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Business Systems Laboratory - 7th International Symposium
SOCIO-ECONOMIC ECOSYSTEMS:
CHALLENGES FOR SUSTAINABLE DEVELOPMENT IN THE DIGITAL ERA
January 22-24, 2020
University of Alicante
Alicante,Spain
Please send to abstract-submission@bslab-symposium.net
Reliable Signals and Limit Conditions
using Trigonometric Interpolation
for Algorithmic Capital Investments
Cristian Păuna
Ph.D. Candidate,
Economic Informatics Doctoral School,
Academy of Economic Studies Affiliation, Romania.
e-mail: cristian.pauna@ie.ase.ro
EXTENDED ABSTRACT
Algorithmic capital investment procedures became the essential tools to make a profit in the
volatile price markets of the 21st century. A large number of market participants, private
traders, companies, or investment funds are buying and selling on thousands of markets every
day to make a profit. After the 2010 year, algorithmic trading systems became a significant part
of the capital investment environment. The price evolution is analyzed today in real-time by
powerful computers. To buy cheap and to sell more expensive is a simple idea, but to put it on
practice is not easy today in very volatile price markets. The orders are built and set almost
instantly today by artificial intelligence software using special mathematical algorithms. These
procedures automatically decide the best moments to buy and to sell on different financial
markets depending on the price real-time movements. This paper will present a specific
methodology to analyze the time price series of any capital market. The model will build reliable
trading signals to enter and to exit the market to make a profit. The presented method uses
trigonometric interpolation of the price evolution to build a significant trend line called here the
Trigonometric Trend Line. It will be mathematically proved that this function is in a positive and
direct correlation with the price evolution. The Trigonometric Trend Line will be used to build
and automate capital investment signals. Besides, the introduced function will be used in order
to qualify the actual price trend and to measure the trend power in order to decide if the price
makes an important evolution or not. Limit conditions will be imposed in the financial market to
avoid trading in non-significant price movement and to reduce the risk and capital exposure.
Comparative trading results obtained with the presented methodology will be included in the last
part of this paper to qualify the model. Each trading signal type presented in the paper was
traded separately to have a qualitative image. Also, all capital trading signals built with the
Trigonometric Trend Line were traded together in order to obtain a better risk to reward ratio.
To classify the presented methodology, the presented results were compared with real trading
profits obtained with the other three well-known capital investment strategies. With all of these,
it was found that using the Trigonometric Trend Line reliable automated trading procedures can
7th International Symposium
“SOCIO-ECONOMIC ECOSYSTEMS "
January 22-24, 2020
University of Alicante , Spain
Please send to: abstract-submission@bslab-symposium.net
be made and optimized for each financial market to obtain good results in the capital investment.
Being exclusively a mathematical model, the Trigonometric Trend Line methodology presented
in this paper can be applied with good results for any algorithmic trading and high-frequency
trading software. The functional parameters can be optimized for each capital market and for
each timeframe used in order to optimize the capital efficiency and to reduce the risk. The
optimization methods will use the historical time price series in order to catch the price behavior
and specificity of each market. The reduced number of parameters and the simplicity of the
presented method recommend the Trigonometric Trend Line model to be used in any advanced
algorithmic trading software.
Keywords: financial markets, capital investment, trigonometric interpolation, algorithmic
trading, automated trading systems.
1. Introduction
Capital investment is a key activity nowadays. A large number of market participants,
traders, investors, companies, private or public investment funds, all are buying and selling on
thousands of free markets, every day, with a single purpose: to make a profit. “Due to the
increasing diversity of financial index-related instruments as well as the economic growth
experienced during the past years, the extent of global investment opportunities for both
individual and institutional investors has broadened.” [1] After the 2010 year, the algorithmic
trading systems became a significant part of the capital investment environment. High-frequency
trading, those procedures which make a large number of trades with small profits, have “gained
some significant attention due to the flash crash in the U.S. on May 6, 2010.” [2] The field of
automated trading systems is developing fast today. Trading and mathematical investment
models are research subjects with high-interest today. This paper will present one of these
models based on trigonometric interpolation.
The reason for everyone is, of course, the profit. To buy cheap and to sell more expensive is
a simple idea, but to put it on practice is not easy today, in very volatile price markets. All capital
investors are searching for answers to the questions when to buy and when to sell? The founded
answers are making the differences between the market participants, between the profit and
losses. A third question became significant in the last time: when to stay away from the market
risk to save the capital? The answer to the last question mark is the most precious one for those
who want to limit their capital exposure and to make a profit with a reduced and controllable
risk. The methodology presented in this paper will answer clearly all these three questions as we
will see later in this article.
The practice and experience of others give us some indications. There are a large number of
papers in literature with trading and investment advice, but few are quite precise. For example,
trading ideas like: “buy the market after it’s dropped; not after it’s risen” [3] can not be
automated. Rules like that can not even be traded by hand. When the market is dropped enough?
Also, when the price is too high in order to change the direction? How to determine all of these
to have a supportable and measurable risk? For these questions, we need precise mathematical
models to put in practice. The method presented in this paper will offer exact answers to all these
questions. The entry and exit conditions will be built through mathematical algorithms, based on
a trigonometric interpolation function. Moreover, all of these conditions can be automated to be
integrated into automated capital investment software.
7th International Symposium
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High price market volatility pushes algorithmic trading and high-frequency trading
procedures to become significant parts of the current research activity. A reliable trading strategy
makes the difference between the profit and the loss. A good trading model is more valuable than
the available capital today. At this moment, algorithmic trading “represents 52% of market order
volume and 64% of nonmarketable limit order volume.” [4]. There are a tremendously high
number of published papers with so-called trading strategies related to capital investment.
Besides, few of them include models that can be used in algorithmic trading in order to automate
the investment decisions. Significant studies and working methods that can be used in
algorithmic trading for the stock markets can be found in [3], [5], [6], and [7]. For the currency
and commodity markets, reliable trading models that can be automated can be found in [8], [9],
and [10]. Original investment models and methodologies especially optimized for automated
capital investment software can be found in [11], [12], [13], [14], [15], and [16]. Essential
psychological strategies that can be used for better adaptation to the context of actual markets
can be found in [17], [18], and [19]. Considering the academic literature related to the trading
strategies used by the algorithmic trading environment, the subject about how the trigonometric
interpolation can be used in order to automate the trading decisions is not treated anymore. This
is the gap filled by the current paper.
A common concept in financial markets is the price trend. Trading in the direction of the
main trend is an essential idea for many working trading strategies. To automate the trading
decisions for these cases, we need first a precise method to identify the current trend and a model
to measure the significance of the current price movements. “Which trend is your friend?” [20].
This question is very actual, especially in highly volatile markets, where long trends are rare
today. One sustained mathematical method to detect if a trend is continuing or not is valuable for
algorithmic trading. The model presented in this paper will permit us to detect the current price
trend, and to estimate the power of it, in direct correlation with the amplitude of the price
movement. As we will see, for not enough powerful trends cases, to stay away from the market
risk is the best idea. This step can also be automated using the model revealed in this paper.
Moreover, the method presented in this paper permits to automate the exit decisions if the current
price trend is finished or if the price is preparing to change its direction. With all of these, the
presented capital investment methodology is a reliable one. To prove this and to compare the
presented model with other algorithmic trading methodologies, capital investment results are
presented in the last part of this article.
2. Trigonometric interpolation
The interpolation is the process to find a function which describes a phenomenon given by
points. Having a measurable variable over the space or overtime, the interpolation process is the
method to find a function which defines accurate enough that phenomenon. Whether it is about a
physical variable as a pressure field in a defined space, the temperature evolution over the time, a
process measurable in different points of its defined space, or a financial process defined as the
evolution of the price in time, the interpolation methods are the same.
The evolution of the price in time is also a phenomenon defined by points. From time to
time, in any financial markets, the price is given by its values. Even the price is delivered in a
repetitive process, and the time intervals are small, in electronic trading, the function price(time)
is not a continuous function; the values for this function are defined from time to time.
We propose in this paper to find a function which describes the evolution of the price in time
based on the interpolation methods. Once we found out this function, a significant amount of
7th International Symposium
“SOCIO-ECONOMIC ECOSYSTEMS "
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information can be considered about the variation of the price in time, even some predictions for
the next intervals. We will have a mathematical method to know if the price is in an ascending or
descending evolution, we can establish the local or global minimal and maximal points, and we
can predict the value of the price for the next time intervals with a measurable error. For this, we
will consider the price as a function defined by points given with:
Nitfp ii ,1),(
(1)
where (pi) are the price values in each (ti) moment of time and (N) is the total number of the
measured points. The function (f) is unknown. We have only the values of this function for each
(ti) moment of time. The approximation process wants to find another function (g), which is
close enough to the function (f). The advantage is that the function (g) will be complete known,
continuous, and mathematically defined in order to be analyzed. The conditions to find the
approximation function (g) are imposed using the known price values:
Niptg ii ,1,)(
(2)
There are multiple methods of interpolation, depending on how the function g is described.
The general purpose is to minimize the error and to have the interpolation function as close as
possible to all the points (1) given as input data. Because we want the function (g) to be as close
as possible to the function (f), a method to find the solution is to minimize the distance between
the two functions. The most popular method for that is the least-squares method. Considering the
error for each point as:
Nitgperr iii ,1,
(3)
the sum of all square errors will be:
min
1
2
N
ii
errS
(4)
To solve the minimum problem defined by relation (4), we have to set the time gradient to
zero for each moment of time. This will define a system with N linear equation with N unknown,
where N is the number of the measured values:
Ni
ttf
err
t
err
err
t
SN
ii
i
i
N
ii
i
i
i
,1,022 11
(5)
To find the function (g), we have to solve this equation system and to solve this system.
Depending on the form of the function
)( ii tfp
, solving the system (5) can have particular
solutions. The function (f) can have many forms. Good results can be met with polynomial
functions, especially for non-volatile markets. Also, logarithmic or exponential functions can be
used. All of these models can give us a good approximation function, but when it is about a very
volatile market, with fast price changes, a new interpolation method was searched to build a
more accurate approximation model for the price evolution. In engineering and physics practice,
7th International Symposium
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January 22-24, 2020
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fast processes are better defined by trigonometric interpolations. Starting from this idea, in this
paper, we apply the trigonometric interpolation to describe the price evolution.
The trigonometric interpolation implies that the approximation function is defined as:
N
k
N
kikiki Nitktktp 1 1
0,1),sin()cos()(
(6)
The idea of the trigonometric interpolation is to reduce the price space to a periodic space
using an orthogonal function set as sin(kx) and cos(kx). The orthogonal property of these
functions is already proved in [18]. The function defined by formula (6) is periodic in the
interval [0; 2π]. For simplicity, we will consider an additional transformation to reduce the
function to the [0; 1] interval [21] using the relation:
12,0,
2 Mj
M
j
ti
(7)
Now, the time moments from the price space are reduced to 2M points in the trigonometric
space. Using this transformation, and the price values that are given as input data, the
approximation function can be known once all αi and βi parameters are computed. “The
trigonometric interpolation is convergent for continuous functions or for a finite discontinued
number“ [21] which gives us enough reasons to use it for an approximation function for the price
evolution. In addition, price behavior can be considerate a periodic one. The price can be
associated with “a wave with variable wavelengths” [16], and the price cyclicality will be well
approximated by the trigonometric interpolation, better than the linear interpolation, especially
for very volatile markets, where fast changes in the price direction are usually met.
An additional condition for the trigonometric interpolation is regarding the number of points
considered. For a low number of points, the approximation will describe a function with slower
variations. A significantly higher number of points will permit to approximate a function with
faster variations. The question is: how much is the minimum number of points in order to have a
good approximation. The answer is given by the Nyquist criteria [21], which said N=2M≥1/Δt
where Δt is the minimum interval in which the price has a significant movement. Having all of
these, in the next chapters will be presented the results obtained for trigonometric interpolation
of the price in financial markets.
Solving the equation system (5) with trigonometric functions (6), after transformation (7) is
a standard numerical task in the space of [0;1] interval. Several numerical methods can also be
found in [21], and they are not the subject of this paper. After solving the equation system, the
function (g) will be completed defined. One important note is that the system (5) for historical
price periods must be solved only one time. Because the price is not changing in the historical
time price series, the values of αi and βi are the same. This is a huge advantage regarding the
computational power needed. In fact, the only value that needs to be calculated every time when
the price is changing are the values of the coefficients αi and βi for the last time period. Starting
from this fact, optimization of the computational procedures is welcome in any automated
trading or investment software system.
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3. Trigonometric Price Line
In figure 1, it is presented the trigonometric interpolation function for a daily evolution of
Frankfurt Stock Exchange Deutscher Aktienindex DAX30 [22] between February and May
2018. We will call this function to be the Trigonometric Price Line (TPL).
Figure 1. Trigonometric Price Line for different numbers of convergence points.
In figure 1, it was drawn the Trigonometric Price Line obtained using 20, respectively 10
convergence points for the trigonometric interpolation in order to see differences. For a more
stable interpolation line, as can be seen, more convergence points should be used. As we can
observe, the TPL function monotony intervals indicate the direction of the price movement.
When the interpolation price line is ascending, the main action of the price is to go up. When the
price line turns into the red zones, the descending price values will form a downtrend. The model
presented can be applied to different timeframes in order to analyze the price evolution on longer
or shorter time intervals. In figure 2, we can see two Trigonometric Price Lines computed for
four hours and weekly timeframe with the same number of convergence points. The data source
for the price evolution presented in figure 1 and figure 2 is [22].
Figure 2. Trigonometric Price Line interpolation applied for different timeframes.
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The number of points used in the trigonometric interpolation model is at our choice. This
can be an optimization criterion depending on the market behavior and on the proposed scope. In
practice, the number of points is between 10 and 40. A lower number of points will invalidate the
Nyquist criterion described in the last chapter. For a small number of points, the Trigonometric
Price Line makes fast movements, proper for a close analysis in very volatile markets. For those
models who manage investments on the large price movements, a higher number of points must
be used.
To trust the trigonometric Trend Line, we also need a mathematical validation. This
confirmation is given by Pearson's correlation coefficient [23] between two statistical series. The
correlation coefficient can be adapted for our case with the formula:
N
ii
N
ii
N
iii TPLTPLppTPLTPLppr 1
2
1
2
1/
(8)
where we have noted with (TPL) the Trigonometric Price Line values and with (p) the average
price values for each time interval. In formula (8)
p
and
TPL
are the average values for p and
TPL for the N considered time intervals number. Computing the correlation coefficient for the
leading stock exchange indices between 2010 and 2019, we have obtained positive values
between 0.551 and 0.999. The analyzed indices were Frankfurt Stock Exchange 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). The
time frames used were 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 for all mentioned capital markets. With these values
obtained for the correlation coefficient, it can be said that a direct and strong correlation between
the price evolution and the Trigonometric Price Line exists.
Analyzing the time price series mentioned above, it was found that on a strong price
movement, the distance between two consecutive points of the Trigonometric Price Line is
higher than the case when the price moves slowly or changes its preferred direction. The gradient
of the Trigonometric Price Line is a good indicator for the amplitude of the price movement, and
it will be assimilated with the power of the current trend:
1
iii TPLTPLTrendPower
(9)
It was found that when the distance between two consecutive points of the TPL function is
decreasing, the power of the trend is decreasing, the price will not make significant movements
in the near future, or it is preparing to change its direction. For the long trends, these cases are
usually met when investors start to close their buy positions to mark the profit. In these intervals,
the price evolution slows, and the trend has big chances to change the direction. This behavior is
also mathematically confirmed using the formula (8). The correlation coefficient that confirms
the formula (9) is in a direct and strong correlation with the near future of the price movement
amplitude is calculated between 0.568 and 0.971 for all markets mentioned above. The
7th International Symposium
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timeframe used was also one hour (1H), four hours (4H), and daily timeframe. Higher values are
obtained for higher timeframes for all the markets.
One important characteristic of the presented model is that the approximation function will
remain the same for old intervals once more intervals are added. For each moment (i), the values
for the Trigonometric Price Line (TPLi) are calculated considering the historical price values
existed at that moment, regarding the formula (6). The power of the trend also remains
unchanged for the historical price periods, no matter the price is moving in the present. This fact
permits a personalized computational optimization; at each price movement, only the current
time interval must be recalculated. Moreover, this will permit to take some decisions in the
current moment of time, based on the trend power computed for the historical intervals behind,
decisions that can also be analyzed after a time, once they remain unchanged.
Figure 3. Difference between the Trigonometric Price Line and polynomial least square line.
To see the differences between the Trigonometric Price Line interpolation and the
polynomial least square interpolation line, we present figure 3. As we can see, the trigonometric
interpolation turns faster than the other function when the trend is changing. This is the main
reason we prefer the trigonometric interpolation for the price movement analysis in the financial
market instead of other interpolation functions. Any delay in the interpolation method will
reduce capital investment efficiency. With the Trigonometric Price Line, we have obtained
reliable results as we can see in the next chapters.
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4. Trading signals
Looking at the presented graphs of the Trigonometric Price Line in figure 1. and figure 2,
the computed function does not seem to be quite self-explanatory when it is about the trading
decisions. The ascending and descending intervals are quite obvious, but how to trade this line?
In order to answer, we will present in this chapter three types of trading signals. We will develop
and explain all these signals to generate buy trades, the most used trades in the stock market. For
sell trades, the trading signals can be built in the same way for those markets where short
positions can be considered. Being exclusively mathematical formulas, all the trading signals
presented in this paper can be automated and easily integrated into automated trading software
systems.
A trading signal is a Boolean variable which is true when a trade can be opened in the signal
conditions defined by the current market price movement, and historical price evolution. These
variables permit to automate the trade decisions and are computed using different functions
derived by the time price series. Looking at the graphs, in our case, when the Trigonometric
Price Line is turning from a descending interval into an ascending evolution, a buy signal can be
considered. We have to mention from the beginning that not all these cases are good to be traded
in order to obtain a reasonable risk. Additional conditions must be met as we will see.
We will note with TPL the Trigonometric Price Line and with (TPLi) the values of this
function for each (i) time interval considered. One case is when one of these values of the TPL
engulfing two, three or more past values from the past. We will call this case as to be an
engulfing trend line signal given by:
NNiiNii TPLTPLTPLTPLTPLTPLBuySignal 121 ...
(10)
where N is the number of the intervals considered in the signal assembly. With other words, after
a descending interval of the Trigonometric Price Line, when the last value exceeded the values
from one, two or more time intervals, this is considered a significant change in the price
behavior, and can be assimilated with a new long trend beginning, at least for a short period of
time, until new market conditions will take place and will change the TPL function monotony
again. From practice, higher values for N will give an indication for a better new long trend. A
too much higher value for N will insert a significant delay and will increase the risk significantly.
The optimal N value can be found using statistical methods applied for each market, and for each
time frame used. This parameter can be optimized for each market using a numerical
optimization method. The “gradient method” [21] is one of those methods who need small
convergence iterations number.
It is important to note that the (10) trading signal conditions are only entry signals, the
conditions to make new trades when the TPL function has changed its monotony, and a new
trend is installed. The exit conditions may vary depending on the trading strategy adopted. Later
in this paper, some exit conditions also based on the Trigonometric Price Line will be presented.
Another trading signal based on the TPL function is regarding the power of the trend. As we
have mentioned, the TPL function gradient, meaning the distance between two consecutive
points of the TPL function is a good measure for the amplitude of the price movements in the
near future. A higher distance will be produced by an important price movement, which is a good
sign for a longer trend. Trading signals using the trend power notion can be built using the next
form:
1iii TPLTPLBuySignal
(11)
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where ξ is a functional parameter that can be optimized for each market in order to reduce the
risk and to optimize capital efficiency. For a significant value of the ξ, the trend will be
considered strong enough in order to open a buy trade. We will call ξ the minimal power of the
trend to open new long trades. The signals made by (11) could be called strong trend signals,
while new trades are opened only if the power of the trend exceeded an imposed value. As
mentioned before, this is only an entry signal on the market; the exit signals will be treated
separately. The optimal value for the ξ will also be found by numerical methods using the
historical time price series of each capital market and for each timeframe used.
The third type of trading signals built with the Trigonometric Price Line is targeting the
cases when the price is under the trend line, but the trend is strong enough in the ascending
period. In these cases, the price made a correction move down, but it will recover after a short
period of time. The TPL gradient gives the indication that a strong long trend is still present even
the price level is lower, and this will give us a prediction for a next up movement of the price.
Entries on a low level of the price in these conditions will be good opportunities for lower-risk
trades, signals that can be automated using the formula:
iiiii TPLpTPLTPLBuySignal 1
(12)
in which (pi) is the current level of the price and δ is the displacement of the price under the
Trigonometric Price Line, a functional parameter in order to optimize the signal for each traded
market. When the parameters ξ and δ are well optimized, the signals (12) will give us very good
entries, with a low risk to reward ratio (RRR) as we will present later in the results chapter. All
of these considerations made with TPL answer now to the question: when to enter the market?
5. Limit conditions
There are several methodologies in order to establish the presence of a trend. The most
known indicator of the price trend is the Average Directional Movement Index (ADX) [24]. In
common practice, a delay is obtained between the time when the price behavior is changed and
the time when the used indicator displays the changes. When it is about a good and steady trend
in a more extended period of time, the ADX can give us some relevant indications. When it is
about to exit the trade in order to prevent the trend change, the ADX and many other indicators
are too slow. In this case, the significant delay increases the risk drastically, and decrease capital
efficiency because of a too late exit from the market. After the price is changing its direction and
the trade is closed too late. In order to avoid these cases, we will use additional limit conditions
impose in the gradient of the trigonometric Price Line.
A good indication for the exit signals can be found using the Trigonometric Price Line.
Starting from the assumption that the distance between two consecutive points of the TPL
function is a good measure for the power of the trend, when this distance is low or it is
decreasing, the current trend approaches to the finish or makes a pause. Measuring the power of
the trend with the presented method in chapter 3, produces no delay. In any moment of time, we
can know if the current trend line goes higher, or if its gradient is slowing down without any
delay. This is a very strong indication to build an exit signal.
The exit conditions can be set considering a limit for the power of the trend. From this
reason, we will call these conditions as to be limit conditions in order to exit the trades or to stay
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away from that market to reduce the risk involved. The limit conditions to limit the trades using
the Trigonometric Price Line can be automated using formula:
1iii TPLTPLExit
(13)
where ψ is a functional parameter to be optimized for each traded market. The ψ parameter is the
minimal power of the trend for which the transactions are kept still open. Under this minimal
value, the current trend is considered not strong enough, and all opened trades will be closed.
Thinking of (11) and (13) together we can say that, with this model, an up-trend is traded if the
power of the trend is higher than ξ value until the power of the trend is decreasing at ψ value.
These parameters are dependent on the traded market and on the used timeframe; they are the
subject of numerical optimization and machine-learning process for each case.
The limit conditions imposed by formula (13) can also be used in order to exit the trades made
by any other trading strategy. Considering that a trend is not strong enough if the power of the
trend is under the ψ value, any other open trade can be closed with this criterion. Besides, this
can be a filtering condition in order not to open new trades, with any other strategy, when the
price trend slows down and the power of the price movement decrease under the ψ. All of these
answer now to the questions: when to exit a trade and when to stay away from the market risk?
6. Results using Trigonometric Price Line
In this section are presented trading results obtained with the signals built with the
Trigonometric Price Line presented above. These results were obtained using DaxTrader [25], an
automated trading system that uses the TPL function in order to generate buy trades for DAX30
index [22]. The results from table 1 were obtained for the period 01.07.2016 – 30.06.2019 using
a fixed target of 10 points for each trade. In addition, the exit conditions (13) were imposed in
order to exit the trades and not to open new trades when these conditions are met.
The DAX30 index was traded as a contract for differences (CFD) with a spread of 1 point.
The exposed capital involved and the risk management were managed using the “Global Slot
Loss Method” [26]. The Trigonometric Price Line was assembled for four-hour timeframe
interval using 20 convergence points. An additional condition was imposed regarding the hourly
intervals of the executed trades between 8:00 and 16:00 coordinated universal time (UTC) in
order to ensure the liquidity on the market.
In table 1. are presented the trading results for: a). engulfing trend line entry signals given by
(10) with N=2 and N=3; b). strong trend line entry signals given by (11) with ξ =30; c). price
under the trend line entry signals given by (12) with ξ=30, δ=5; d). all above signals assembled
together. The limit conditions made by (13) were included for ψ=10.
Figure 4. Capital evolution due to the trades made by Trigonometric Price Line signals.
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Trigonometric
Price Line signals
Number
of trades
Profit
Drawdown
Risk to
reward ratio
Engulfing trend line (10)
42
10,131
5,481
1:1.84
Strong trend line (11)
67
16,160
10,186
1:1.58
Price under trend line (12)
48
11,578
4,133
1:2.80
All signals together
157
37,869
10,186
1:3.71
Table 1. Trading results obtained with Trigonometric Price Line methodology
As we can see in table 1, for all signal types presented, a good parameter set can assure a
positive income with a reasonable risk to reward ratio (RRR). The lowest capital exposure is
obtained with the signals made by (12) when the price in under the trend line and a significant
uptrend is present. All signals assembled together can generate a considerable number of trades
and can constitute a reliable trading solution with a good RRR.
In order to compare the presented capital investment methodology with other known trading
strategies, comparative trading results are included in table 2. These are obtained using
DaxTrader [26] between 01.07.2016 and 30.06.2019 for Frankfurt Stock Exchange DAX30
Index [22] with ten points as the target for each trade. The results obtained using the
Trigonometric Trend Line signals are compared with the ones obtained with “Moving averages
perfect order methodology” [8], “Parabolic stop and reverse methodology” [24] and “Relative
strength index methodology” [6]. Each method was optimized to obtain the best efficiency for
the traded financial markets. In table 2, it can be seen that the signals made with the
Trigonometric Trend Line make a significantly larger number of trades with a higher risk and
reward ratio. These results are an additional confirmation that the presented method gives us a
reliable trading methodology for algorithmic trading.
Trading signals
Number
of trades
Profit
Drawdown
Risk to
reward ratio
All Trigonometric
Price Line signals
157
37,869
10,186
1:3.71
Moving averages
perfect order signals
96
18,045
8,712
1:2.07
Parabolic stop and
reverse signals
102
18,816
6,011
1:3.47
Relative Strength
Index signals
68
16,372
5,901
1:2.77
Table 2. Comparison between Trigonometric Price Line and other known trading methodologies
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7. Combined investment methodologies
In the previous chapter, comparative trading results were presented in order to qualify the
Trigonometric Price Line methodology and to compare it with other known capital investment
strategies that can be automated. Being a very good estimation of the price behavior, the
Trigonometric Price Line can be combined with different other functions in order to obtain
reliable trading signals. A combined investment methodology is presented in this chapter.
Figure 5. Combined signals built with trigonometric Price Line and Price Prediction Line.
In Figure 5, it is presented a time price series of DAX30 Frankfurt Stock Exchange
Deutscher Aktienindex [22] between June and September 2019. Over the price, with blue and red
is drawn the Price Prediction Line introduced by [16], and noted here with PPL. Also over the
price, with pink and purple is drawn the Trigonometric Price Line, introduced in this article in
chapter 3, noted with TPL. As we can observe, when the price changes the main direction, the
two functions are intersecting. Usually, the TPL is over the PPL when the price evolves in an
uptrend. This is the case met on August 2019, a strong and clear buy signal highlighted with a
green circle in Figure 5. Also, when the price changes the evolution into a downtrend, the TPL
will intersect the PPL and will descend faster. The market seems to be near such moment at the
end of September or beginning of October 2019.
As it can be observed in Figure 5, not all the intersections between TPL and PPL are reliable
signals. For example, the intersection met in July 2019 was a false buy signal. Even the TPL
exceeded in values the PPL, because of a strong descending PCY function behavior, this was not
a buy signal. All of these considerations can be automated using the formula:
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ikikiiii PCYPPLTPLPPLTPLBuySignal
(14)
where ξ is the limit of PCY function, having the purpose to filtering the false signals. The
parameter ξ will be optimized using numerical methods for each market and for each timeframe
used. In Formula (14), k is also a functional parameter. Usually, it can be used as k=1, k=2, or
k=3 in order to catch the signals in different volatility market conditions.
With investment signals made by formula (14), applied for DAX30 between 01.07.2016 and
30.06.2019 on the same conditions for results in Table 1 and Table 2 were obtained, the
combined investment methodology presented above made a risk to reward ratio of 1:4.13. This is
a very good result once a capital with a 30% allocated risk will obtain 30:123.9, meaning more
than double in three years of investment.
8. Conclusions
The trigonometric interpolation method presented in this paper builds a price approximation
function called Trigonometric Price Line. It was found that a positive and strong correlation
exists between the price movement and this interpolation function. On the monotony intervals
when the Trigonometric Price Line is ascending, the main tendency of the price is to go up.
These intervals will be assimilated with the uptrend. For the descending intervals of the TPL
function, a downtrend can be considered.
By analyzing the extreme points of the TPL function, we can decide about the price
tendency. After a minimum point of the TPL, an up-trend is present, moments when a buy trade
can be considered. Besides, a descending interval of the TPL is a good indicator short trade on
those markets to close the long positions or to open short positions on those markets where sell
trades can be considered.
It was found that the gradient of the TPL function is a measure assimilated with the power of
the trend. When the distance between two consecutive points of the TPL function is small, the
trend is weak, the amplitude of the price is small. On these cases, the price direction is
undecided, and the trend is preparing to change its behavior or to make a pause. These intervals
will be assimilated with the condition when no significant trend is present. Exit conditions can be
automated, starting from this principle.
When the gradient of the trend line is higher than a specified value, the price will make
important moves. This can be associated with strong long signals if the TPL function is
ascending or with short trading signals if the TPL is descending. In addition, the price distance
from the TPL values can be a good indicator in order to open new trades with lower risk.
All of these considerations can be automated using Boolean variables given by (10), (11),
(12), or (14). These conditions will be repetitively computed, and when one of the signals take
the “true” value, the trading software will build and automatically send the buy or sell orders to
the brokerage informatics system. The exit decisions can also be automated using a fixed target
and limit conditions made by formula (13). The functional parameters included in the trading
signals above will be optimized using numerical algorithms applied on historical time price
series of each market, in order to maximize the profit and to reduce the capital risk. The proper
parameter set will be obtained for each traded market, for each timeframe used. A machine-
learning process can be organized in order to adjust the parameters to the new market behavior.
Looking at the compared results included in Table 2, we can say that the signals obtained
using the Trigonometric Price Line are reliable trading signals. The risk to reward ratio values,
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the number of trades, and the profit obtained with the presented methodology being a good
confirmation for this assumption. Moreover, the Trigonometric Price Line presented in this paper
is a significant part of many other capital investment strategies as it is the one presented in
chapter 7.
The capital investment methodology presented here is exclusively a mathematical model. It
can be applied with good results in any algorithmic trading and high-frequency trading software.
All the limit conditions presented can be used in order to restrict any other trading strategy to
reduce the risk and to optimize capital efficiency. The Trigonometric Price Line methodology
presented in this paper can also be used for manual trading or capital investment.
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