In this study, we proposed an improvement to a previous strategy of the work called VolTS. It is a module of the AITA framework that integrates statistical analysis with machine learning techniques for forecasting stock market trends. The main goal is to verify the cointegration in volatility-based trading strategies within ifnancial markets, and then to devise a methodology that leverages market dynamics to yield profits. This methodology improvement incorporates the Dynamic Time Warping to assess the similarity of trend sequences, even when they exhibit temporal misalignment coupling it. With the K-Nearest Neighbors algorithm, which identifies the most akin price patterns, we construct a sophisticated model spotlighting stocks among the nine largest ones listed on both the NYSE and Nasdaq exchanges that exhibit analogous price movements to our portfolio stocks. From volatility assets pointofview, it is applied the Granger Causality Test to the dataset containing the same mid-range volatility clusters of the chosen stocks, thus identifying them with robust predictive relationships. These “predictor” stocks were pivotal in shaping our trading strategy, serving as trend indicators to inform decisions on target stock trades. The empirical findings demonstrated the efectiveness of our method in identifying, small but not rare, profitable day-trading opportunities. This success was attributed to the predictive insights from volatility clusters, the Granger causality relationships, and Co-integration trends identified among the stocks. In conclusion, our research has significantly contributed to the realm of volatility-based trading strategies by introducing a methodology that mixes statistical techniques with machine learning.
enced stocks, can Dynamic Time Warping (DTW) [2] be paired with K-Nearest Neighbours (KNN) to This article explores the emerging field of volatility- spotlight stocks that “fluctuate” to the same tune based trading strategies, which is encountering exploiting the delay among them? promising growth in the financial sector. Artificial This study aimed to formulate trading strategies intelligence (AI) has emerged as a pivotal player, based on predictive connections afecting influenofering robust tools for analysing market volatil- tial stocks as trend indicators. The proposed time ity and leveraging it for profitable outcomes. AI series study, related to the prices and volatilitymodels, trained to estimate mean volatility, ofer driven trading strategy was then rigorously evaluvaluable insights into the inherent uncertainty and ated through backtesting and performance analysis risk linked to individual securities or the overall to validate its reliability. Through empirical evimarket [1]. dence, this research has given an improvement to
Our research was primarily driven by the follow- the previous volatility-based trading strategies [3]. ing key questions: 1: Can k-means clustering The augmented AI trading strategy utilised kefectively determine the mean volatility of promi- means clustering of average volatility data [4]. This nent stocks? 2: defined the stocks with the same data encompassed nine major stock markets. Our mid-term volatility, can the Granger Causality Test initial objective was to identify distinct volatility be employed to identify predictive influences be- patterns within the market and subsequently group tween stocks? 3: Established the subset of influ- assets accordingly. Following this, the Granger
tIetallli-gIeAnc2e0,2o4r:ga4ntihzedNabtyioCnIaNlIC, oMnfaeyre2n9c–e30o,n 2A0r2t4i,ficiNalapInle-s, stocks that significantly predicted others within our Italy analysis. Then, we pairwise compare two time series * Corresponding author. to find the closest match between them via DTW. $ ivan.letteri@univaq.it (I. Letteri) Then, the future trend is predicted as the average 0000-0©0200224-3C8op4y3r-ig3h8t6foXr t(hIis. pLaeptetrebryi)its authors. Use permitted under of the trends of the neighbours. These predictive
Creative Commons License Attribution 4.0 International (CC BY relationships were then utilised to establish buy, sell, CPWrEooUrckReshdoinpgs IhStpN:/c1e6u1r3-w-0s.o7r3g 4C.0E).UR Workshop Proceedings (CEUR-WS.org) or hold trading decisions.
⎯ ⎸ = ⎷⎸ 4 1 (2) ︁∑ (︂ =1 (ℎ ) )︂ 2 ( ) .
volatility of the asset is proportional to the logarithm ( ) of the ratio of the high (ℎ ) and low ( ) observations.
Garman-Klass (GK) estimator that
price distributed 12 (ℎ ) )︂ 2 ︂( ( ) movements calculated
are as
follows: − ︀∑ =1 (2 (2) − 1) ( ) )︂ 2 ︂( ( ) aslog
ity (HV), and in turn, HV can provide insights into ments, such as wide trading ranges, breakouts, or future PA. When the PA exhibits strong price move- −1 =1 rapid directional changes, it tends to increase.
VolTS-Aug, as an improvement of the VolTS mod- −1 =1 ule within the AITA framework, adheres to these principles. Low HV signifies a period of consolidation or low price volatility, indicating a potential performance across a broad spectrum of scenarios, including those characterised by jumps and nonnormality in the data. However, this estimator has upcoming spike in volatility or a shift in the PA. limitations, and its efectiveness may be constrained On the other side, high HV suggests a higher proba- in certain contexts. + 2 ( ) ( ) − ( ) )︂ 2 ( ) ( ) ( ) − −1 ( ) )︂ 2 ( ) −1 + (1 − ) 2 , 2 , and ℎ . YZ exhibits notable ing strategies that emphasised technical indicators [1],[6]. This current exploration delves, first into
tifying medium volatility selecting stocks for the
[7]. Second, within the context of investment timing, we predict it via DTW paired with KNN.
summarize fundamental concepts within our AITA framework [8], highlighting the Volatility Trading
Section 3 outlines the improvement with the VolTS Augmented (in short VolTS-Aug) module, which analyses securities’ volatility averages and price trends establishing predictive relationships. It then delves into the implementation of the trading strategy and includes a thorough empirical analysis of its performance and robustness. Section 4 presents practical findings achieved through backtesting followed by a discussion. Finally, Section 5 concludes the study by summarising the efectiveness and applicability of the proposed method. 2. BACKGROUND
assets. ( ( )
, and bility of sharp market movements or trend changes.
For each timeframe , the OHLC of an asset is represented as a 4-dimensional vector
= (ℎ ) , ( ), ( )) , where ( ) > 0,
( ) < (ℎ ) ( )
( ) , ∈ [
( ), (ℎ )]. 2.2. Historical Volatility Time Series The construction of the dataset is composed by time (TF) strategy. It is one way to engage in trend series from the following HV estimators:
In this research, our attention is centred on midvolatility. This focus allows us to either close open positions or refrain from entering a position when the anticipated volatility coeficient is high, thereby mitigating the risk of losses. On the other hand, if the expected volatility is too low, it does not ofer any potential for gains.
In this experiment, we used the Trend Following trading, where a trader initiates an order in the direction of the breakout after the price surpasses the resistance line as follows: let the price at ︃√ ︃√ prices of
The sumes normally 1 ∑︀ =1 1 ∑︀
=1 ︀√ 2 ℎ 1 ∑︀ 1 ∑︀ (︂ ︂(
The Rogers-Satchell (RS) estimator uses the range of prices within a given time interval as a proxy for the volatility of the asset as follows: ︂( (ℎ ) )︂ ( ) ︂( (ℎ ) )︂ ( ) + ︂( ( ) )︂ ( )
interval is a good proxy for the volatility of the asset, additionally, the estimator may be sensitive to outliers and extreme price movements.
The
Yang-Zhang (YZ) estimator [9] incorporates OHLC prices as follows: where = 0.34/1.34 + −+11 , 2
= ︂( ( ) )︂ ( ) = = = position. benchmark. time , and let
denote the Moving Average of the asset price over a certain period. If ≥ indicates an upward trend to take a long position otherwise it is a downward trend to take a short
2.4. Backtesting Metrics .
risk metrics to evaluate the potential profitability of investments and manage risk exposure.
Drawdown (DD). It is a measure of the peak-totrough decline in the value of a trading account before a new peak is attained. DD is defined as follows: or peak of the portfolio.
= − , where is the highest value is the lowest value or trough after the peak.
Maximum Drawdown (MDD) is the most significant loss from peak to trough during a specific period calculated as follows: = ︁( − : ≤ ≤ )︁ , where is the highest value or peak of the portfolio time . is the lowest value or trough after the peak up to time is the total number of data points.
Sortino ratio (SoR). It is a risk-adjusted profit measure, which refers to the return per unit of deviation as follows: the expected portfolio return, the risk-free rate = − , where is 3. THE EXPERIMENT 3.1. Asset Collections
via an internal Python library connected to an API, using the MetaTrader5 (MT5)1. The data collected for this study includes the OHLC prices of the stocks listed in Table 1. List of the main 9 stocks selected for the experimentation.
Ticker MSFT GOOGL MU NVDA AMZN META QCOM IBM INTC
Company Microsoft Corporation
Alphabet Inc.
Micron Technology, Inc.
NVIDIA Corporation Amazon.com, Inc.
Meta Platforms, Inc.
QUALCOMM Incorporated Int. Business Machines Corp.
Intel Corporation
Market Nasdaq Nasdaq Nasdaq NYSE NYSE NYSE Nasdaq NYSE
NYSE 3.2. Historical Volatility Time Series The History Volatility Clustering process of our approach determines the stocks with intermediate volatility. First calculate the average of historical volatility time series among the aforementioned estimators (see sect. 2.2). Next, the resulting volatility of return, and denotes the downside deviation of series are clusterized using the KMeans++ algothe portfolio returns.
rithm. In particular, we split into three clusters
= 3) high, middle, and low volatility.
Calmar ratio (CR) is another variant of the risk- noting that, the main region is in the time window To check the goodness of trades, we mainly fo- from the intermediate cluster (confined between adjusted profit measure, which applies as a risk measure:
= − where is the standard deviation of the portfolio return. adjusted profit measure, which applies MDD as risk measure:
= − . cused on the Total Returns
( ) for each stock (
Furthermore, we analyzed the standardized re- 3.3. Causality Analysis plot of the time series belonging to the middle cluster where we are focused on our strategy. It is worth from 1st January 2021 to 1st March 2024.
the two red dashed lines in fig 1). The candidate assets selected are NVDA, META, AMZN, MU and
ear combination between two or more time series, although individual series may be non-stationary.
one asset and the future volatility of another asset 3.4. The Algorithm
Regression step:
For each pair of time series ( , ), where ̸= , we construct a linear regression model: = 0, + 1, + , where 0, is the intercept, 1,
is the regression coeficient, and is the error term. We calculate the F-statistic to evaluate the overall adequacy of the model.
GCT step: For each pair of time series ( , ), we perform the Granger causality test. The model for the Granger test can be expressed as ( ) = + ︀∑ =1 , ( − ) + ∑︀ =1 , ( − ) + ( ), where ( ) is the current value of , ( − ) and ( − ) are the lagged values of and , respectively, and ( ) is the error term. If the coeficients ,
are statistically diferent from zero, we reject the null hypothesis and conclude that
DTW-KNN step: (i) For each pair of time selength , the DTW distance ( , ) is given by ︀∑ =1 ︀∑ =1 (, ), where (, ) is the distance between points [ ] and [ ], and the optimization is performed overall possible alignments. (ii) The process of finding the best parameters delta time involves the KNN whit optimization through grid search.
Profit and risk metrics are pivotal considerations in trading AITA framework evaluates the following, for the potential profitability of the investments and to manage the risk exposure.
(i) The Maximum drawdown (MDD) measures the period, to show the worst case, as follows: = that as follows: by applying the following steps:
Step 1. Significant Granger causality: Let and be the pair stocks time series volatility to check, where and
represents the potential causal variable represents the potential efect variable. The null hypothesis (H0) states that does not Granger cause , while the alternative hypothesis (H1) states does Granger cause . The F-test is defined − = [( [( ( ) − ( ) )/( − − )] ( ) )/ ] , where
is the Residual Sum of Squares for the two AutoRegressive models: ( ) = + 1 * ( − 1) + 2 * ( − 2) + · · · + * ( − ) + ( ), and
: ( ) = ( − 2) + · · · + the lag order, + * 1 * ( − 1) +
2 * ( − ) +
( ), with the number of observations, and the target time series.
Step 2.
Causality Direction: If the volatility of
Granger causes the volatility of Stock , it suggests that changes in Stock be used to predict changes in Stock volatility can
volatility. A low p-value suggests the presence of a causal relationship between the time series.
Step 3. Delta Time Trends: VolTS-Aug performs the DTW paired with KNN to examine the interval time necessary for profitable trades: (i) The
DTW distance between two time series is the sum of diferences between their corresponding points, optimally aligned. (ii) The KNN classifier, aimed at finding based on DTW distance. the number of parameters in the models. It in- ries ( , ), where has length and has dicates how much the regression coeficients of the lagged time series help to explain the variation in ( , ) = the most similar neighbours for each observation largest decline from the peak in the whole trading ∈(0, )[ ∈(0, ) − ].
(ii) The Sharpe ratio (SR) is a risk-adjusted profit measure, which refers to the return per unit of deviation as follows:
= E[[ ]] . risk measure:
= E[ ] . risk-adjusted profit measure, which applies DD as of the risk-adjusted profit measure, which applies
=
= ( ) = ( +Δ )− ( ) , and furthermore analysing the standardized returns = (
− )/ , with ( = 1, ..., ), where is the standard deviation of , e denote the average overtime for the studied period. ( ) 4. RESULTS AND
(iii) The Sortino ratio (SoR) is a variant of the ing, also considering the fixed commission of 9$ per (iv) The Calmar ratio (CR) is another variant reused for each trade. mance metrics of the strategy and shows how the total amount in the portfolio is increased to 3241.57$ (8.05%), which is a positive sign of profitable tradtrade.
presented in figure 3 about META co-integration.
trend given a profit of 59.98$, with a winrate of 75%, a MDD of 0.035%, and a return of 5.98%, which outperforms the B&H strategy with return of 0.998%.
The trades of MU bought following the META trend given a profit of 146.68$, with a winrate of 100%, a MDD of 0%, and a return of 34.91%, which outperforms the B&H strategy with return of 3%. The trades of NVDA bought following the META trend given a profit of 39.11$, with a winrate of 75%, a MDD of 0.11%, and a return of
market opening, starting with an initial budget of VolTS-Aug handle volatility in trading strategy comment of the AITA framework module calle VolTS. bining causality by the Historical Volatility Granger
Stock META ->AMZN, NVDA, MU a profitable stock pairings improving on previous work [10, 1].
date this claim, we applied our methodology to nine assets reduce to four after filtering by our methodology. The results shows a promising potential of this approach with a gain of 241.57$ (8.05%) in 21 days. In future works, we will further test its reliability with more refined assets selection (e.g., [ 11, 12]) and balancing (buy, sell and hold trades) strategies (e.g., [13, 14]).