A hybrid procedure with feature selection for resolving stock/futures price forecasting problems

Abstract

Stock/futures price forecasting is an important financial topic for individual investors, stock fund managers, and financial analysts and is currently receiving considerable attention from both researchers and practitioners. However, the inherent characteristics of stock/futures prices, namely, high volatility, complexity, and turbulence, make forecasting a challenging endeavor. In the past, various approaches have been proposed to deal with the problems of stock/futures price forecasting that are difficult to resolve by using only a single soft computing technique. In this study, a hybrid procedure based on a backpropagation (BP) neural network, a feature selection technique, and genetic programming (GP) is proposed to tackle stock/futures price forecasting problems with the use of technical indicators. The feasibility and effectiveness of this procedure are evaluated through a case study on forecasting the closing prices of Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) futures of the spot month. Experimental results show that the proposed forecasting procedure is a feasible and effective tool for improving the performance of stock/futures price forecasting. Furthermore, the most important technical indicators can be determined by applying a feature selection method based on the proposed simulation technique, or solely on the preliminary GP forecast model.

Appendix: Descriptions and definitions of technical indicators used in this study

1.1 Notations

i :

The day i

${\text{HP}}_{i}$ :

The highest price of day i

${\text{LP}}_{i}$ :

The lowest price of day i

${\text{OP}}_{i}$ :

The opening price of day i

${\text{CP}}_{i}$ :

The closing price of day i

${\text{TV}}_{i}$ :

The trade volume of day i

• 1. 10-day moving average

  The 10-day moving average is the mean price of the futures over the most recent 10 days and is calculated by:

  $$ {\text{MA}}\_10_{i} = \frac{{\sum\nolimits_{j = i - 9}^{i} {{\text{CP}}_{j} } }}{10}. $$

  (18)

• 2. 20-day bias

  The 20-day bias is the deviation between the closing price and the 20-day moving average (MA_20) and is calculated by:

  $$ {\text{BIAS}}\_20_{i} = \frac{{{\text{CP}}_{i} - {\text{MA}}\_20_{i} }}{{{\text{MA}}\_20_{i} }} . $$

  (19)

• 3. Moving average convergence/divergence

  The moving average convergence/divergence is a momentum indicator that shows the relationship between two moving averages. First, define the demand index (DI) as:

  $$ {\text{DI}}_{i} = {{\left( {{\text{HP}}_{i} + {\text{LP}}_{i} + 2 \times {\text{CP}}_{i} } \right)} \mathord{\left/ {\vphantom {{\left( {{\text{HP}}_{i} + {\text{LP}}_{i} + 2 \times {\text{CP}}_{i} } \right)} 4}} \right. \kern-\nulldelimiterspace} 4} . $$

  (20)

  Next, define the 12-day exponential moving average (EMA_12) and 26-day exponential moving average (EMA_26) as:

  $$ {\text{EMA}}\_12_{i} = \frac{11}{13} \times {\text{EMA}}\_12_{i - 1} + \frac{2}{13} \times {\text{DI}}_{i} $$

  (21)

  and

  $$ {\text{EMA}}\_26_{i} = \frac{25}{27} \times {\text{EMA}}\_26_{i - 1} + \frac{2}{27} \times {\text{DI}}_{i}, $$

  (22)

  respectively. Then, the difference between EMA_12 and EMA_26 can be calculated by:

  $$ {\text{DIF}}_{i} = {\text{EMA}}\_12_{i} - {\text{EMA}}\_26_{i} . $$

  (23)

  Hence, the moving average convergence/divergence can be defined by:

  $$ {\text{MACD}}_{i} = \frac{8}{10} \times {\text{MACD}}_{i - 1} + \frac{2}{10} \times {\text{DIF}}_{i} . $$

  (24)

• 4. 9-day stochastic indicator K

  The 9-day stochastic indicator K is defined as:

  $$ \text{K}\_9_{i} = \frac{2}{3} \times \text{K}\_9_{i - 1} + \frac{1}{3} \times \frac{{{\text{CP}}_{i} - {\text{LP}}\_9_{i} }}{{{\text{HP}}\_9_{i} - {\text{LP}}\_9_{i} }} \times 100. $$

  (25)

  where ${\text{LP}}\_9_{i}$ and ${\text{HP}}\_9_{i}$ are the lowest and highest prices of the previous 9 days, i.e., days i, i − 1, …, i − 7, and i − 8, respectively.

• 5. 9-day stochastic indicator D

  The 9-day stochastic indicator D is defined as:

  $$ \text{D}\_9_{i} = \frac{2}{3} \times \text{D}\_9_{i - 1} + \frac{1}{3} \times {\text K}\_9_{i} . $$

  (26)

  where $\text{K}\_9_{i}$ is the 9-day stochastic indicator K of day i, as previously defined.

• 6. 9-day Williams overbought/oversold index

  The 9-day Williams overbought/oversold index is a momentum indicator that measures overbought and oversold levels and is calculated by:

  $$ {\text{WMS}}\% \text{R}\_9_{i} = \frac{{{\text{HP}}\_9_{i} - {\text{CP}}_{i} }}{{{\text{HP}}\_9_{i} - {\text{LP}}\_9_{i} }} $$

  (27)

  where ${\text{LP}}\_9_{i}$ and ${\text{HP}}\_9_{i}$ are the lowest and highest prices of the previous 9 days, i.e., days i, i − 1, …, i − 7, and i − 8, respectively.

• 7. 10-day rate of change

  The 10-day rate of change measures the percent changes of the current price relative to the price of 10 days ago and is calculated by:

  $$ {\text{ROC}}\_10_{i} = \frac{{{\text{CP}}_{i} - {\text{CP}}_{i - 10} }}{{{\text{CP}}_{i - 10} }} \times 100. $$

  (28)

• 8. 5-day relative strength index

  The relative strength index is a momentum oscillator that compares the magnitude of recent gains to the magnitude of recent losses. First, define the gain of day i as:

  $$ \text{G}_{i} = \left\{ {\begin{array}{*{20}c} {{\text{CP}}_{i} - {\text{CP}}_{i - 1} } \hfill & {{\text{if}}\quad {\text{CP}}_{i} > {\text{CP}}_{i - 1} } \hfill \\ 0 \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right..$$

  (29)

  Similarly, the loss of day i is calculated by:

  $$ \text{L}_{i} = \left\{ {\begin{array}{*{20}c} {{\text{CP}}_{i - 1} - {\text{CP}}_{i} } \hfill & {{\text{if}}\quad {\text{CP}}_{i} < {\text{CP}}_{i - 1} } \hfill \\ 0 \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right..$$

  (30)

  Next, the 5-day average gain (AG_5) and 5-day average loss (AL_5) can be calculated by:

  $$ {\text{AG}}\_5_{i} = \frac{4}{5} \times {\text{AG}}\_5_{i - 1} + \frac{1}{5} \times \text{G}_{i} $$

  (31)

  and

  $$ {\text{AL}}\_5_{i} = \frac{4}{5} \times {\text{AL}}\_5_{i - 1} + \frac{1}{5} \times \text{L}_{i} , $$

  (32)

  respectively. Hence, the 5-day relative strength index can be defined by:

  $$ {\text{RSI}}\_5_{i} = \frac{{{\text{AG}}\_5_{i} }}{{{\text{AG}}\_5_{i} + {\text{AL}}\_5_{i} }} \times 100. $$

  (33)

• 9. 24-day commodity channel index

  The commodity channel index is used to identify cyclical turns in commodities. First, define the typical price (TP) as:

  $$ {\text{TP}}_{i} = \frac{{{\text{HP}}_{i} + {\text{LP}}_{i} + {\text{CP}}_{i} }}{3}. $$

  (34)

  Next, calculate the 24-day simple moving average of the typical price (SMATP_24) by:

  $$ {\text{SMATP}}\_24_{i} = \frac{{\sum\nolimits_{j = i - 23}^{i} {{\text{TP}}_{j} } }}{24}. $$

  (35)

  Then, the 24-day mean deviation (MD_24) can be calculated by:

  $$ {\text{MD}}\_24_{i} = \frac{{\sum\nolimits_{j = i - 23}^{i} { | {\text{TP}}_{j} - {\text{SMATP}}\_24_{i} |} }}{24}. $$

  (36)

  Hence, the 24-day commodity channel index can be defined as:

  $$ {\text{CCI}}\_24_{i} = \frac{{{\text{TP}}_{i} - {\text{SMATP}}\_24_{i} }}{{0.015 \times {\text{MD}}\_24_{i} }}. $$

  (37)

• 10. 26-day volume ratio

  The 26-day volume ratio is defined by:

  $$ {\text{VR}}\_26_{i} = \frac{{{{{\text{TVU}}\_26_{i} - {\text{TVF}}\_26_{i} } \mathord{\left/ {\vphantom {{{\text{TVU}}\_26_{i} - {\text{TVF}}\_26_{i} } 2}} \right. \kern-\nulldelimiterspace} 2}}}{{{{{\text{TVD}}\_26_{i} - {\text{TVF}}\_26_{i} } \mathord{\left/ {\vphantom {{{\text{TVD}}\_26_{i} - {\text{TVF}}\_26_{i} } 2}} \right. \kern-\nulldelimiterspace} 2}}} \times 100\% . $$

  (38)

  where ${\text{TVU}}\_26_{i}$, ${\text{TVD}}\_26_{i}$, and ${\text{TVF}}\_26_{i}$ represent the total trade volumes of stock prices rising, falling, and holding, respectively, from the previous 26 days, i.e., days i, i − 1, …, i – 24, and i − 25.

• 11. 13-day psychological line

  The psychological line is a volatility indicator based on the number of time intervals that the market was up during the preceding period. The 13-day psychological line is defined by:

  $$ {\text{PSY}}\_13_{i} = \frac{{{\text{TDU}}\_13_{i} }}{13} \times 100\% . $$

  (39)

  where ${\text{TDU}}\_13_{i}$ is the total number of days with regard to stock price rises of the previous 13 days, i.e., days i, i − 1, …, i – 11, and i − 12.

• 12. 14-day plus directional indicator

  First, define plus directional movement (+DM) and minus directional movement (−DM) as:

  $$ + {\text{DM}}_{i} = {\text{HP}}_{i} - {\text{HP}}_{i - 1} $$

  (40)

  and

  $$ - {\text{DM}}_{i} = {\text{LP}}_{i - 1} - {\text{LP}}_{i}, $$

  (41)

  respectively. The plus true directional movement (+TDM) can be calculated by:

  $$ + {\text{TDM}}_{i} = \left\{ {\begin{array}{*{20}c} { + {\text{DM}}_{i} } \hfill & {{\text{if}}\quad + {\text{DM}}_{i} > - {\text{DM}}_{i} \quad {\text{and}}\quad + {\text{DM}}_{i} > 0} \hfill \\ 0 \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right.. $$

  (42)

  Similarly, the minus true directional movement (−TDM) can be calculated by:

  $$ - {\text{TDM}}_{i} = \left\{ {\begin{array}{*{20}c} { - {\text{DM}}_{i} } \hfill & {{\text{if}}\quad + {\text{DM}}_{i} < - {\text{DM}}_{i} \quad {\text{and}}\quad - {\text{DM}}_{i} > 0} \hfill \\ 0 \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right.. $$

  (43)

  Hence, the 14-day plus directional movement (+DM_14) can be calculated by:

  $$ + {\text{DM}}\_14_{i} = \frac{13}{14} \times \left( { + {\text{DM}}\_1 4_{i - 1} } \right) + \frac{1}{14} \times \left( { + {\text{TDM}}_{i} } \right). $$

  (44)

  Similarly, the 14-day minus directional movement (−DM_14) can be calculated by:

  $$ - {\text{DM}}\_14_{i} = \frac{13}{14} \times \left( { - {\text{DM}}\_14_{i - 1} } \right) + \frac{1}{14} \times \left( { - {\text{TDM}}_{i} } \right). $$

  (45)

  Next, define the true range (TR) as:

  $$ {\text{TR}}_{i} = {\text{Max}}\{ {\text{HP}}_{i} - {\text{LP}}_{i} ,|{\text{HP}}_{i} - {\text{CP}}_{i - 1} |,|{\text{LP}}_{i} - {\text{CP}}_{i - 1} |\} . $$

  (46)

  The 14-day true range (TR_14) can be calculated by:

  $$ {\text{TR}}\_14_{i} = \frac{13}{14} \times {\text{TR}}\_14_{i - 1} + \frac{1}{14} \times {\text{TR}}_{i} . $$

  (47)

  Therefore, the 14-day plus directional indicator can be defined as:

  $$ + {\text{DI}}\_14_{i} = \frac{{ + {\text{DM}}\_14_{i} }}{{{\text{TR}}\_14_{i} }}. $$

  (48)

• 13. 14-day minus directional indicator

  The 14-day minus directional indicator is defined as:

  $$ - {\text{DI}}\_14_{i} = \frac{{ - {\text{DM}}\_14_{i} }}{{{\text{TR}}\_14_{i} }}. $$

  (49)

  where $- {\text{DM}}\_14_{i}$ and ${\text{TR}}\_14_{i}$ are, respectively, the 14-day minus directional movement and 14-day true range of day i, as previously defined in (45) and (47).

• 14. 26-day buying/selling momentum indicator

  The 26-day buying/selling momentum indicator is defined as:

  $$ {\text{AR}}\_26_{i} = \frac{{\sum\nolimits_{j = i - 25}^{i} {\left( {{\text{HP}}_{j} - {\text{OP}}_{j} } \right)} }}{{\sum\nolimits_{j = i - 25}^{i} {\left( {{\text{OP}}_{j} - {\text{LP}}_{j} } \right)} }}. $$

  (50)

• 15. 26-day buying/selling willingness indicator

  The 26-day buying/selling willingness indicator is defined as:

  $$ {\text{BR}}\_26_{i} = \frac{{\sum\nolimits_{j = i - 25}^{i} {\left( {{\text{HP}}_{j} - {\text{CP}}_{j - 1} } \right)} }}{{\sum\nolimits_{j = i - 25}^{i} {\left( {{\text{CP}}_{j - 1} - {\text{LP}}_{j} } \right)} }}. $$

  (51)