Final Year Project Analysis of Bollinger Bands and Triple ...

Final Year Project

Analysis of Bollinger Bands and Triple

Exponentially Smoothed Average for Hang Seng

Index

Supervisor: Liao, Li Zhi

WANG Jiawei

17250706

Hong Kong Baptist University

Abstract

With the development of stock market and its corresponding theory, nowadays more

and more people try to derive their own trading strategy to make profits. However, can

we really outperform the stock market as a whole in the long run based on their past

trends and some mathematical models? In technical analysis, which assumes that the

movements of stock prices are determined by the forces of demand and supply in the

whole stock market, it shows people that we can predict the future stock’s values and

changes by using its past information and relative mathematical models. In order to test

whether the stock market can be analyzed by the mathematical methods / models and

find out how well their effects are, I will develop a trading strategy by using the

combination of Bollinger Bands (BOLL) and Triple Exponentially Smoothed Average

(TRIX) to simulate the Hang Seng Index through MATLAB. In this trading strategy, I

set different cases to decide the signals for buying in as well as selling out. When I use

this strategy based on the historical closing price of the last 20 years’ Hang Seng Index,

the final outcome is quite well, as the Stock Yield is 7.4145 and the Maximum

Retracement Rate is 0.2734.

Introduction

As the stock market is composed of different types of numerical values and these

numbers have formed lots of unique patterns, it feels natural for us to believe that there

may be some regular patterns within. And with the deepening of people's understanding

of the stock market, nowadays more and more people choose to use mathematical

methods/models to analysis those patterns and wish to find out a perfect trading strategy

for the stocks them have invested. Therefore, in this project, through exploring the

meaning of three different mathematical stock models (BOLL, TRIX, ATR) and

analyzing the connection between them and the Hang Seng Index, I will try to derive

my own trading strategy on Hang Sang Index.

For the setting of this project, I will use daily closing price of the Hang Seng Index to

be the market daily capital, and then manipulate it through trading strategy to get the

personal daily capital. The time length of chosen historical data is from 2000/01/03 to

2020/12/24, and the commission fee I set include a 0.1% stamp tax and a 0.3%

brokerage charge. (In the following report, I will denote the nth day’s closing price as

Close(n), opening price as Open(n), highest price as High(n), and lowest price as

Low(n)).

The first stock model I use is the Bollinger Bands (BOLL), which is created by John

Bollinger and it is based on the principle of statistics. BOLL consists of three lines:

The middle one called Middle Band (MB) is the Simple Moving Average of N days:

𝑀𝐵 =∑ 𝐶𝑙𝑜𝑠𝑒(𝑖)𝑁

𝑖=1

𝑁.

The upper line is called Upper Band (UB) and is k standard deviations (SD) larger than

the Middle Band:

𝑆𝐷 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑁 𝑐𝑜𝑛𝑠𝑒𝑐𝑢𝑡𝑖𝑣𝑒 𝑑𝑎𝑦𝑠′ 𝑐𝑙𝑜𝑠𝑖𝑛𝑔 𝑝𝑟𝑖𝑐𝑒,

𝑈𝐵 = 𝑀𝐵 + 𝑘 ∗ 𝑆𝐷.

The lower line is called Lower Band (LB) and is k standard deviations smaller than the

Middle Band:

𝐿𝐵 = 𝑀𝐵 − 𝑘 ∗ 𝑆𝐷.

According to Bollinger (2000), by using the assumption that the movement of stock

price always revolves around a certain value center (such as moving average, cost line,

etc.) within a certain range, we set Bollinger Bands in order to provide a relatively

specific range for the fluctuate of stock prices. It means that under normal

circumstances, the stock price should always move inside this range. And if the stock

price deviates from the range, it indicates that the market is in an extreme state.

About the functions of each line in BOLL, normally, the Upper Band and Lower Band

are to show the highest and the lowest price of the safe operation of the stock price.

Also, Bollinger (2000) states that sometimes the Lower Band and Middle Band play a

supporting role to the operation of stock price, while the Upper Band and Middle Band

sometimes exert pressure on the operation of stock price, and thus, we will know the

stock price is in a strong (weak) trend if the stock price moves above (below) the Middle

Band.

The second mathematical model I choose is the Triple Exponentially Smoothed

Average (TRIX), which smooth an average three times and then forecasts the long-term

trend of the stock price according to the change of the moving average. The reason that

I decide to combine this model with Bollinger Bands is that, according to Xu (2017),

TRIX can filter the interferences of short-term fluctuations, so as to avoid mistakes and

losses caused by frequent stock operation. Besides this, as TRIX is the most suitable

for the long-term trend of the market, it will do great help to the simulation of last 20

years’ historical data of Hang Seng Index.

Generally speaking, two lines are contained in TRIX model, one is TRIX, the other is

TRMA. The algorithm for each line is as follows: (EMA means Exponential Moving

Average)

𝑇𝑅 = 𝐸𝑀𝐴 (𝐸𝑀𝐴(𝐸𝑀𝐴(𝑁 𝑑𝑎𝑦𝑠′ 𝐶𝑙𝑜𝑠𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒))),

𝑇𝑅𝐼𝑋 =(𝑇𝑅(𝑖) − 𝑇𝑅(𝑖 − 1))

𝑇𝑅(𝑖 − 1),

𝑇𝑅𝑀𝐴 =∑ 𝑇𝑅𝐼𝑋(𝑖)𝑀

𝑖=1

𝑀.

The third stock model I used in the process of deriving my trading strategy (not used in

the final strategy) is the Average True Range (ATR) developed by Welles Wilder, for

which is usually used to stop loss or stopping profit during the stock trading. Carr (2019)

indicates that with a higher the value of ATR, the more likely the trend of stock price

Methodology

To derive a successful trading strategy for the stock index of 20 years, the most

important idea is to avoid the huge trend of sharp fall and keep up with the huge trend

of sharp rise of the stock market. If achieving this, our total revenue will increase

substantially, and the small fluctuations within the market will not affect it generally.

Therefore, in this section, I will explain the basic selling and buying conditions for each

of the three stock models.

Bollinger Bands:

When using Bollinger Bands in a trading strategy, we mainly focus on the relationships

between candlestick chart and each band (MB, UB, LB) within the BOLL. As for the

trading signals, there are mainly 3 kinds of situations.

First, we let k =1 and 2 and set two upper and lower bands, according to Mitchell (2020),

when the stock price is moving between the two upper bands, it means an upward trend

has formed and dealers should buy in and hold. On the contrary, Mitchell (2020) points

out if the price is moving between two lower bands, or if 75% of candlestick’s body is

below the two upper bands as well as the body is red (Close(n) < Open(n)), it indicates

the stock value is going down and dealers should sell out and wait.

The following graph is an example of the first situation: (The red and blue line are UB

and LB when k=2; the yellow lines are UB and LB when k=1)

Second, to ensure the happening of a trend, it feels safer and better to set the trading

condition to be extreme. From the above, we get that in normal situation, the stock price

would move between Upper Band and Lower Band, so here we can set the buying signal

to be the case when the candlestick’s body is above the Upper Band as well as the

closing price of that day is higher than its opening price. And set the selling condition

to be the case that the candlestick’s body is below the Lower Band as well as the

opening price of that day is higher than its closing price.

The following graph is an example of the second situation:

The third trading situation is aimed to solve the time lag problem which also means to

shorten the time distance between the happening of a trend and the trading action. The

reason for this strategy is that the happening of an extreme case is relatively low when

dealing with Bollinger Bands, and it is very hard to predict when the ongoing trend is

going to change after the extreme case (e.g.: the body of candlestick moves outside the

Bollinger Bands) happens. Also, as an extreme case often indicates the ongoing trend

has already last for a while, so if we make trading decision till then, a large amount of

money will be lost if the trend is going down, or we will miss a huge opportunity of

stock price appreciation if the trend is moving up. Apart from this, to avoid the

misjudgments which will cause the problem so called “buy high sell low”, we also need

some safety and lower the risk when decide this trading condition. On these grounds, I

choose to use and reform the basic idea that the stock price will go up (down) if the

daily candlestick charts move from below (above) the Middle Band to above (below)

the Middle Band. The general structure of the improved version is as follows (let today

be day t, and m, n, x, y be constants): Buying Signal: 1: The lowest prices of the last m

days are higher than their Middle Band. 2: The highest prices of the last n days before

the day t-m-1 are lower than their Middle Band. Selling Signal: 1: The highest prices

of the last x days are lower than their Middle Band. 2: The lowest prices of the last y

days before the day t-x-1 are higher than their Middle Band.

The following graph is an example of the third situation:

Besides these trading conditions, the M-Tops pattern and W-Bottoms pattern (Bollinger,

2000) are also very popular and relatively practical signals when using the Bollinger

Bands. However, as these two are too abstract and thus very difficult to build when

derive specific trading strategy, I will not use and explain them in this report.

Triple Exponentially Smoothed Average:

In the introduction, I introduce that there are two lines within the TRIX model: one is

TRIX, and another one is TRMA. During the stock transaction, the general usage of

these two lines mainly focuses on their intersection. As we combine Triple

Exponentially Smoothed Average with Bollinger Bands, we will avoid some false

trading points and thus get a more accurate outcome.

According to Xu and Wang (2017), once TRIX breaks through the TRMA from the

bottom to the top and forms a "golden fork", it indicates that the stock price begins to

enter a strong rising stage, and investors should buy in and hold. On the contrary, once

the TRIX breaks through the TRMA at a high level and forms a "dead fork", it indicates

that the strong rise of stock price has ended, and investors should resolutely sell the

remaining stocks and leave the market in time.

The following graph is an example of the “golden fork” and “dead fork”: (The red line

is the TRIX line; the blue line is the TRMA line)

However, when deriving a trading strategy, it is impossible for us to set an exact date

for the time of these two intersections. Instead, we will use a time range which contains

the intersection to ensure that the intersection of two lines occurs.

For example: (set the time range to be 5, and today is day t)

The buying signal:

𝑇𝑅𝐼𝑋(𝑡 − 6) < 𝑇𝑅𝑀𝐴(𝑡 − 6) && 𝑇𝑅𝐼𝑋(𝑡 − 1) > 𝑇𝑅𝑀𝐴(𝑡 − 1)

The selling signal:

𝑇𝑅𝐼𝑋(𝑡 − 6) > 𝑇𝑅𝑀𝐴(𝑡 − 6) && 𝑇𝑅𝐼𝑋(𝑡 − 1) < 𝑇𝑅𝑀𝐴(𝑡 − 1)

When getting the final strategy in this project, I use TRIX model as an auxiliary

indicator, which means that I only use this trading situation to help to narrow the range

of buying and selling signals set by Bollinger Bands. But I do try to just use TRIX to

simulate the total return, and as its only one trading situation is too broad to detail the

signals, the outcome is not good, for which the final total return is less than

corresponding closing price and the stock yield is -0.2369.

Average True Range:

As I only use this indicator in the process of getting the final trading strategy, I will

only briefly introduce how to use it during the stock transactions, and the reason that I

use it at first is to wish to stop loss from the trading of BOLL and TRIX. According to

Hall (2020), ATR is an indicator that measures the volatility of stock price and is used

to decide when to entry or exiting the stock market or used to stop loss or stopping

profit. The trading signals are as follows: (r is a constant, and today is day t)

The buying condition:

𝐶𝑙𝑜𝑠𝑒(𝑡) > 𝐶𝑙𝑜𝑠𝑒(𝑡 − 1) + 𝑟 ∗ 𝐴𝑇𝑅(𝑡 − 1).

The selling (stop loss) condition:

𝐶𝑙𝑜𝑠𝑒(𝑡) < 𝐶𝑙𝑜𝑠𝑒(𝑡 − 1) − 𝑟 ∗ 𝐴𝑇𝑅(𝑡 − 1).

Numerical Experiment

During the process of getting the final strategy, I have manually tried hundreds of

different combinations of parameters within those selling and buying situations and

choose the best strategy by comparing the total yield of each outcome. In this section,

we will begin with three examples from my deduced process first, and then introduce

the final trading strategy. And for these examples, the general idea of their buying and

selling signals are from the 5 situations I explained before, while their specific

structures are the result of many attempts.

Example 1: BOLL

The first example is one of my best early trading strategies which I only used the

Bollinger Bands to define the trading conditions. I use the Simple moving average for

the Middle Band and set N = 20 and k = 2. Apart from that, by reforming the second

and third trading situations I introduced for Bollinger Bands in the methodology, I

allocate two conditions to each of the buying and selling signal:

Buying signal:

𝐶𝑎𝑠𝑒1 = 𝐿𝑜𝑤(𝑡 − 1) > 𝑈𝐵(𝑡 − 1) ∩ 𝐶𝑙𝑜𝑠𝑒(𝑡 − 1) > 𝑂𝑝𝑒𝑛(𝑡 − 1),

𝐶𝑎𝑠𝑒2 = 𝐶𝑙𝑜𝑠𝑒(𝑡 − 1) < 𝑂𝑝𝑒𝑛(𝑡 − 1) ∩

𝑂𝑝𝑒𝑛(𝑡 − 2) < 𝑀𝐵(𝑡 − 2) ∩

𝑂𝑝𝑒𝑛(𝑡 − 1) > 𝑀𝐵(𝑡 − 1) ∩

𝐶𝑙𝑜𝑠𝑒(𝑡 − 1) > 𝑀𝐵(𝑡 − 1) + (𝑈𝐵(𝑡 − 1) − 𝐿𝐵(𝑡 − 1)

100),

𝐵𝑢𝑦𝑖𝑛𝑔 𝑆𝑖𝑔𝑛𝑎𝑙 = 𝐶𝑎𝑠𝑒1 ∪ 𝐶𝑎𝑠𝑒2.

Selling signal:

𝐶𝑎𝑠𝑒3 = 𝐶𝑙𝑜𝑠𝑒(𝑡 − 1) < 𝑂𝑝𝑒𝑛(𝑡 − 1) ∩ 𝑂𝑝𝑒𝑛(𝑡 − 1) < 𝐿𝐵(𝑡 − 1),

𝐶𝑎𝑠𝑒4 = 𝐻𝑖𝑔ℎ(𝑡 − 1) < 𝐿𝐵(𝑡 − 1) ∩

𝐻𝑖𝑔ℎ(𝑡 − 2) < 𝑀𝐵(𝑡 − 2) ∩

𝐻𝑖𝑔ℎ(𝑡 − 3) < 𝑀𝐵(𝑡 − 3) ∩

𝐿𝑜𝑤(𝑡 − 4) > 𝑀𝐵(𝑡 − 4),

𝑆𝑒𝑙𝑙𝑖𝑛𝑔 𝑆𝑖𝑔𝑛𝑎𝑙 = 𝐶𝑎𝑠𝑒3 ∪ 𝐶𝑎𝑠𝑒4.

The results of the first example are listed on the table:

Number of

Buying

Number of

Selling

Total Yield Maximum

Retracement

Rate

Total Return

/

Closing Price

27 26 1.8945 0.3740 1.7457

From the table, the total number of transactions is quite large, which indicates that I

will lose certain amount of money on the commission fee. The total yield is not high

enough, only 1.8945, and the final total return is 1.7457 times bigger than that day’s

closing price. The maximum retracement rate is a little bit high, and it shows that

according to historical data, my future total revenue may fall by 37.40% at worst.

The following graph shows the total return of stock in 20 years:

Starting from here, in the graph of “The Return of Stock”, the red line indicates the

original closing price of Hang Seng Index in 20 years; the blue line shows the total

return after subtracting the commission fee; and the yellow line indicates the total return

without the commission fee. We can see that from 2000 to around 2011, the blue line

generally avoids the sharp fall in 2008 and keeps up with the rising trend of closing

price in 2004~2007 and 2009~2011. However, ever since the huge increase in 2010, the

overall trend of blue line keeps falling while the general trend of original closing price

keeps moving up. And at the same time, the length between the yellow line and blue

line starts to increase after 2013, which means there are too many useless trading actions

since 2013.

The following graph shows the specific times of buying and selling actions in 20 years:

This trading graph can reflect the specific trading situation more accurately, and if we

combine this picture with the last one, then we can confirm that there are too many

unnecessary buying and selling actions since 2013, for which reduce the total revenue

because of commission fee. Besides this, some trading points appear to “buy high” and

then “sell low”. Also, for certain points, there exist a huge time lag between the

happening of a trend and the corresponding trading action.

Example 2: BOLL+TRIX+ATR

In order to fix the problems in the first example, I need to find a way to make the trading

condition more specific and accurate. Therefore, I add two more stock indicators (TRIX

and ATR) to further narrow the range of buying and selling signal. As TRIX indicator

is through the intersection of two lines to determine the buying and selling time, when

we combine the intersection time in TRIX with the trading conditions in BOLL, it will

make the signals more accurate. And at the same time, with the help of ATR indicator

to stop loss, I think the total revenue will increase.

Based on the general trading conditions I derived in the first example, here I use

Bollinger Bands as a principal model and add two more supporting stock indicators

(TRIX, ATR) to analysis Hang Seng Index. After trying different values for the

parameters in each stock indicator: For BOLL, I use simple moving average to be the

Middle Band and set N = 20 and k = 2.05; for TRIX, I let N = 12 and M = 9; and for

ATR, I set N = 20.

The trading signals are as follows:

Buying signal:



𝐶𝑙𝑜𝑠𝑒(𝑡 − 1) < 𝑂𝑝𝑒𝑛(𝑡 − 1) ∩

𝑂𝑝𝑒𝑛(𝑡 − 2) < 𝑀𝐵(𝑡 − 2) ∩

𝑂𝑝𝑒𝑛(𝑡 − 1) > 𝑀𝐵(𝑡 − 1) ∩


100),

𝐶𝑎𝑠𝑒3 = 𝑇𝑅𝐼𝑋(𝑡 − 6) < 𝑇𝑅𝑀𝐴(𝑡 − 6) ∩ 𝑇𝑅𝐼𝑋(𝑡 − 1) > 𝑇𝑅𝑀𝐴(𝑡 − 1),

𝑆𝑖𝑔𝑛𝑎𝑙𝐵𝑢𝑦 = (𝐶𝑎𝑠𝑒1 ∩ 𝐶𝑎𝑠𝑒3) ∪ (𝐶𝑎𝑠𝑒2 ∩ 𝐶𝑎𝑠𝑒3).

Selling signal:


𝐶𝑎𝑠𝑒5 = 𝐻𝑖𝑔ℎ(𝑡 − 1) < 𝐿𝐵(𝑡 − 1) ∩

𝐻𝑖𝑔ℎ(𝑡 − 2) < 𝑀𝐵(𝑡 − 2) ∩

𝐻𝑖𝑔ℎ(𝑡 − 3) < 𝑀𝐵(𝑡 − 3) ∩

𝐿𝑜𝑤(𝑡 − 4) > 1.05 ∗ 𝑀𝐵(𝑡 − 4),

𝐶𝑎𝑠𝑒6 = 𝑇𝑅𝐼𝑋(𝑡 − 5) > 𝑇𝑅𝑀𝐴(𝑡 − 5) ∩ 𝑇𝑅𝐼𝑋(𝑡 − 1) < 𝑇𝑅𝑀𝐴(𝑡 − 1),

𝐶𝑎𝑠𝑒7 = 𝑂𝑝𝑒𝑛(𝑡 − 1) < 𝑀𝐵(𝑡 − 1) ∩ 𝐶𝑙𝑜𝑠𝑒(𝑡 − 1) < 𝑀𝐵(𝑡 − 1) ∩

𝑂𝑝𝑒𝑛(𝑡 − 2) < 𝑀𝐵(𝑡 − 2) ∩ 𝐶𝑙𝑜𝑠𝑒(𝑡 − 2) < 𝑀𝐵(𝑡 − 2) ∩




𝑂𝑝𝑒𝑛(𝑡 − 6) < 𝑀𝐵(𝑡 − 6) ∩ 𝐶𝑙𝑜𝑠𝑒(𝑡 − 6) < 𝑀𝐵(𝑡 − 6),

𝐶𝑎𝑠𝑒8 = 𝐶𝑙𝑜𝑠𝑒(𝑡 − 1) < 𝐶𝑙𝑜𝑠𝑒(𝑡 − 2) − 2 ∗ 𝐴𝑇𝑅(𝑡 − 2),

𝑆𝑒𝑙𝑙𝑖𝑛𝑔 𝑆𝑖𝑔𝑛𝑎𝑙 = (𝐶𝑎𝑠𝑒4 ∩ 𝐶𝑎𝑠𝑒6) ∪ (𝐶𝑎𝑠𝑒5 ∩ 𝐶𝑎𝑠𝑒6) ∪

(𝐶𝑎𝑠𝑒7 ∩ 𝐶𝑎𝑠𝑒6) ∪ (𝐶𝑎𝑠𝑒8 ∩ 𝐶𝑎𝑠𝑒6).

These different cases are derived from the methodology and first example, as well as

many attempts.

The results of the second example are listed on the table:

Number of

Buying

Number of

Selling

Total Yield Maximum

Retracement

Rate

Total Return

/

Closing Price

11 10 3.8005 0.2734 2.8954

Compared with the first example, the results have improved a lot. Not only the number

of transactions has decreased by more than half, but also the total yield doubled this

time. Besides, the maximum retracement rate decreases from 0.3740 to 0.2734, which

indicates that this trading strategy is a lot safer and stabler than before. From here we

can draw a preliminary conclusion that after adding TRIX and ATR to the BOLL, the

signals do become more accurate.


If we ignore the small declines in the past 20 years, we can see that the total return (blue

line) is increasing in general this time. And it is also very clear that as the signals

become more accurate and the number of transactions decreases, the losses from those

useless buying and selling actions have also significantly reduced, which lead greatly

increased to the total return. Nevertheless, there are still some problems in this trading

strategy. The first one is about the sensitivity and accuracy. For the blue line, too many

unnecessary horizontal line segments are contained within these 20 years, which means

that when the overall trend of the original closing price is moving up slowly, the buying

signal is not triggered, and thus make the total return unchanged. For example, from

2016 to around 2018, the closing price is rising while the line of total return remains

horizontal. The second problem is still about the commission fee. From the graph, at

the end of 2020, the length between the yellow line and blue line is too big to be ignored,

which means that even though we only have 21 transactions, some of them are still

useless.


From the transaction plot, we can observe that there are many useless trading actions

around 2000, 2004, and 2020, and this confirms the second problem mentioned above.

Also, this plot shows us that the buying action around 2006 as well as the selling action

around late 2015 are too slow, and there is no buying during the upward trend starts

from 2016. These phenomena confirm the sensitivity problem of this trading strategy.

Example 3: BOLL+TRIX

To fix the sensitivity and useless trading problems from the second example, my next

step here is to use the third method for Bollinger Bands that I explained in the

methodology to shorten the time lag and thus improve the sensitivity and accuracy.

During the process of finding a way to reduce the unnecessary transactions, while I was

trying to abandon the ATR in order to shorten the conditions for the selling signal, the

outcome improved significantly. Therefore, in this third example, I still choose BOLL

as my main model, but only use TRIX as the supporting indicator. Here, the values of

those parameters within BOLL and TRIX are the same as those in the second example.

I set the trading situations as follows:

Buying signal:



𝐶𝑙𝑜𝑠𝑒(𝑡 − 1) < 𝑂𝑝𝑒𝑛(𝑡 − 1) ∩

𝑂𝑝𝑒𝑛(𝑡 − 2) < 𝑀𝐵(𝑡 − 2) ∩

𝑂𝑝𝑒𝑛(𝑡 − 1) > 𝑀𝐵(𝑡 − 1) ∩


100),

𝐶𝑎𝑠𝑒3 = 𝑂𝑝𝑒𝑛(𝑡 − 1) > 𝑀𝐵(𝑡 − 1) ∩ 𝐶𝑙𝑜𝑠𝑒(𝑡 − 1) > 𝑀𝐵(𝑡 − 1) ∩

𝑂𝑝𝑒𝑛(𝑡 − 2) > 𝑀𝐵(𝑡 − 2) ∩ 𝐶𝑙𝑜𝑠𝑒(𝑡 − 2) > 𝑀𝐵(𝑡 − 2) ∩






𝑂𝑝𝑒𝑛(𝑡 − 8) > 𝑀𝐵(𝑡 − 8) ∩ 𝐶𝑙𝑜𝑠𝑒(𝑡 − 8) > 𝑀𝐵(𝑡 − 8),


𝑆𝑖𝑔𝑛𝑎𝑙𝐵𝑢𝑦 = (𝐶𝑎𝑠𝑒1 ∩ 𝐶𝑎𝑠𝑒4) ∪ (𝐶𝑎𝑠𝑒2 ∩ 𝐶𝑎𝑠𝑒4) ∪ (𝐶𝑎𝑠𝑒3 ∩ 𝐶𝑎𝑠𝑒4).

Selling signal:


𝐶𝑎𝑠𝑒6 = 𝐻𝑖𝑔ℎ(𝑡 − 1) < 𝐿𝐵(𝑡 − 1) ∩

𝐻𝑖𝑔ℎ(𝑡 − 2) < 𝑀𝐵(𝑡 − 2) ∩

𝐻𝑖𝑔ℎ(𝑡 − 3) < 𝑀𝐵(𝑡 − 3) ∩

𝐿𝑜𝑤(𝑡 − 4) > 1.05 ∗ 𝑀𝐵(𝑡 − 4),







𝑂𝑝𝑒𝑛(𝑡 − 6) < 𝑀𝐵(𝑡 − 6) ∩ 𝐶𝑙𝑜𝑠𝑒(𝑡 − 6) < 𝑀𝐵(𝑡 − 6),

𝑆𝑒𝑙𝑙𝑖𝑛𝑔 𝑆𝑖𝑔𝑛𝑎𝑙 = (𝐶𝑎𝑠𝑒5 ∩ 𝐶𝑎𝑠𝑒7) ∪ (𝐶𝑎𝑠𝑒6 ∩ 𝐶𝑎𝑠𝑒7) ∪ (𝐶𝑎𝑠𝑒8 ∩ 𝐶𝑎𝑠𝑒7).

Because extreme situations often take a long time to happen after a major trend begins,

instead of using ATR, this time I focus on specifying the relationships between the

Middle Band and candles in the candlestick chart, like the comparison of closing price

and opening price with corresponding MB.

The results of the third example are listed on the table:

Number of

Buying

Number of

Selling

Total Yield Maximum

Retracement

Rate

Total Return

/

Closing Price

16 15 5.2648 0.2734 3.7787

Although the number of buying and selling has gone up from 21 to 31, the total yield

as well as the total return increase a lot, and the maximum retracement rate remains the

same. From here, we can see the effectiveness and accuracy of the third method for

Bollinger Bands in the methodology.


This total revenue plot reflects us some problems of this trading strategy that the last

table cannot show. First, since late 2010, too many small vibrations in the blue line

which cause the total return falls a lot of times. Second, the blue line is too easy to

fluctuate substantially with the small undulate of the original closing price. Third, the

difference between the yellow line and blue line starts to increase since 2011.


As we take those three problems to the transaction plot, it shows us that despite some

useless transactions around 2000 and 2004, the performance of the first ten years is

quite well. Therefore, we need to draw our attention to the last ten years. We can find

out around 2011 as well as between 2015 and 2020, there are still some transaction

points with the “buy high” and “sell low” situation. These late transactions can not only

cause the blue line to fluctuate, but also create a large amount of unnecessary

commission fee.

Final Trading Strategy: BOLL+TRIX

To solve the late transaction problem in the third example, it is important to focus on

the patterns that happened before a trend has formed. Based on the former trading

signals and the third method for BOLL, I add one more trading case for each of the

buying and selling signal. In these two new cases, I detail the patterns before and during

the change of a trend of the closing price.

In my final trading strategy for Hang Seng Index, I set N = 20, k = 2.05 for Bollinger

Bands, and assign N = 12, M = 9 for TRIX indicator. The trading situations I derived

are as follows:

Buying signal:


𝐶𝑎𝑠𝑒2 = 𝐶𝑙𝑜𝑠𝑒(𝑡 − 5) < 𝑂𝑝𝑒𝑛(𝑡 − 5) ∩ 𝐶𝑙𝑜𝑠𝑒(𝑡 − 1) < 𝑂𝑝𝑒𝑛(𝑡 − 1) ∩

𝑂𝑝𝑒𝑛(𝑡 − 2) < 𝑀𝐵(𝑡 − 2) ∩ 𝑂𝑝𝑒𝑛(𝑡 − 1) > 𝑀𝐵(𝑡 − 1) ∩


100),

𝐶𝑎𝑠𝑒3 = 𝑂𝑝𝑒𝑛(𝑡 − 1) > 𝑀𝐵(𝑡 − 1) ∩ 𝐶𝑙𝑜𝑠𝑒(𝑡 − 1) > 𝑀𝐵(𝑡 − 1) ∩







𝑂𝑝𝑒𝑛(𝑡 − 8) > 𝑀𝐵(𝑡 − 8) ∩ 𝐶𝑙𝑜𝑠𝑒(𝑡 − 8) > 𝑀𝐵(𝑡 − 8),

𝐶𝑎𝑠𝑒4 = 𝐿𝑜𝑤(𝑡 − 1) > 𝑀𝐵(𝑡 − 1) ∩ 𝐿𝑜𝑤(𝑡 − 2) > 𝑀𝐵(𝑡 − 2) ∩

𝐿𝑜𝑤(𝑡 − 3) > 𝑀𝐵(𝑡 − 3) ∩ 𝐿𝑜𝑤(𝑡 − 4) > 𝑀𝐵(𝑡 − 4) ∩

𝐿𝑜𝑤(𝑡 − 5) < 𝑀𝐵(𝑡 − 5) ∩ 𝐶𝑙𝑜𝑠𝑒(𝑡 − 5) < 𝑂𝑝𝑒𝑛(𝑡 − 5) ∩

𝐶𝑙𝑜𝑠𝑒(𝑡 − 3) < 𝑂𝑝𝑒𝑛(𝑡 − 3) ∩ 𝐶𝑙𝑜𝑠𝑒(𝑡 − 2) < 𝑂𝑝𝑒𝑛(𝑡 − 2) ∩

𝐶𝑙𝑜𝑠𝑒(𝑡 − 1) > 𝑂𝑝𝑒𝑛(𝑡 − 1) ∩ 𝐶𝑙𝑜𝑠𝑒(𝑡 − 4) > 𝑂𝑝𝑒𝑛(𝑡 − 4),


𝑆𝑖𝑔𝑛𝑎𝑙𝐵𝑢𝑦 = (𝐶𝑎𝑠𝑒1 ∩ 𝐶𝑎𝑠𝑒5) ∪ (𝐶𝑎𝑠𝑒2 ∩ 𝐶𝑎𝑠𝑒5) ∪


Selling signal:


𝐶𝑎𝑠𝑒7 = 𝐻𝑖𝑔ℎ(𝑡 − 1) < 𝐿𝐵(𝑡 − 1) ∩ 𝐻𝑖𝑔ℎ(𝑡 − 2) < 𝑀𝐵(𝑡 − 2) ∩

𝐻𝑖𝑔ℎ(𝑡 − 3) < 𝑀𝐵(𝑡 − 3) ∩ 𝐿𝑜𝑤(𝑡 − 4) > 1.05 ∗ 𝑀𝐵(𝑡 − 4),







𝐶𝑙𝑜𝑠𝑒(𝑡 − 7) > 𝑀𝐵(𝑡 − 7),







𝐶𝑙𝑜𝑠𝑒(𝑡 − 8) > 𝑀𝐵(𝑡 − 8) ∩ 𝐿𝑜𝑤(𝑡 − 9) > 𝑀𝐵(𝑡 − 9),


𝑆𝑖𝑔𝑛𝑎𝑙𝑆𝑒𝑙𝑙 = (𝐶𝑎𝑠𝑒6 ∩ 𝐶𝑎𝑠𝑒10) ∪ (𝐶𝑎𝑠𝑒7 ∩ 𝐶𝑎𝑠𝑒10) ∪


In these 10 cases, Case 4 and Case 9 are newly derived in the final strategy, while the

general structures of other cases come from the earlier examples.

From the Case 4, I set the lowest price of last 4 days to be higher than their

corresponding Middle Band, which is the trend after the closing price starts to move up.

And I let Low(t-5) < MB(t-5) and Close(t-5) < Open(t-5) to be the patterns happened

during the change of the trend, which means during the time when the candlestick is

trying to break through the Middle Band from bottom to top. And as for the

relationships between the opening prices and closing prices of last 4 days, at first, I set

all their closing prices be larger than opening prices in order to strengthen and narrow

the range for the upper trend. However, as I change Close(t-2) > Open(t-2) and Close(t-

3) > Open(t-3) to Close(t-2) < Open(t-2) and Close(t-3) < Open(t-3) in order to try

different combinations, the result suddenly improves a lot, then I keep this pattern.

From Case 9, in order to define the downward trend after the changes, I set both the

opening prices and closing prices of the last 6 days to be smaller than their

corresponding Middle Band. And for the patterns before the changes, from the third

trading situation for BOLL, we conjecture that all or part of the candlesticks will be

above the Middle Band. And from many manually attempts, I let Close(t-8) > MB(t-8)

and Low(t-9) > MB(t-9).

The results of the final trading strategy are listed on the table:

Number of

Buying

Number of

Selling

Total Yield Maximum

Retracement

Rate

Total Return

/

Closing Price

13 12 7.4145 0.2734 5.0739

From the table, there are 25 trading actions totally, and because the basic structure and

idea are similar for both the buying and selling signal, the number of buying and selling

are nearly the same. As we shorten the time lag and improve the sensitivity, the final

total yield has now increased from 5.2648 to 7.4145, with the maximum retracement

rate remains still at 0.2734. On December 24th, 2020, the total return is 5.0739 times

bigger than that day’s closing price.

The following table shows the yearly stock yield from 2000 to 2020:

2000 2001 2002 2003 2004 2005

-0.0702 0.1047 -0.0455 0.3662 0.0495 0.0449

2006 2007 2008 2009 2010 2011

0.3363 0.3792 0.0618 0.4941 0.0700 -0.0116

2012 2013 2014 2015 2016 2017

0.1974 0.0283 -0.0080 0.0538 0.0129 0.3002

2018 2019 2020 Average Max -Min

0.0205 0.0509 -0.0169 0.1152 0.5643

From 2000 to 2020, the average yearly stock yield is 0.1152, and there are 5 years that

have negative values, which means the total returns at year end are lower than the

corresponding investment capital at the beginning of those years. Although the total

yield is 7.4145, the yearly yield is not stable enough and the maximum range of its

change reaches 0.5643.


By using the final trading strategy, the number of fluctuates and small declines on the

blue line has decrease a lot when we compare it with the third example. Besides, thanks

to the shorter decision time, which leads to more timely buying and selling, the range

of rise on the blue line increases as well as the range of decline shortens. If we look at

the yellow line, it shows that before 2019, the cost on commission fee has also

decreased a lot compared to the former examples, which means the decline of the

number of unnecessary transactions, and once again it indicates the accuracy of this

final strategy.


Despite a few unnecessary transaction points, most of the buying and selling actions

from this graph are both timely and accurate, and follow the so called “buy low and sell

high” standard. So next, I will choose to illustrate some of the representative trading

points from this transaction plot:

For the buying actions start from year 2000, the third buying happens after a relatively

small upward trend has risen by 15.1780%. As for the fourth one, it happens during a

short downward wave, and that means it should be a useless transaction. However, the

time of this buying is right before the big rising trend starts around 2003/04, which

makes the total revenue increase more than 50%. The fifth buying keeps up with the

huge upward trend between 2004/05 and 2007/10. Although this transaction happens

after the closing price has risen by 18.6873%, it is still very timely when we look at the

whole trend. The sixth buying takes place on 2008/12/04, after the closing price has

risen by 8.4270%. This trading action catches the start of the sharp rise between 2008/11

and 2009/12, and leads to a significant increase on the total return. During the period

starts from 2011/12 and ends on 2015/04, the total trend of closing price increases

slowly through many fluctuates, and the seventh buying happens after the closing price

has risen by 3.7383%. As for the ninth buying action, if we choose to look at it through

the perspective of small fluctuates, it is not a good transaction point as it is at the highest

of the fluctuates well as the closing price has risen by 16.2664%. But if we put it in the

general trend, we can see that this transaction keeps up with the sharp rise started from

2016/02. The tenth buying happens after the closing price has risen by 6.3514%, and

the meaning of this transaction is to make up for the selling happened previously. For

the eleventh buying, although it is a little slow, it still keeps up with the small upward

trend after the closing price has risen by 6.7837%.

Now we turn to the selling points within these 20 years. The second selling happens

after the closing price has fall by 8.3474%. Even though 8.3474% is not a small number,

this transaction still let the total return avoid most of the downward trend between

2000/09 and 2001/09 successfully. Next, the performance of the fourth selling is not

good, the reasons are that not only did this transaction happen after the closing price

has fall by 11.0632%, but also the time lag between the start of the trend and the selling

action is too long. As for the fifth selling, it saves the total return from falling sharply.

Because of the sharp decline within a very short time, this selling action happens after

the closing price has fall by 11.0936%. For the eighth selling, same as the last one, due

to the sharp fall of closing price, it happens after the closing price has fall by 12.1351%.

Nevertheless, this selling still stops the loss between 2015/07 and 2016/02 in a way.

The tenth selling action happens after the general downward trend has formed for 2

months as well as the closing price has fall by 7.5415%. Although the result of this

transaction is not bad, the time lag is long, which indicates that it is not sensitive enough

to catch the start. For the eleventh selling, it is to stop the fast decline from 2020/01 to

2020/04, and it happens after the closing price has fall by 9.5664%.

Other Applications

Although the result is good when using my final strategy on Hang Seng Index from

2000/01/03 to 2020/12/24, we do not know whether it will also perform well when the

trading time is changed or in thousands of the other indexes and stock prices. Therefore

in this section, I will use the same final trading strategy to different indexes and different

time period of Hang Seng Index.

SSE Composite Index: (2000/01/03 – 2020/12/21)

The results are listed on the table:

Number of

Buying

Number of

Selling

Total Yield Maximum

Retracement

Rate

Total Return

/

Closing Price

13 13 2.4536 0.5766 1.4791

From the outcome, there are totally 26 transactions throughout the 20 years, and as the

final yield is 2.4536, our revenue is more than twice as much as the beginning capital.

But the huge fluctuations have caused the total daily return (blue line) very unstable

and thus give us a large maximum retracement rate that reaches 0.5766.


By using my trading strategy on 20 years’ SSE Composite Index, although the final

total return is not as good as the result before, the blue line is still higher than the red

line (daily closing price) in general, which means my strategy has outperformed the

market.


As we change the index, the specific structure of my strategy does not match the new

patterns here contained in the SSE Composite Index. A lot of the buying and selling are

too slow to catch the changing trend and not accurate enough, and cause a large amount

of losses.

Nasdaq Composite Index: (2000/01/03 – 2020/12/23)


Number of

Buying

Number of

Selling

Total Yield Maximum

Retracement

Rate

Total Return

/

Closing Price

15 14 1.9428 0.5317 0.8958



When using the strategy on 20 year’s historical data of Nasdaq Composite Index, the

general pattern of the blue line (total daily return) after 2004 is similar to that of the red

line (daily closing price), and it helps the total daily return to keep up with the general

rising trend of the closing price after around 2009 and thus makes the final total yield

reaches 1.9428. However, not only the blue line does not outperform the market this

time, but also there are lots of useless selling and buying actions from the transaction

plot, which indicates that the specific structure and parameters of this strategy does not

applicable to this Nasdaq Composite Index.

Dow Jones Industrial Average: (2000/01/03 – 2020/12/23)


Number of

Buying

Number of

Selling

Total Yield Maximum

Retracement

Rate

Total Return

/

Closing Price

14 13 1.1917 0.2971 0.8331



For applications on Dow Jones Industrial Average, which is also an US stock index, the

results are also not so well. The total yield is 1.1917, but the final capital return is lower

than its corresponding closing price. From its transaction plot, despite I have solved the

accurate and sensitivity problem when deriving this strategy in Hang Seng Index, there

are still many false buying and selling actions as we change to a different index data.

Hang Seng Index: (1987/03/13 – 2020/12/24)


Number of

Buying

Number of

Selling

Total Yield Maximum

Retracement

Rate

Total Return

/

Closing Price

21 20 19.1335 0.5312 2.1250



In my prior numerical experiments, I use the historical data of Hang Seng Index from

2000/01/03 to 2020/12/24 to derive my trading strategy. And here, I try my strategy on

the whole historical data start from 1987. Compared to the results that starts from 2000,

because the starting capital becomes lower this time, the total yield has increased a lot

from 7.4145 to 19.1335. There are totally 41 transactions and the maximum retracement

rate has gone up to 0.5312, and it means that as the time period starts to increase, the

risk and instability of the investment has also become bigger. And if we look at the final

total return on 2020/12/24, it has decrease from 5.0739 times bigger than that day’s

closing price to 2.1250 times, and the reason for this decrease may be that, from the

return plot, the blue line has never outperformed the red line of closing price for the

first 22 years. As we dig further into the transaction plot, we can see lots of transactions

before 2001 are not accurate enough and “buy at a high price” as well as “sell at a low

price”.

Hang Seng Index: (2010/01/04 – 2020/12/24)


Number of

Buying

Number of

Selling

Total Yield Maximum

Retracement

Rate

Total Return

/

Closing Price

8 7 0.8330 0.1939 1.4934



After trying the whole time period for HSI, here I start from the 2010/01/04. Because

of the relatively short time period, the final outcome is not as high as the last one, the

total yield is only 0.8330. But from the return plot, the daily capital line generally avoids

the huge downward trend and rises with the upward trend. And for the transaction plot,

apart from the first and fourth buying as well as the second and fourth selling, the rest

of the trading actions are basically accurate.

Conclusion

With the rapid development of the stock market and the maturity of the stock trading

theory, more and more investors begin to make profits by using mathematical methods

and models to analyze the stock market. In this paper, I began with a basic introduction

of the project settings, and then for the three stock indicators: Bollinger Bands, Triple

Exponentially Smoothed Average and Average True Range, I introduced the usage of

each of them and gave out their general mathematical forms.

I showed and explained five fundamental methods to determine the trading signals by

using these three stock indicators: three methods for BOLL, one method for TRIX, and

one method for ATR. Each of these methods can be derived into many different and

specific buying and selling conditions. And then, there are hundreds of ways to form

these conditions into different combinations, and each combination will have a unique

effect on the stock data.

Before getting the final trading strategy, I discussed three examples to show the process

of deriving the best buying and selling conditions. In these examples, I analyzed the

plots of each result, and summarized their advantages, shortcomings as well as how to

improve them. Finally, I derive a trading strategy that can make the stock yield reach

7.4145 and the maximum retracement rate reach 0.2734 when used in the 20 years’

historical data of Hang Seng Index. This trading strategy can avoid a lot of downward

trend within these 20 years and keep up with the most upward trends.

At last, I used my final strategy into three other different indexes and two different time

period of Hang Seng Index. Although their outcomes were not as good as the one when

using the strategy to Hang Seng Index from 2000/01/03 to 2020/12/24, the total yields

of them still bigger than zero as well as the total daily capital could keep up with the

major upward trend of the closing price.

Source Code (MATLAB)

% When to start to extract the data:

StartingDateNumber = 3221;

%% Extract historical data of Hang Seng Index (HSI):

opts = delimitedTextImportOptions("NumVariables", 7);

opts.DataLines = [StartingDateNumber, Inf];

opts.Delimiter = ";";

opts.VariableNames = ["Date", "Open", "High", "Low", "Close", "Var6",

"Var7"];

opts.SelectedVariableNames = ["Date", "Open", "High", "Low", "Close"];

opts.VariableTypes = ["datetime", "double", "double", "double",

"double", "string", "string"];

opts.ExtraColumnsRule = "ignore";

opts.EmptyLineRule = "read";

opts = setvaropts(opts, ["Var6", "Var7"], "WhitespaceRule",

"preserve");

opts = setvaropts(opts, ["Var6", "Var7"], "EmptyFieldRule", "auto");

opts = setvaropts(opts, "Date", "InputFormat", "dd/MM/yyyy");

% Import the data

HangSengIndex = readtable("C:\Users\xianc\OneDrive\桌面\^HSI (1).csv",

opts);

% Historical data extraction of Hang Seng Index completed

% Extraction of Date, Closing Price, Opening Price, Highest Price,

Lowest Price from HSI:

Close = HangSengIndex.Close;

Date = HangSengIndex.Date;

Open = HangSengIndex.Open;

High = HangSengIndex.High;

Low = HangSengIndex.Low;

% Establishing the Bollinger Bands (N = 20, K = 2.05):

N = 20;

kUP = 2.05;

kDOWN = 2.05;

MovingAverage = zeros(length(Close),1);

StandardDeviations = zeros(length(Close),1);

UP = zeros(length(Close),1);

DOWN = zeros(length(Close),1);

for t = N:length(Close)

MovingAverage(t) = mean(Close(t-N+1:t));

StandardDeviations(t) = (((Close(t-N+1)-

MovingAverage(t)).^2+(Close(t-N+2)-MovingAverage(t)).^2+(Close(t-

N+3)-MovingAverage(t)).^2+(Close(t-N+4)-











MovingAverage(t)).^2+(Close(t-N+20)-MovingAverage(t)).^2)/20).^0.5;

UP(t) = MovingAverage(t)+kUP*StandardDeviations(t);

DOWN(t) = MovingAverage(t)-kDOWN*StandardDeviations(t);

end

% Establishing the Triple Exponentially Smoothed Average (N = 12, M =

9):

Period1 = 12;

Period2 = 9;

EMA1 = zeros(length(Close),1);

EMA2 = zeros(length(Close),1);

TR = zeros(length(Close),1);

TRIX = zeros(length(Close),1);

TRMA = zeros(length(Close),1);

EMA1(1)=Close(1);

for t=2:length(Close)

EMA1(t)=Close(t)*(2/(Period1+1))+EMA1(t-1)*((Period1-

1)/(Period1+1));

end

EMA2(1)=EMA1(1);


EMA2(t)=EMA1(t)*(2/(Period1+1))+EMA2(t-1)*((Period1-

1)/(Period1+1));

end

TR(1)=EMA2(1);


TR(t)=EMA2(t)*(2/(Period1+1))+TR(t-1)*((Period1-1)/(Period1+1));

end

for t = 2:length(Close)

TRIX(t) = ((TR(t)-TR(t-1))/TR(t-1))*100;

end

for t = Period2+1:length(Close)

TRMA(t) = mean(TRIX(t-Period2+1:t));

end

% Draw the TRIX:

figure(1);

plot(Date,TRIX,'r');

hold on;

plot(Date,TRMA,'b');

datetick('x','yyyymmdd');

xlabel('Date');

ylabel('TRIX & TRMA');

title('TRIX Indicator');

hold off;

% Total initial capitial:

InitialCapital = Close(N-1);

% Set position: pos = 1 means hold stock，pos = 0 means hold money:

pos = zeros(length(Close),1);

% Set Total Returen and Total Return with no tax:

TotalReturn = zeros(length(Close),1);

TotalReturnNoTax = zeros(length(Close),1);

% Draw and mark the specific date of buying and selling:

figure(3);

plot(Date,Close,'r');


xlabel('Date');

ylabel('Close Price');

title('Time Series of Stock');

hold on;

% Trading Strategy：

for t = N:length(Close)

% Set different cases for trading strategy：

% Cases for Bollinger Bands:

Case1 = Low(t-1)>UP(t-1) && Close(t-1)>Open(t-1);

Case2 = Close(t-1)<Open(t-1) && Open(t-1)<DOWN(t-1);

Case2a = High(t-1)<DOWN(t-1) && High(t-2)<MovingAverage(t-2) &&

High(t-3)<MovingAverage(t-3) && Low(t-4)>1.05*MovingAverage(t-4);

Case2b = Open(t-1)<MovingAverage(t-1) && Close(t-

1)<MovingAverage(t-1) && Open(t-2)<MovingAverage(t-2) && Close(t-





6)<MovingAverage(t-6) && Close(t-7)>MovingAverage(t-7);

Case2c = Open(t-1)<MovingAverage(t-1) && Close(t-






6)<MovingAverage(t-6) && Close(t-8)>MovingAverage(t-8) && Low(t-

9)>MovingAverage(t-9);

Case3b = Close(t-5)<Open(t-5) && Close(t-1)<Open(t-1) && Open(t-

2)<MovingAverage(t-2) && Open(t-1)>MovingAverage(t-1) && Close(t-

1)>MovingAverage(t-1)+((UP(t-1)-DOWN(t-1))/100);

Case3c = Open(t-1)>MovingAverage(t-1) && Close(t-

1)>MovingAverage(t-1) && Open(t-2)>MovingAverage(t-2) && Close(t-







8)>MovingAverage(t-8);

Case3d = Low(t-1)>MovingAverage(t-1) && Low(t-2)>MovingAverage(t-

2) && Low(t-3)>MovingAverage(t-3) && Low(t-4)>MovingAverage(t-4) &&

Low(t-5)<MovingAverage(t-5) && Close(t-5)<Open(t-5) && Close(t-

3)<Open(t-3) && Close(t-2)<Open(t-2) && Close(t-1)>Open(t-1) &&

Close(t-4)>Open(t-4);

% Cases for TRIX:

Case7 = TRIX(t-6)<TRMA(t-6) && TRIX(t-1)>TRMA(t-1);

Case8 = TRIX(t-5)>TRMA(t-5) && TRIX(t-1)<TRMA(t-1);

% Set buying and selling signal：

SignalBuy= ((Case1 == 1 && Case7 == 1) || (Case3b == 1 && Case7 ==

1) || (Case3c == 1 && Case7 == 1) || (Case3d == 1 && Case7 == 1));

SignalSell = ((Case2 == 1 && Case8 == 1) || (Case2a == 1 && Case8

== 1) || (Case2b == 1 && Case8 == 1) || (Case2c == 1 && Case8 == 1));

if (SignalBuy==1 && pos(t-1)==0)

pos(t) = 1;

text(Date(t),Close(t),'\leftarrowbuy');

plot(Date(t),Close(t),'go');

elseif (SignalSell==1 && pos(t-1)==1)

pos(t) = 0;

text(Date(t),Close(t),'\leftarrowsell');

plot(Date(t),Close(t),'bo');

else

pos(t) = pos(t-1);

end

end

% Total Return minus the commission fee: 0.1% stamp tax (only happen

when selling the stock); 0.3% brokerage charge:

TotalReturn(N-1) = InitialCapital;

for t=N:length(Close)

if pos(t)==0 && pos(t-1)==0

TotalReturn(t) = TotalReturn(t-1);

continue;

end

% Buy in

if pos(t)==1 && pos(t-1)==0

TotalReturn(t) = TotalReturn(t-1)*(1-0.003);

continue;

end

if pos(t)==1 && pos(t-1)==1

TotalReturn(t) = TotalReturn(t-1)*(Close(t)/Close(t-1));

continue;

end

% Sell out

if pos(t)==0 && pos(t-1)==1

TotalReturn(t) = TotalReturn(t-1)*(Close(t)/Close(t-1))*(1-

0.004);

continue;

end

end

% Total return with no commission fee:

TotalReturnNoTax(N-1) = InitialCapital;

for t=N:length(Close)

if pos(t)==0 && pos(t-1)==0

TotalReturnNoTax(t) = TotalReturnNoTax(t-1);

continue;

end

% Buy in

if pos(t)==1 && pos(t-1)==0

TotalReturnNoTax(t) = TotalReturnNoTax(t-1);

continue;

end

if pos(t)==1 && pos(t-1)==1

TotalReturnNoTax(t) = TotalReturnNoTax(t-1)*(Close(t)/Close(t-

1));

continue;

end

% Sell out

if pos(t)==0 && pos(t-1)==1

TotalReturnNoTax(t) = TotalReturnNoTax(t-1)*(Close(t)/Close(t-

1));

continue;

end

end

hold off;

% Draw the position:

figure(4);

plot(Date,pos,'b');


xlabel('Date');

ylabel('The state of your account');

InitialCapitalLine = Close;

% One red line for closing price; One blue line for total return; One

yellow line for total return with no tax:

figure(5);

plot(Date,InitialCapitalLine,'r');

hold on

plot(Date,TotalReturn,'b');

hold on

plot(Date,TotalReturnNoTax,'y');


xlabel('Date');

ylabel('Total Return');

title('The Return of Stock');

% Stock Yield:

StockYield = (TotalReturn(length(Close))-

InitialCapital)/InitialCapital

% Maximum Retracement Rate with its date:

RetracementRate = zeros(length(Close),1);

for t1 = N:length(Close)

for t2 = N:t1

RetracementRate(t1) = (max(TotalReturn(N:t1))-

TotalReturn(t1))/max(TotalReturn(N:t1));

end

end

[m,index]=max(RetracementRate);

DateNumber = Date(index)

MaximumRetracementRate = max(RetracementRate)

Reference

Bollinger, J. (2000). Bollinger on Bollinger Bands. Blacklick, USA: McGraw-Hill

Professional Publishing.

Mitchell, C. (2020). Using Bollinger Bands to Gauge Trends. Investopedia Stock

Analysis. Retrieved November 16, 2020, from

https://www.investopedia.com/trading/using-bollinger-bands-to-gauge-trends/

Staff, I. (2020). Advantages of Triple Exponential Average (TRIX). Investopedia Stock

Analysis. Retrieved August 5, 2020, from

https://www.investopedia.com/articles/technical/02/092402.asp

徐英, & 王素霞. (2017). MACD和 TRIX在股票投资中的应用。泰山学院学报，

39(3), 40-43.

Chen, J. (2019). Triple Exponential Average (TRIX). Investopedia Stock Analysis.

Retrieved July 21, 2019, from

https://www.investopedia.com/terms/t/trix.asp

Carr, M. (2019). Measure Volatility With Average True Range. Investopedia Stock

Analysis. Retrieved November 8, 2019, from

https://www.investopedia.com/articles/trading/08/average-true-range.asp

Hall, M. (2020). Enter Profitable Territory With Average True Range. Investopedia

Stock Analysis. Retrieved September 2, 2020, from

https://www.investopedia.com/articles/trading/08/atr.asp

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