by an increased values of Pmaxi and
Pmini on an increasing interval of the PCYi.
Due to the way the Price Prediction Line (PPL) function is
built, (5), usually the minimum values of the price are attached
to the PPL levels on the ascending intervals, and the maximum
values of the price are usually attached with the PPL levels on
the descending intervals.
Fig. 2 Clear point for the price behavior change
Intentionally the PPL is plotted in two colors depending on
the price cyclicality ascending or descending intervals. As we
can see, the monotony intervals of the prediction line are not
the same as the monotony intervals of the cyclicality function.
The maximum and minimum points of the prediction line are
delayed. The PPL is plotted in blue for the ascending periods
of the PCY function, in order to keep the advantage of the
monotony of the cyclicality function. Similarly, the PPL is
plotted in red for the descending intervals of the PCY function.
With all of these, we will have a clear point for the price
behavior change, highlighted by the changing in color or the
prediction line.
As we can see in Fig. 2, the price change behavior point is
revealed before the maximum point of the PPL. In the moment
when the PPL line changes its color, meaning the PCY
function changed its monotony, the exit decisions from the
buy positions can be met since a price descending interval is
about to arrive. The color of the PPL is associated with the
trend behavior property. In the same way, in Fig. 3, we can see
that the ascending interval is announced by the prediction line
before its minimum point. In addition, once the color of the
line is changed, the prediction line values will be under the
price level, as a minimum value for the buy price level where
the trend behavior can be changed again.
Fig. 3 Buy price level decided by the PPL
Considering the above, one can think that the PPL is only a
colored line function depending on the PCY. After detailed
studies, it was found that the PPL makes a better prediction in
comparison with other models as moving averages. In Fig. 4,
it is presented a comparison between the PPL and usually used
moving averages. In Fig. 4 are drawn together the PPL line
with a period of 20-time intervals and the simple and
exponential price moving averages for the same period, on a
time price series expressed on a daily-bases interval for a 52
days evolution of the Frankfurt Stock Exchange Deutscher
Aktienindex DAX30.
As we can see, the PPL is a better approach to real-time
price series values. Sometimes the simple or exponential
moving averages monotony can predict the price behavior.
However, a comparison of the current price level with the
values of these moving averages in order to make a trade
decision can involve a considerable delay. The PPL values are
more close to the real price values, especially in the longer
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trend intervals. Besides, the specificity of the PPL can give us
more accurate information about the price change behavior
before the moving averages to start to change their monotony.
Fig. 4 Comparison between the PPL and moving averages
To trust the PPL, we also need a mathematical validation.
This confirmation is given by Pearson's correlation coefficient
[22] between two statistical series. Once our model tries to
predict the minimal value of the price for the ascending
intervals and the maximum values of the price for the
descending intervals, the correlation coefficient can be
adapted to this logic for our case with:
N
i
i
N
i
i
N
i
ii
PPLPPLqq
PPLPPLqq
r
1
2
1
2
1 (6)
where
1
if
iiii PCYPCYlowq (7)
and
1
if
iiii PCYPCYhighq (8)
in which the values (lowi) are the minimum price level and
(highi) represent the maximum price level for each (i) time
interval included in the time price series. Comparing if the
next price values are higher than the previous minimum price
interval for an ascending period of the PPL, and evaluating if
the next price values are lower than the previous high price
value for a descending period of the PPL, the correlation
coefficient has values between 0,583 and 0,999 for the
DAX30 market depending on the time frame used and the
period used to build the model. The values mentioned were
obtained for periods (N) between 10 and 50-time intervals.
The time frames used were 5 minutes (5M), 15 minutes
(15M), 30 minutes (30M), one hour (1H), 4 hours (4H) and
one day (1D). The historical time series computed were
between 01.01.2010 and 31.06.2019. Higher values for the
correlation coefficient were obtained for longer time frames.
With these values for the correlation coefficient, it can be said
that a direct and strong correlation between the price evolution
and the PPL exists. The PPL values will predict the minimal
local price values for the ascending intervals. Somewhere on
these values, the price will change the behavior to a
descending interval. Similarly, on descending price
movements, the PPL will predict the local maximal price
values.
In Fig. 5 is presented a code sample of the MetaQuotes
Language [23] for the PPL. For simplicity, the code includes
an attenuation process of the PPL values using an exponential
moving average. Also, for simplicity, the Signal vector takes 0
and 1 values depending on if the price behavior is up or down.
Sample code for the PCY can be found in [17].
IV. PRICE PREDICTION
The majority of the prediction models for algorithmic
trading are indicating a price level for the next time intervals.
After a buy trade was opened, the traders or the automated
trading software procedures are waiting the price to rich that
level in order to close the trade. Sometimes the predicted price
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level is never reached and the trade is turning into a loss trade.
The model presented in this paper will work differently in
order to assure the profit expectancy.
Fig. 5 MetaQuotes Language code for PPL
First of all, this prediction model will predict the next price
behavior and the moment when the price changes its tendency.
When the PPL turns from the red to the blue zone, an
ascending interval for the price is arrived. The minimal price
level where the price can change again its behavior is defined
by future values of the PPL. When the PPL turns from the blue
interval into the red values, a decreasing price interval will be
on the next scenario. The price will descend under the PPL
values, and that is the last good moment to close the long
trades. In this way, an automated trading procedure can be
organized in order to automate the capital investment process.
Similarly, for the markets where short trades can be
considered, in the presence of a significant descending trend,
sell signals can be built when the PPL turns from the blue
zone to the red one. The maximal entry price level will be
defined by the values of the PPL. The exit points will be met
when the PPL turns again from the red interval into the blue
one. Of course, the red and blue color properties were used
here for the visual presentation. In algorithmic trading, these
are properties defined as buy and sell internal properties of the
PPL class.
Fig. 6 Buy positions managed by the PPL
In Fig. 6 are displayed more consecutive buy trades
managed by the PPL. As we can see, using the exit conditions
provided only by the PPL intervals, not all trades are
profitable. Additional conditions must be met for this target, as
we will see in the next chapters. However, on a well-
capitalized market, the PPL catches early the most important
trades. The exit decisions are a separate subject later in this
paper.
V. INVESTMENT SIGNALS
By trading or investment signal, we understand a Boolean
variable with true value if a trade can be opened. Depending
on different conditions imposed with the PPL and PCY
functions, in direct correlation with the price evolution and
market behavior, these variables will permit to automate the
trading process. A buy signal is a signal that will decide if a
buy trade can be opened. A sell signal will be the signal for a
short trade. According with the PPL significance, the buy
signals can be defined as:
01 1 iiiii PPLPPLPPLpBuySignal (9)
where (pi) are the price values and PPLi are the PPL values for
each (i) time interval. In (9) i
PPL are the values for the
property associated with the color of the PPL providing the
information about the PCY function monotony. These last
values can be set in many ways. Boolean variables are the
easiest way. We consider here 1 values for the ascending
periods of the PCY function and 0 values for descending
intervals of the PCY.
The trading signals made by (9) open trades when the local
price trend is changed from a descending interval to a new
increasing-price interval, in order to buy the equity near a
local minimum point and to wait for the price growth. This
scenario is for those algorithmic trading procedures that ask a
relatively small number of trades and try to keep the opened
positions as long as possible to catch the big movements of the
market. For high-frequency trading procedures, the signal
made by (9) makes a reduced number of trades, and because in
these cases the profit target is a small one. For high-frequency
trading procedures, an additional signal type is needed as long
an uptrend is present:
11 1 iiiii PPLPPLPPLpBuySignal (10)
Equation (10) makes new trades even on the last section of
a trend which is not a good setup. Additional limit conditions
must be imposed to avoid these cases, as we will see in the
next chapter. In this section we have presented buy investment
signals, the most used trading signals in the stock markets. For
those markets where sell trades can be considered profitable,
especially for contract for differences (CFD) investment
procedures, the sell trading signals are built similarly with (9)
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and (10), considering the descending periods of the PPL.
VI. LIMIT CONDITIONS
Automated trading systems involve different types of
capital investment strategies and methodologies. One of the
most used algorithmic trading procedures opens a trade and
wait for the market as long as possible if the buy signal is
present. For these, the PPL gives proper entry and exit signals.
The entries will be made when the PPL turns in blue and will
exit the trades when the PPL turns into the red zone. The price
level for entry and exists are given by the PPL values. These
trading signals are made by (9). Not all trades will be
profitable as we will see in the last chapter, but the
methodology tries to catch some big trades in order to cover
all the small losses and to make a profit.
For high-frequency trading procedures, which make a
significantly large number of tiny trades, the trading entries
are made by the trading signals (10). Because these signals
open trades even on the last part of the trend, when the price
prepares to turn into the descending zone, additional limit
conditions must be imposed in order to ensure that the entry
point is good enough in order to make that small profit and to
avoid essential losses.
A. Entry Limit Conditions
In practice there are cases when the high price volatility can
produce exception cases. One of these cases is presented in
Fig. 7. This is the case of the DAX30 Index market at 7 June
2016 when the price made a significant correction in the up
direction in a very small time interval. The PPL turns form the
red zone into the blue at very high values of the price. The
trading signals made by (9) produced an entry point on the top
level price figured in Fig. 7. After that, the market reversed
strongly in down direction, fact signalized by the PPL which
produced a small gradient evolution and turned into the red
zone after a while, without the price to recover the losses on
those days. The strange evolution is due significant volatility
on the currency market following the Brexit news [24].
Fig. 7 Volatility exceptions with the PPL
To avoid cases like the one presented above, a limit
condition imposed for the distance of the price from the PPL
are used:
iii
iiii
PPLpPPL
PPLPPLpBuySignal
0
1
1
(11)
and for high-frequency trading:
iii
iiii
PPLpPPL
PPLPPLpBuySignal
1
1
1
(12)
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where δ is the distance from the PPLi accepted to be viable for
opening a trade into the risk level accepted. The functional
parameter δ can be optimized for each market, depending on
the exposure capital level, using the historical time price series
for a long interval of time. The limit conditions imposed by
ii PPLp will filter the trading signals made by (9) and
will allow only those trades with the price level under a limit.
From this reason, we call these conditions as to be limit
conditions. A more convenient graphic representation of these
types of limit conditions is presented in Fig. 8.
Looking Fig. 8, we can see that, on a long trend, the price
has the tendency to go up and to make new maximum points
together with the PPL. Due to the construction of the PPL, the
minimal values of the price will be in the zone of the PPLi
values, and the price tries to exceed these values and to make
new local picks. On a normal trend zone, after a new
minimum in the narrow of the PPLi values, a new maximum
will be made in one of the next time intervals if the price does
not change its behavior. Using this idea the trading signals
made by (12) work.
Fig. 8 Limit conditions depending on the PPL and PCY functions
When the price decreases under the δ distance from the
PPLi, the high-frequency trading procedure will open a new
trade. This position will be closed on one of the next time
intervals when the price makes a new local point, once the
profit target is small. The high-frequency trading procedure
will wait for the price to turn back into the δ distance from the
PPLi and will open a new trade, and so on, until the behavior
of the price will be changed. With a good optimization for the
δ, the strategy works for a large number of markets.
The second type of limit conditions imposed with the PP
line is regarding the gradient of this function. As we have
resented in Fig. 7, evolution with a small grading of the PPL
indicates a weakness of the price trend. A limit condition in
the gradient of the PPL function is imposed to avoid opening
trades in cases like these:
11 1
1
iii
iiii
PPLPPLPPL
PPLPPLpBuySignal (13)
where ξ is the minimum gradient value for the PPL function to
be considered strong enough in order to open a trade at a
specified risk level. The ξ parameter can be considered in
direct correlation with the power of the price trend. The filter
in the power of the trend is imposed by the term
1ii PPLPPL . This can be used with good results in
order to filter any other trading signal. Trading results for
high-frequency trading procedures made with these formulas
will be presented in the last chapter.
For high-frequency trading procedures, limit conditions can
be imposed using the PCY function as presented in [17]. To
exit from the trades and to stop making new trades in high-
frequency trading, the PCY function is limited in values as it is
displayed in Fig. 8. However, these conditions cut a
significant number of trades, especially for those cases when a
long-time trend is present when the asymptotic section of the
PCY function is long.
More limit conditions can be described with additional
algorithms based on the direct of inverse Fished transform of
the price, presented in [11] and [12]. In these cases also the
limits are too restrictive for long trends because they are
imposed only in the asymptotic stage of those indicators. The
PPL function permits a new kind of limit conditions which can
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be combined with all the others mentioned.
B. Exit Limit Conditions
Based on this PPL behavior and the price movement near
the prediction function levels, the exit conditions can be
imposed as:
0
1
1
i
iiii
PPL
PPLpPPLBuyExit
(14)
where φ is a functional parameter that can be optimized for
each traded capital market using a repetitive procedures with a
large historical time price series interval. The parameter φ is
the maximal distance of the price from the PPL function
where the price is considered high enough in order to wait the
next local minimum point. The limit conditions (14) can be
used with good results to automate the exit trading decisions
for buy trades made with any other trading strategy both for
algorithmic trading and high-frequency trading procedures.
For those market cases in which sell trades can be considered,
the limit conditions in order to exit the short trades can be
expressed with:
1
0
1
i
iiii
PPL
PPLpPPLSellExit
(15)
The limit conditions imposed by (14) and (15) on daily or
four-hour time frames can also be combined with the limit
conditions presented in [17], [11], and [12] imposed in a
smaller time frame as one hour or even lower, in order to filter
all overbought and oversold trades.
VII. TRADING RESULTS
In this section we will present some trading results obtained
with the trading signals presented in above. The exit
conditions presented in previous chapter were also included in
the algorithmic trading procedures. The results were obtained
using TheDaxTrader [25], an automated trading system that
uses the PPL in order to generate buy trades for DAX30 index
[19].
The results presented in Table I were obtained for the time
period between 1 June 2015 and 31 May 2018. An additional
condition was imposed for the entry trades regarding the
hourly intervals between 8:00 and 16:00 coordinated universal
time (UTC), Monday to Friday, in order to ensure the liquidity
on the market. The DAX30 index was traded as a CFD with a
spread of 1 point. For simplicity, the trades were made into an
account with no leverage. The capital risk was managed using
the “Global Slot Loss Method” [26]. The trading signals in
Table I were assembled into four-hour time frame interval.
TABLE I
TRADING RESULTS FOR THE PPL METHODOLOGY
Signal Target Positions Profit Drawdown RRR
(9) 10 points 203 1,230 1,495 1:0.82
(11)+(14) 10 points 179 1,939 845 1:2.29
(11)+(14) 20 points 122 2,683 846 1:3.17
(11)+(14) 50 points 60 3,318 1,032 1:3.21
(13)+(14) 10 points 101 1,199 796 1:1.50
(13)+(14) 20 points 101 2,423 796 1:3.04
Fig. 9 Capital evolution with and without PPL limit conditions
The results presented in Table I are obtained using (11) for
δ=10 and (14) for φ=100. In Fig. 9 is represented the capital
variation for the trading case with (9) without the limit entry
conditions and the same variation for the case for (11) using
the limit conditions. As we can see, the limit conditions have
successfully filtered the exception case presented in Fig. 7
which produced a heavy loss in the trading case with (9)
without limit conditions. Even the number of trades is lower
using the limit conditions, the profit is higher and the risk to
reward ratio (RRR) is considerably improved for all cases
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using the limit conditions for the trading entries.
Trading with (11) and (14) for targets higher than 20 points
we do not obtain a significant increase in RRR. In addition,
using longer targets will produce longer trades. Values higher
than 20 pips can be used for automated investment system
instead high-frequency trading systems.
Working with signals made by (13) and (14) on volatile
markets need sometimes additional conditions. In a volatile
market, after a new trend was established and the PPL changes
its color, entries can be made on points too far away the PPL
or in the last part of the trend where the price is near the
turning point. Additional limit conditions to entry and to exit
the trades can be imposed. In this section are presented results
obtained using the same PPL strategy applied in two time
frames in the same time. The results in Table I were obtained
applying (13) and (14) on one and four-hour time frames.
When the buy trading signals were true for the both time
frames, a new trade was opened with a ten points target.
VIII.
CONCLUSIONS
The PPL can be built based on the local minimum and
maximum points of the price time series using the PCY and a
transformation function into the price space. The strong and
direct correlation between the PPL function and the price
movement indicates that the monotony of the PPL function
can give us a strong indication about the price direction for the
next intervals. Besides, the PPL has an additional property
which is given by the monotony of the PCY function used to
build the transformation. Reading the evolution of the PPL
function together with the significance of the PCY function,
trend decision can be made automatically by building logical
trading signals.
The PPL function predicts the direction of the next price
evolution and the moment when the trend will be changed.
The values of the PPL function are more close to the price
level than the moving averages and the PPL can be used to
estimate the next possible price level where the new trend will
start. The gradient of the PPL function is also a good indicator,
being in strong relation with the power of the existing price
trend and with the amplitude made by the price movements.
High values for the gradient will indicate a strong price
tendency. Low values of the gradient will indicate weak price
movements. An exit signal can also be automated based on the
PPL function gradient.
Limit conditions in order to filter the entry trades can be
built using the PPL function, which can be used as price level
reference. A higher value of the price, too far away the PPL
levels can indicate a local maximum point and is a reasonable
limit signal for exiting the trades. The limit conditions and the
exit conditions based on the PPL function can be automated
using functional parameters that are optimized for each traded
market in order to reduce capital exposure and to maximize
the profit. All PPL limit conditions can be used in order to
filter any other type of trading signal. The model presented
can be applied in any time frame for any financial market. The
PPL can be easily automated in order to be included in any
automated trading system, the model being exclusively a
mathematical model based on the time price series.
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