Introduction

This article explores the emerging field of volatilitybased trading strategies, which is encountering promising growth in the financial sector. Artificial intelligence (AI) has emerged as a pivotal player, offering robust tools for analysing market volatility and leveraging it for profitable outcomes. AI models, trained to estimate mean volatility, offer valuable insights into the inherent uncertainty and risk linked to individual securities or the overall market [1].

Our research was primarily driven by the following key questions: 𝑅𝑄1: Can k-means clustering effectively determine the mean volatility of prominent stocks? 𝑅𝑄2: defined the stocks with the same mid-term volatility, can the Granger Causality Test be employed to identify predictive influences between stocks? 𝑅𝑄3: Established the subset of influ-enced stocks, can Dynamic Time Warping (DTW) [2] be paired with K-Nearest Neighbours (KNN) to spotlight stocks that "fluctuate" to the same tune exploiting the delay among them?

This study aimed to formulate trading strategies based on predictive connections affecting influential stocks as trend indicators. The proposed time series study, related to the prices and volatilitydriven trading strategy was then rigorously evaluated through backtesting and performance analysis to validate its reliability. Through empirical evidence, this research has given an improvement to the previous volatility-based trading strategies [3].

The augmented AI trading strategy utilised kmeans clustering of average volatility data [4]. This data encompassed nine major stock markets. Our initial objective was to identify distinct volatility patterns within the market and subsequently group assets accordingly. Following this, the Granger Causality Test (GCT) [5] was leveraged to pinpoint stocks that significantly predicted others within our analysis. Then, we pairwise compare two time series to find the closest match between them via DTW. Then, the future trend is predicted as the average of the trends of the 𝐾 neighbours. These predictive relationships were then utilised to establish buy, sell, or hold trading decisions.

Our previous research focused on technical trading strategies that emphasised technical indicators [1], [6]. This current exploration delves, first into Historical Volatility estimators as a dataset for identifying medium volatility selecting stocks for the Granger Causality Test asset cointegration approach [7]. Second, within the context of investment timing, we predict it via DTW paired with KNN.

This paper is organised as follows: Section 2 summarize fundamental concepts within our AITA framework [8], highlighting the Volatility Trading System (VolTS) module [3]. 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 effectiveness and applicability of the proposed method.

BACKGROUND

Price Action

The price action (PA) influences Historical Volatility (HV), and in turn, HV can provide insights into future PA. When the PA exhibits strong price movements, such as wide trading ranges, breakouts, or rapid directional changes, it tends to increase.

VolTS-Aug, as an improvement of the VolTS module within the AITA framework, adheres to these principles. Low HV signifies a period of consolidation or low price volatility, indicating a potential upcoming spike in volatility or a shift in the PA. On the other side, high HV suggests a higher probability of sharp market movements or trend changes. Also into VolTS-Aug, the PA is encoded as OHLC, i.e., the open, high, low, and close prices of the assets.

For each timeframe 𝑡, the OHLC of an asset is represented as a 4-dimensional vector 𝑋𝑡 = (𝑥

(𝑜) 𝑡 , 𝑥 (ℎ) 𝑡 , 𝑥 (𝑙) 𝑡 , 𝑥 (𝑐) 𝑡 ) 𝑇 , where 𝑥 (𝑙) 𝑡 > 0, 𝑥 (𝑙) 𝑡 < 𝑥 (ℎ) 𝑡 and 𝑥 (𝑜) 𝑡 , 𝑥 (𝑐) 𝑡 ∈ [𝑥 (𝑙) 𝑡 , 𝑥 (ℎ) 𝑡 ].

Historical Volatility Time Series

The construction of the dataset is composed by time series from the following HV estimators:

The Parkinson (PK) estimator incorporates the stock's daily high and low prices as follow:

𝑃 𝐾 = ⎯ ⎸ ⎸ ⎷ 1 4𝑁 𝑙𝑛(2) 𝑁 ∑︁ 𝑖=1 (︂ 𝑙𝑛 𝑥 (ℎ) 𝑡 𝑥 (𝑙) 𝑡 )︂ 2 .

It is derived from the assumption that the true volatility of the asset is proportional to the logarithm (𝑙𝑛) of the ratio of the high 𝑥 prices of 𝑁 observations. The Garman-Klass (GK) estimator assumes that price movements are lognormally distributed calculated as follows:

√︃ 1 𝑁 𝑁 ∑︀ 𝑖=1 1 2 (︂ 𝑙𝑛 𝑥 (ℎ) 𝑡 𝑥 (𝑙) 𝑡 )︂ 2 − 𝑁 ∑︀ 𝑖=1 (2𝑙𝑛(2) − 1) (︂ 𝑙𝑛 𝑥 (𝑐) 𝑡 𝑥 (𝑜) 𝑡 )︂ 2

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:

𝑅𝑆 = √︃ 1 𝑁 𝑁 ∑︀ 𝑡=1 𝑙𝑛 (︂ 𝑥 (ℎ) 𝑡 𝑥 (𝑐) 𝑡 )︂ 𝑙𝑛 (︂ 𝑥 (ℎ) 𝑡 𝑥 (𝑜) 𝑡 )︂ + 𝑙𝑛 (︂ 𝑥 (𝑙) 𝑡 𝑥 (𝑐) 𝑡 )︂ 𝑙𝑛 (︂ 𝑥 (𝑙) 𝑡 𝑥 (𝑜) 𝑡 )︂

RS assumes that the range of prices within the 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:

𝑌 𝑍 = √︀ 𝜎 2 𝑂𝑣𝑒𝑟𝑛𝑖𝑔ℎ𝑡𝑉 𝑜𝑙 + 𝑘𝜎 2 𝑂𝑝𝑒𝑛𝑇 𝑜𝐶𝑙𝑜𝑠𝑒𝑉 𝑜𝑙 + (1 − 𝑘)𝜎 2 𝑅𝑆 , where 𝑘 = 0.34/1.34 + 𝑁 +1 𝑁 −1 , 𝜎 2 𝑂𝑝𝑒𝑛𝑇 𝑜𝐶𝑙𝑜𝑠𝑒𝑉 𝑜𝑙 = 1 𝑁 −1 𝑁 ∑︀ 𝑖=1 (︂ 𝑙𝑛 𝑥 (𝑐) 𝑡 𝑥 (𝑜) 𝑡 − 𝑙𝑛 𝑥 (𝑐) 𝑡 𝑥 (𝑜) 𝑡 )︂ 2 , and 𝜎 2 𝑂𝑣𝑒𝑟𝑛𝑖𝑔ℎ𝑡𝑉 𝑜𝑙 = 1 𝑁 −1 𝑁 ∑︀ 𝑖=1 (︂ 𝑙𝑛 𝑥 (𝑜) 𝑡 𝑥 (𝑐) 𝑡−1 − 𝑙𝑛 𝑥 (𝑜) 𝑡 𝑥 (𝑐) 𝑡−1 )︂ 2

. YZ exhibits notable performance across a broad spectrum of scenarios, including those characterised by jumps and nonnormality in the data. However, this estimator has limitations, and its effectiveness may be constrained in certain contexts.

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 coefficient is high, thereby mitigating the risk of losses. On the other hand, if the expected volatility is too low, it does not offer any potential for gains.

Trading Strategy

In this experiment, we used the Trend Following (TF) strategy. It is one way to engage in trend trading, where a trader initiates an order in the direction of the breakout after the price surpasses the resistance line as follows: let 𝑃𝑡 the price at time 𝑡, and let 𝑀 𝐴 denote the Moving Average of the asset price over a certain period. If 𝑃𝑡 ≥ 𝑀 𝐴𝑡 indicates an upward trend to take a long position otherwise it is a downward trend to take a short position.

Then, our strategy is compared only with the Buy and Hold (B&H) strategy considering it as a benchmark.

Backtesting Metrics

AITA framework considers the following profit and 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: 𝐷𝐷 = 𝑃 −𝑇 𝑇 , where 𝑃 is the highest value or peak of the portfolio. 𝑇 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:

𝑀 𝐷𝐷 = 𝑚𝑎𝑥𝑖 (︁ 𝑃 𝑖 −𝑚𝑖𝑛 𝑗:𝑖≤𝑗≤𝑁 𝑇 𝑗 𝑃 𝑖 )︁

, where 𝑃𝑖 is the highest value or peak of the portfolio time 𝑖. 𝑇𝑗 is the lowest value or trough after the peak up to time 𝑗. 𝑁 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:

𝑆𝑜𝑅 = 𝑅𝑝−𝑅 𝑓 𝜎 𝑑

, where 𝑅𝑝 is the expected portfolio return, 𝑅 𝑓 the risk-free rate of return, and 𝜎 𝑑 denotes the downside deviation of the portfolio returns.

Sharpe ratio (SR). It is a variant of the riskadjusted profit measure, which applies 𝜎𝑝 as a risk measure:

𝑆𝑅 = 𝑅𝑝−𝑅 𝑓 𝜎𝑝

where 𝜎𝑝 is the standard deviation of the portfolio return.

Calmar ratio (CR) is another variant of the riskadjusted profit measure, which applies MDD as risk measure: 𝐶𝑅 = 𝑅𝑝−𝑅 𝑓 𝑀 𝐷𝐷 .

To check the goodness of trades, we mainly focused on the Total Returns 𝑇 𝑅 𝑘 (𝑡) for each stock (𝑘 = 1, ..., 𝑝) in the time interval (𝑡 = 1, ..., 𝑛) with the price 𝑃 , defined as follows:

𝑇 𝑅 𝑘 (𝑡) = 𝑃 𝑘 (𝑡+Δ𝑡)−𝑃 𝑘 (𝑡) 𝑃 𝑘 (𝑡)

. Furthermore, we analyzed the standardized returns 𝑟 𝑘 = (𝑇 𝑅 𝑘 −𝜇 𝑘 )/𝜎 𝑘 , with (𝑘 = 1, ..., 𝑝), where 𝜎 𝑘 is the standard deviation of 𝑇 𝑅 𝑘 , e 𝜇 𝑘 denote the average overtime for the studied period.

THE EXPERIMENT

Asset Collections

AITA automatically downloads the OHLC prices 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.

Table 1

List of the main 9 stocks selected for the experimentation.

Ticker

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 series are clusterized using the KMeans++ algorithm. In particular, we split into three clusters (𝐾 = 3) high, middle, and low volatility.

Figure 1 shows the results displayed through a plot of the time series belonging to the middle cluster where we are focused on our strategy. It is worth noting that, the main region is in the time window from 1st January 2021 to 1st March 2024.

So, we use this interval as the dataset, and then from the intermediate cluster (confined between the two red dashed lines in fig 1). The candidate assets selected are NVDA, META, AMZN, MU and QCOM.

Causality Analysis

Co-integration refers to the long-term stable linear combination between two or more time series, although individual series may be non-stationary.

In the context of volatility-based trading, the VolTS-Aug module performs the GCT to examine this relationship between the lagged volatility of one asset and the future volatility of another asset by applying the following steps:

Step 1. Significant Granger causality: Let 𝑋 and 𝑌 be the pair stocks time series volatility to check, where 𝑋 represents the potential causal variable and 𝑌 represents the potential effect variable. The null hypothesis (H0) states that 𝑋 does not Granger cause 𝑌 , while the alternative hypothesis (H1) states that 𝑋 does Granger cause 𝑌 . The F-test is defined as follows:

𝐹 − 𝑡𝑒𝑠𝑡 = [(𝑅𝑆𝑆 𝑌 (𝑡) − 𝑅𝑆𝑆𝑌 𝑋 (𝑡) )/𝑝] [(𝑅𝑆𝑆𝑌 𝑋 (𝑡) )/(𝑛 − 𝑝 − 𝑘)] ,

where 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 differences between their corresponding points, optimally aligned. (ii) The KNN classifier, aimed at finding the most similar neighbours for each observation based on DTW distance.

The Algorithm

Three are the main steps followed by VolTS-Aug: Regression step: For each pair of time series (𝑋𝑖, 𝑌𝑗), where 𝑖 ̸ = 𝑗, we construct a linear regression model: 𝑋𝑖 = 𝛽0,𝑖𝑗 + 𝛽1,𝑖𝑗𝑌𝑗 + 𝜖𝑖𝑗, where 𝛽0,𝑖𝑗 is the intercept, 𝛽1,𝑖𝑗 is the regression coefficient, and 𝜖𝑖𝑗 is the error term. We calculate the F-statistic to evaluate the overall adequacy of the model.

Metrics observed

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 largest decline from the peak in the whole trading period, to show the worst case, as follows: 𝑀 𝐷𝐷 = 𝑚𝑎𝑥 𝜏 ∈(0,𝑡) [𝑚𝑎𝑥 𝑡∈(0,𝜏 )

𝑛 𝑡 −𝑛𝜏 𝑛 𝑡

]. (ii) The Sharpe ratio (SR) is a risk-adjusted profit measure, which refers to the return per unit of deviation as follows: 𝑆𝑅 = E [𝑟] [𝑟] . (iii) The Sortino ratio (SoR) is a variant of the risk-adjusted profit measure, which applies DD as risk measure: 𝑆𝑜𝑅 = E [𝑟] 𝐷𝐷 . (iv) The Calmar ratio (CR) is another variant of the risk-adjusted profit measure, which applies MDD as risk measure: 𝐶𝑅 = E [𝑟] 𝑀 𝐷𝐷 . To check the goodness of trades, we mainly focused on the (v) Total Returns 𝑅 𝑘 (𝑡) for each stock (𝑘 = 1, ..., 𝑝) in the time interval (𝑡 = 1, ..., 𝑛), where 𝑇 𝑅 = 𝑅 𝑘 (𝑡) = 𝑍 𝑘 (𝑡+Δ𝑡)−𝑍 𝑘 (𝑡)

𝑍 𝑘 (𝑡)

, and furthermore analysing the standardized returns 𝑟 𝑘 = (𝑅 𝑘 − 𝜇 𝑘 )/𝜎 𝑘 , with (𝑘 = 1, ..., 𝑝), where 𝜎 𝑘 is the standard deviation of 𝑅 𝑘 , e 𝜇 𝑘 denote the average overtime for the studied period. The VolTS algorithm iterates the daily lags in a range from 2 to 30 days to determine the best testing range time. The best result is achieved with lags=21, where 'best' is considered when there is direction coherency among the stocks with the lowest p-value, with the maximum cardinality of the set of stocks previously filtered, and not create cyclic graph in the connections. On other words, the GCT suggests buying AMZN, MU and NVDA when META has a positive trend and vice versa (see fig. 2).

RESULTS AND DISCUSSIONS

The Experiment

The experiment results indicate that the strategy resulted in a total gain of 241.57$ in 21 days of market opening, starting with an initial budget of 1000$ per stock.

Tab. 2 contains further details about the performance 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 trading, also considering the fixed commission of 9$ per trade. Notice that, the managing of the budget is set in compounded mode, so the full amount is reused for each trade.

Backtesting

The analysis of individual stocks' performance is presented in figure 3 about META co-integration. The trades of AMZN bought following the META 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 3.98%, which is similar to the B&H strategy with return of 4.02%. Figure 3 shows all the 7 positions applied at the same time on NVDA, MU and AMZN with buying/selling trades following the co-integration with META trend determined with VolTS-Aug algorithm. The trades on MU stock outperforms all the others stocks increasing notably the portfolio gain (see Table 2).

Conclusions

In this paper, we propose VolTS-Aug an improvement of the AITA framework module calle VolTS. VolTS-Aug handle volatility in trading strategy combining causality by the Historical Volatility Granger Causality Test and DTW & KNN to determine a profitable stock pairings improving on previous work [10,1]. The novelty of the approach implemented in VolTS-Aug lies in a better trades timing. To validate 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]).