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DOI:
Price Probability Predictor.
Capital investments assisted by a probability field.
Cristian PĂUNA1
Economic Informatics Doctoral School, Academy of Economic Studies, Bucharest, Romania
cristian.pauna@ie.ase.ro
Abstract. Capital investment is a sustained activity nowadays. After the worldwide release of the
electronic trading systems, automated decision-making investment software is the new trend in
financial speculation. A significant part of capital trading is fully computerized today. The buying and
selling orders are made and sent automatically, almost in real-time. The price evolution is analyzed by
servers using advanced mathematical algorithms. This paper will present one of these models named
Price Probability Predictor. It is a method to build a probability field based on the price history and the
real-time price action. The revealed function will generate the current probability of a price growth in
the next time intervals. Automated entry and exit signals and market limit conditions will be built
using the new indicator, in order to automate the whole investment process. Capital investment results
will also be included in the current paper to qualify the presented trading methodology and to
compare it with other similar models. In conclusion, it was found that the Price Probability Predictor is
a reliable mathematical algorithm that can assist any trading decisions, in both ways, manual or
automatic capital investments.
Keywords: capital investment, price action, price probability field, investment decision, trading
signals, limit conditions, automated investment software
Introduction
Capital investment has become a sustained and collective activity nowadays. An impressive
number of market participants are buying and selling every day to make a profit. Private or
public investment funds, financial companies, professional investors, traders, or just
speculating people, all are part of an unstopped dispute with the price action. Using
different strategies, everyone tries to predict the price evolution in order to buy cheap and
sell more expensive. After the global scale of the electronic trading systems
implementation, access to the capital markets is at a couple of clicks distance. This fact has
increased the liquidity in all markets and also has grown the price volatility once the
psychological factors can influence the market price behavior in seconds.
The question this paper will answer to is: how much is the current probability for
price growth in the next period of time? The interest in an accurate answer at this question
is huge. Studying a probability field for price growth in a specific market will permit us to
exit investments when the probability is low, and to initiate new positions when the
probability is high. Moreover, a lower price growth probability can avoid buying near the
maximal price levels and will inform investors to wait until the market presents a
significant decrease. The evolution of a price probability field will also give us additional
information about market behavior. The Ph.D. research behind this paper has found a
probability prediction function, based only on the price time series that will give us the
answer to the formulated question. It is called here as the Price Probability Predictor.
1 This paper was financed by Algorithm Invest company (https://algoinvest.biz).
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In the electronic trading environment, a significant part of the investment orders is
made and sent today by automated decision-making computer software. The online price
quotes transmission permits today a real-time price analysis through advanced
mathematical algorithms to fundament the investment decisions. A simple trading strategy
is not useful today if it can not be set in the algorithmic trading environment. One of the
highest advantages of the Price Probability Predictor is the fact that it can be automated.
Based on the revealed function, automated trading signals and limit conditions can be built
and incorporate in any algorithmic trading software system, for any market.
In the last part of this paper, real capital investment results will be included, in order
to qualify the presented trading methodology and to compare it with other similar
mathematical models. At the end of this paper has found that the Price Probability Predictor
is a reliable probability field for a price growth that can be used with excellent results to
build a stable investment strategy and also to optimize the capital efficiency of any other
strategy. Any reader will find the originality and universality of the presented method.
Literature review
There are few studies in the literature about price growth probability even the probability
term appears in many titles of trading books. The most are trying to estimate the
probability that the price will touch a precise level in the future. The author of the current
research has decided not to cite any of these studies. During the research conducted behind
this paper, it was found that in more than 2200 trading books and papers read, no
probability field method with sustained and repetitive characteristics when it is about the
algorithmic capital investment was found. Several studies indicate entry points in the
market based on probability estimate but have no criteria for a technical exit decision.
Other studies are made for particular markets, having no proved possible generalization.
There are no reasons to cite uncompleted strategies or unproved price probability models.
This will be the gap filled by the current paper.
Regarding the subject of capital investment strategies that can be automated,
working and proved models for the stock markets can be found in (Connors, 1999),
(Connors & Sen, 2004), (Bernstein & Bernstein, 2008), (Connors & Alvarez, 2009a), or
(Connors & Alvarez, 2009b). Even these papers are not presenting the ways to automate the
included strategies; it remains excellent references for stock markets investment strategies.
The currency market trading strategies are also a very actual subject. Effective and proved
strategies that can be automated can be found in (Cheng, 2007), (Lien, 2009), (Lien, 2011),
or (Volman, 2011). Particular approaches for the commodity markets can be found in
(Kleinman, 2009), (Jagerson & Hanson, 2011). Important psychological strategies that can
be easily adapted with any technical analysis investment strategy can be found in (Pring,
1992), (Kiev, 2008), and (Ward, 2010). Some working mathematical models that can be
adapted for algorithmic trading can be found in (Focardi & Fabozzi, 2004). A particular
approach for risk management techniques that can be used in automated decision-making
systems for capital investments can be found in (Vince, 1992). Mathematical models
especially designed and optimized for algorithmic trading with proved and sustained
results in real capital investments can be found in (Păuna & Lungu, 2018), (Păuna, 2018a),
(Păuna, 2018b), (Păuna, 2018c), (Păuna, 2019a), (Păuna, 2019b), (Păuna, 2019c), (Păuna,
2019d), (Păuna, 2019e), (Păuna, 2019f), and (Păuna, 2020).
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Price probability field
In this paper, we will call a price probability field an analytical function that will give us a
measure for price growth in the next time periods. The current research will present how a
price probability field can be built, how it can be optimized to estimate a precise price
growth, and how three different types of price probability fields can be combined in order
to obtain a more accurate mathematical model. Besides, in the second part of this paper,
automated trading signals and limit conditions will be built using the price probability field
introduced to present how this model can be used in automated capital investment
software systems.
To clarify the notion of the price probability growth, we will say that a price presents
zero probability for growth with an established value in the future if the price never riches
that value before to decrease with at least the double of that value. Similarly, a price has a
100% probability for increase with a specified value if it will touch that value in the future
time before to decrease with the double of value considered. The definitions above are the
hypothesis for this study. The probability field can be defined in different other modes. The
current research has established that comparing a price increase with the double of his
decrease gives us the best capital efficiency in the model that will be presented further. An
additional requirement for a price probability field is to be continuously defined in all the
time price series definition intervals. In other words, the price probability field has to give
us the probability value for price growth at any moment of time.
Probability through Price Cyclicality Function
The Price Probability function was introduced in (Păuna & Lungu, 2018). It was proved that
a very strong and direct correlation exists between this function and the price movement.
The function is defined in [0;100] intervals and has minimal points near the minimal price
levels, and maximal values correlated with the maximal price levels. We will define the
probability field through Price Cyclicality Function the next function:
100/100 PCYPROPCY
(1)
where PCY is the Price Probability Function, and φ is a functional parameter that will be
optimized for each capital market analyzed, depending on the way of how the price
probability field is defined, as we will see. Once the PCY function is defined in [0;100], the
PROPCY function will be defined in [0;1] and will be inversely proportional to the price
evolution. When PROPCY has a minimal point, this means the probability for a price increase
is at a minimum, and the price will not make significant up movements. Similarly, when the
PROPCY makes a maximum, the probability for price growth is at maximum, a significant
price increase being expected in that market.
For each market, depending on the price probability field definition, the φ factor will
be computed using the historical price evolution. For example, the probability for a 200
points price increase after a maximal value of the probability field, using the PCY function
with 20 time intervals period, and 20 and 50 periods for the weighted moving averages
used, on a daily time price series of DAX30 Index market, is 98.75%. This means, for the
specified market, the 200 point growth price probability field is defined by φ=0.9875.
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Figure 1. Price probability field using the Price Cyclicality Function for DAX30 Index market.
The value φ=0.9875 was found computing the PCY function and analyzing the price
evolution for the historical time price series of the DAX30 Index market between 2010 and
2019. Considering all the cases when the PCY function made a minimal point lower than 10,
1.25% of cases were not true. In 1,25% of cases, after a minimum point of the PCY function,
the price made a lower movement with more than 400 points before to rich an up value
with 200 points. In this way can be computed the φ parameter for any type of probability
field using any kind of function. To compute the formula (1), the PCY function with n period
will be defined by the recurrence formula:
11 iiii PCYPCYPCY
for
0i
with
0
0PCY
(2)
where α defines the gradient of the PCY function, and
ii
ii
iminmax
max
,
iii maMa
(3)
kk
ni
ik
imaMamax
max
,
kk
ni
ik
imaMamax
max
(4)
using Mak and mak as two weighted moving averages with different periods (Cox, 1969).
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Probability through price Smoothed Heikin-Ashi transform
Smoothed Heikin-Ashi price transform is analytically defined in (Păuna, 2018c):
4/
),,max(
),,min(
2/
11
i
ii
ii
ii
i
i
ii
i
i
iii
CLHOCha
COHHha
COLLha
COOha
(5)
in which
i
O
,
i
H
,
i
L
and
i
C
are the averages values of open, high, low, respectively close
price levels into a specified M number of time intervals. During this research, it was found
that this price function has a direct and positive correlation with the price evolution.
Measuring the price evolution after the moment when
ii OH
, a significant price growth
probability was found.
The probability field through the Smoothed Heikin-Ashi price transform will be
defined analyzing the cyclicality of
i
O
and
i
H
price series:
100/100 SAHPROSHA
(6)
where
11 iiii SAHSHASHA
for
0i
with
0
0SHA
(7)
where β defines the gradient of the SHA function with m period, and
ii
ii
iminmax
max
,
iii maMa
(8)
and the maximal and minimal functions are computed using
i
O
and
i
H
:
kk
mi
ik
iOHmax
max
,
kk
mi
ik
iOHmax
max
(9)
The price probability field measuring the chances for 200 point price increase after
ii OH
on a daily time price series of DAX30 Index market is presented in figure 2. The
functional parameters β and ψ can be found using the historical time price series of any
market. β will define the gradient of the probability field, and ψ will define the maximal
probability value measured in the historical time price series.
As a first observation, the probability variation in figure 1 and the probability field in
figure 2 have significantly different evolutions. There are some time intervals when both
series take maximal or minimal probability values. An important note is that the two
functions are measuring the same phenomenon, which is the price increase with 200
points, but using different methods. Later in this paper, we will see how two or more
probability fields can be combined in order to obtain a better capital efficiency.
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Figure 2. Price probability field using the Smoothed Heikin-Ashi for DAX30 Index market.
Probability through Price Prediction Line and Trigonometric Price Line
The Price Prediction Line is a function introduced in (Păuna, 2019d). It represents the back
transformation into the price space of the Price Cyclicality Function, using the formula:
iiiii PminPminPmaxPCYPPL 100/
(10)
where the terms
i
Pmin
and
i
Pmax
represent the minimum and respectively maximum price
values on the current monotony interval of the Price Cyclicality Function. The Price
Prediction Line is figured with a dotted line in figure 3.
The Trigonometric Price Line is a price interpolation line using trigonometric
functions, presented in (Păuna, 2020). The analytical function is given by the form:
N
k
N
kikiki Nitktktp 1 1
0,1),sin()cos()(
(11)
where
k
and
k
coefficients are computed solving the N equation system, based on the
fact that the values
)( i
tp
are known for the last time moment
i
t
in the price history series.
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Figure 3. Price probability field through Price prediction Line and trigonometric Price Line.
The Trigonometric Price Line is figured in the above picture over the price evolution
with continuous red and blue lines. During the Ph.D. research behind this paper, it was
found that the intersections between the Price Prediction Line and trigonometric Price Line
have an important significance. Once the Trigonometric Price Line values exceed the Price
Prediction Lines values and both functions are increasing, the price will make a significant
up evolution. To define a price probability grow field we will apply the same technique
presented above, considering the cyclicality between the two functions:
100/100 PPLTPLPROPPLTPL
(12)
where
11 iiii PPLTPLPPLTPLPPLTPL
for
0i
with
0
0PPLTPL
(13)
where γ defines the gradient of the PPLTPL function with p period, and
ii
ii
iminmax
max
,
iii PPLTPL
(14)
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and the maximal and minimal functions are computed this time using PPL and TPL:
kk
pi
ik
iPPLTPLmax
max
,
kk
pi
ik
iPPLTPLmax
max
(15)
The price probability field through Price Prediction Line and trigonometric Price
Line is presented in figure 3 for the same example of the DAX30 Index market.
Price Probability Predictor
This research has found that a simple average of the three probability field presented
above, will produce a more stable probability function that is giving us a better capital
efficiency when trading signals are computed using it. This is called the Price Probability
Predictor (PPP) and it is given by the relation:
3/
PPLTPLSHAPCY PROPROPROPPP
(16)
The results of the Price Predictor Probability function for a 200 points chance of price
increase on a daily time price series of DAX30 Index market are presented in figure 3. The n,
m, and p periods of PCY, SHA and PPLTPL functions are also at our disposal to be optimized
for each analyzed market. In figure 4 was used n=m=p=5.
Figure 4. Price probability predictor for DAX30 Index market.
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To prove that the Price Probability Predictor is in a inverse and strong correlation with the
price action, the Paerson correlation coefficient (Andrei, 2003) was computed for a
significant number of capital markets. Values between (-0.999) and (-0.537) were obtaind
for 4-hours and daily timeframes for the next markets: Frankfurt Stock Exchange Deutscher
Aktienindex DAX30, New York Stock Exchange Dow Jones Industrial Average Index DJIA30,
Financial Times London Stock Exchange Index FTSE100, Cotation Assistée en Continu Paris
Stock Exchange Index CAC40, Swiss Market Index SMI20, Standard & Poor’s Index S&P500,
National Association of Securities Dealers Automated Quotations NASDAQ100, Tokyo Stock
Exchange index NIKKEI225, and Australian Security Exchange Index ASX200, spot price of
gold XAUUSD, and brent crude oil. For currency pairs markets, values between (-0.972) and
(-0.518) were obtaind for 4-hours and daily timeframes for all major currencies pairs.
These results confirm that the Price Probability Predictor is in a inverse and strong
correlation with the price tendency. Based on this fact, after a maximum point of the Price
Probability predictor, the price will make a significant up movement.
Automated trading signals and limit conditions
Two types of entry automated trading signals can be built using the Probability Price
Predictor. The first type follows the maximal points made by the PPP function and will
generate one market entry after a maximal point if the PPP values are higher than a
profitability limit level:
iiiiii PPPPPPPPPPPPPPPBuySignal 211
(17)
where λ filter the signals in order to ignore entries when the price growth probability is
lower, in case the PPP function makes a local maximal point at lower values; usually λ=0.5.
The second type of automated trading signals that can be built with the PPP function
is adapted for more volatile markets, initiating multiple entries each time when the price
decrease with a specific δ value:
11 kkiiiik ppPPPPPPPPPBuySignal
for
Nk 1
(18)
where i denotes the time interval, and k denotes the number of opened trade. In formula
(17), the first price entry
1
p
is made for
i
PPP
, hen the market probability riches for the
first time θ value on an ascending probability interval:
iiii PPPPPPPPPBuySignal 11
(19)
In formula (17), N is the maximal number of simultaneously opened trades, having
the purpose of limiting capital exposure; N will make possible risk management. The θ, δ,
and N factors are at the investors' disposal; all can be optimized depending on the traded
market and on the timeframe used for price analysis.
The exit conditions using the Price Probability Predictor function must be related
with the probability field definition considered in the built model. If the probability field
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was computed for a price growth of 200 points, a target higher that this value will have no
sense. Moreover, to be safe of the unpredictable market behavior, using a target lower than
the target considered building the probability field is recommended. As conclusion, the first
exit criteria is a price increase with a Δ distance than the price level met in the maximal
interval of the PPP function, noted here as Ω:
ii pnalExitBuySig
(20)
The exit condition made with formula (20) can be used for one entry made with (17) or for
multiple market entries by formula (18). During this research was found that a better exit
condition is met using the Average True Range (ATR) function introduced in (Wilder,
1978). This will take into consideration the market volatility change during the investment
process. The exit decision can be automated using the relation:
iii ATRpnalExitBuySig
(21)
if ATR value is less than the target considered for the price probability field definition.
Formula (21) will generate exit decisions faster that (20), the capital will stay shorter
allocated in the trades, and the investment efficiency will be significantly improved.
Limit conditions built using the PPP functions can be used to improve any other
investment strategy. Entries made by any mathematical or empirical trading model can be
filtered using the Price Probability Predictor. The capital efficiency can be improved
allowing only entries when the price growth probability is higher than a minimal limit:
ii PPPtionLimitCondi
(22)
Moreover, an increase in the price probability field values will be a positive condition:
iiii PPPPPPPPPtionLimitCondi 1
(23)
In the same idea, exit conditions applied to trades opened with any other trading
strategy can be built when the price growth probability is decreasing under a specific level:
iiii PPPPPPPPPionExitCondit 1
(24)
where the parameters Θ and Ξ can be optimized for each traded market and for each
timeframe used for the price analysis.
Results
In this section, trading results obtained with the signals presented in the last part will be
presented in order to prove the functionality of the method introduces in this paper and to
get the possibility to compare the results with other investment strategies.
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The automated signals made by relation (17) were traded for Frankfurt Stock
exchange Deutscher Aktienindex DAX30 market between 1 January 2019 and 31 December
2019. The Price probability predictor was optimized for a price increase with 200 points
after a maximal probability evolution, using the DAX30 historical time price series between
1 January 2010 and 31 December 2018. The cross-over validation is used in this example,
once the optimization interval does not contain the time interval in which the results are. In
figure 5 is presented the capital evolution due to signals (17), for a particular value of λ=0.5,
using a 4-hours timeframe.
Figure 5. Trading results obtained with (17) automated signals for DAX30 Index market.
In 12 months, signals (17) have obtained a risk to reward ratio of 1:2.64 in a total
number of 39 profitable trades. There was no losing trade, and the most extended trade last
475 hours. The results were obtained with DaxTrader (Păuna, 2010), an automated capital
investment software using the Price Probability Predictor to build automated capital
investment signals. The risk and capital management were assured using the “Global Stop
Loss” method (Păuna, 2018d) with a maximal risk level of 1%.
A significantly large number of trades were made by the signals built with (18)
formula for θ=0.75 and δ=100 points. The results obtained with the same automated
trading software are presented in figure 6. In this case, it was obtained a risk to reward
ratio of 1:2.32, and the longest trade last 522 hours. In this second case, a maximal risk level
of 2% of the available capital was used. In this case, two positions were closed with losses, 2
from 117 trades made, meaning a percentage of success equal with 98,29%.
Figure 6. Trading results obtained with (18) automated signals for DAX30 Index market.
The results above are better understood is both signals are used together. Using the
(17) and (18) automated trading signals for the 2019 period applied for the DAX30 Index
market, with a maximal risk level of 2% of the available capital, a risk to reward ratio of
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1:4.96 was obtained. This is an excellent result, proving that the Price Probability Predictor
is a reliable and profitable model to build trading signals on the capital market.
Another important observation is regarding the specificity of the trades opened with
signals built with Price Probability Predictor. The Quality Trading Coefficient (Păuna,
2018e) of 98% of trades above has a positive and higher value in the interval [0.52;0.96].
This means the trades opened when the probability for a price increase is higher are trades
that are evolving fast and constant to the designed target. The values obtained for the
Quality Trading Coefficient classify the investment methodology using price Probability
predictor in the top trading strategies presented in the Ph.D. research behind this paper.
Conclusions
Price Probability Predictor is a technical indicator built using the time price series and
several particular functions as Price Cyclicality Function, Smoothed Heikin-Ashi price
transformation, Price Prediction Line, and Trigonometric Price Line. An analytical
mathematical model links all of these functions into a continuous probability field,
measuring the chance for price growth in the next time intervals. The model includes
several functional parameters that can be optimized for each market in order to define the
function to indicate the probability of price increase with a specified value.
The Price Probability Predictor can be used to build reliable trading signals and limit
conditions to automate the investment activity. It can be optimized for any analyzed capital
market and for any timeframe used. During this research, the model was tested with
excellent results for a considerable number of financial markets: Frankfurt Stock Exchange
Deutscher Aktienindex DAX30, New York Stock Exchange Dow Jones Industrial Average
Index DJIA30, Financial Times London Stock Exchange Index FTSE100, Cotation Assistée en
Continu Paris Stock Exchange Index CAC40, Swiss Market Index SMI20, Standard & Poor’s
Index S&P500, National Association of Securities Dealers Automated Quotations
NASDAQ100, Tokyo Stock Exchange index NIKKEI225, and Australian Security Exchange
Index ASX200. For more volatile markets as spot gold (XAUUSD) and currency markets, the
signals built with the PPP function must be filtered by additional limit conditions imposed
with the Price Cyclicality Function (Păuna & Lungu, 2018).
Due to how the Price Probability Predictor function is defined and optimized, it
measures the price increase chance. The presented function is also an excellent tool for
educational purposes. When the values of the PPP field are high, the function indicates a
higher price growth probability. Besides, when the PPP function has lower values, it will
mean a lower chance for price growth, but not a higher chance for a price decrease. Based
on this particular observation, any capital trading or investment strategy can be filtered and
improved using the Price Probability Predictors to reduce the allocated risk, to reduce the
time spent into a trade, and, consequently, to increase the capital efficiency.
The Price Probability Predictor function was designed for automated trading
systems. Even the mathematical model is a complex one, the computational power needed
to build the function is a reasonable one. It can be built-in real-time with low latency and
can be included in any automated capital investment software system.
The Price Probability Predictor function has a high degree of efficiency in improving
the results of any other trading strategy. Due to this fact, in the research behind this article,
the PPP function assists any automated capital investment decision. Also, it can be used for
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manual investment decisions. Any investor using the PPP function can simply choose to
invest in a market with an 85% price increase probability and to exit from markets
indicating only a 20% price growth chance or a decreasing probability evolution. With all
the facts presented, the Price Probability Predictor is a reliable investment indicator, giving
us the possibility to measure the chance to record a profit in any market on the next time
interval.
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