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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).

DOI:, pp. , ISSN 2558-9652| Proceedings of the 14th International Conference on Business Excellence 2020

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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).

DOI:, pp. , ISSN 2558-9652| Proceedings of the 14th International Conference on Business Excellence 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.

DOI:, pp. , ISSN 2558-9652| Proceedings of the 14th International Conference on Business Excellence 2020

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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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