mql5/Indicators/MyIndicators/Quant/PairsTrading_Pro.md

Pairs Trading Cointegration Pro Suite (Oscillator & Bands)

1. Summary

The Pairs Trading Cointegration Pro Suite is an institutional-grade, high-performance separate window statistical arbitrage trading suite comprising two advanced indicators: PairsTrading_Pro (Z-Score separate window oscillator) and PairsTrading_Bands_Pro (Main-chart overlay bands). Based on Modern Portfolio Theory and econometric cointegration, the suite decomposes the pricing relationship of two correlated assets into a stationary, volatility-normalized spread.

While traditional retail pairs trading methods rely on simple price correlation (which is highly unstable and prone to structural drift), the PairsTrading_Pro suite utilizes a dynamic rolling Ordinary Least Squares (OLS) mathematical engine. It dynamically calculates the rolling Hedge Ratio (\beta) and Intercept (\alpha) between any two assets to extract the true stationary spread.

Featuring VWAP-style Anchored Resets (Session, Weekly, Monthly, and Custom Session), the indicators can completely isolate intraday/intraweek price relationships from overnight gaps and illiquidity, delivering a highly visual and robust quantitative scanner system.

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2. Mathematical Foundations and Calculation Logic

The statistical calculations operate on synchronized close prices for Asset A (P_{A,t}) and Asset B (P_{B,t}) over an active rolling or anchored window of size N (window_size):

A. Rolling Ordinary Least Squares (OLS)

The calculator computes the rolling mean of Asset A (\bar{A}) and Benchmark B (\bar{B}). It then solves the OLS regression of A on B to find the dynamic Hedge Ratio (\beta) and Intercept (\alpha):

\beta_i = \frac{\text{Covariance}(A, B)}{\text{Variance}(B)} \alpha_i = \bar{A}_i - (\beta_i \times \bar{B}_i)

B. Dynamic Spread and Standard Deviation

The spread at each bar t within the window is calculated. Because we subtract the OLS intercept (\alpha_i), the rolling mean of this spread over the window is algebraically guaranteed to be exactly 0.0:

\text{Spread}_{t} = P_{A,t} - \beta_i P_{B,t} - \alpha_i \quad \text{for } t = i-N+1 \dots i

The sample standard deviation (\sigma_{\text{spread}}) of the spread over the active window is computed as:

\sigma_{\text{spread}} = \sqrt{\frac{1}{N-1} \sum_{k=0}^{N-1} (\text{Spread}_{i-k})^2}

C. Separate Window Z-Score (PairsTrading_Pro)

The final Z-Score is calculated, representing how many standard deviations the current spread has drifted away from its statistical equilibrium of 0.0:

Z_i = \frac{P_{A,i} - \beta_i P_{B,i} - \alpha_i}{\sigma_{\text{spread}}}

• Z \ge 2.0 (OrangeRed Histogram): Spread is extremely overvalued (Short A, Long B).
• Z \le -2.0 (DeepSkyBlue Histogram): Spread is extremely undervalued (Long A, Short B).
• Z \in [-1.5, 1.5] (Gray Histogram): Symmetrical neutral noise zone.

D. Main Chart Cointegration Bands (PairsTrading_Bands_Pro)

By rearranging the spread equation back to the price space of Asset A, we project the dynamic statistical boundaries directly onto the main price chart:

\text{Center Line (Equilibrium / } Z=0.0\text{):} \quad \hat{P}_{A,i} = \beta_i P_{B,i} + \alpha_i \text{Outer Upper Band (Extreme / } Z=+M_{\text{outer}}\text{):} \quad \text{Band}_{\text{up, outer}} = \hat{P}_{A,i} + M_{\text{outer}} \times \sigma_{\text{spread}} \text{Outer Lower Band (Extreme / } Z=-M_{\text{outer}}\text{):} \quad \text{Band}_{\text{low, outer}} = \hat{P}_{A,i} - M_{\text{outer}} \times \sigma_{\text{spread}} \text{Inner Upper Band (Warning / } Z=+M_{\text{inner}}\text{):} \quad \text{Band}_{\text{up, inner}} = \hat{P}_{A,i} + M_{\text{inner}} \times \sigma_{\text{spread}} \text{Inner Lower Band (Warning / } Z=-M_{\text{inner}}\text{):} \quad \text{Band}_{\text{low, inner}} = \hat{P}_{A,i} - M_{\text{inner}} \times \sigma_{\text{spread}}

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3. MQL5 UI & Architecture

• Decoupled Math Engine (PairsTrading_Calculator.mqh): All covariance, variance, rolling OLS, and Z-Score computations are encapsulated inside the highly optimized CPairsTradingCalculator include class. It exposes public getter methods (GetBeta(), GetAlpha(), GetStdDev()) to feed calculated coefficients directly to the main-chart bands wrapper, completely eliminating redundant loops and guaranteeing 100% data alignment.

• Strict O(1) Real-Time Tick Optimization: The calculator uses the platform's prev_calculated parameter to process only the newest incoming bar on every tick. This keeps CPU usage at absolute zero, allowing both the separate-window histogram and the main-chart bands to update live in real-time.

• VWAP-Style Anchored Resets: In addition to standard rolling windows (InpLookback), the indicators support dynamic resets:

  • Daily Reset (ANCHOR_SESSION): Resets daily. Excellent for intraday trading.
  • Weekly Reset (ANCHOR_WEEK): Resets weekly. Ideal for swing trading.
  • Custom Session (ANCHOR_CUSTOM_SESSION): Resets at a user-defined broker-time range (e.g. 09:00 to 18:00). It completely filters out overnight gaps and illiquid trading hours, plotting EMPTY_VALUE during inactive periods to keep statistics pure.

• Advanced Bar-Time Synchronization: Assets do not always share identical trading calendars or liquidity densities. PairsTrading_Pro aligns Symbol A and Symbol B prices perfectly by timestamp using iBarShift(..., false) and iClose, ensuring that different market open/close times do not distort the calculation. To ensure chart-independence, the synchronization uses a dedicated iClose fallback rather than the local chart's close[0].

• Hardlocked Separate Window Scale Bounds [-3.5, 3.5]: To prevent single extreme black-swan spikes (e.g., Z-score hitting -10.0 during weekend gaps) from squishing the entire historical chart into an unreadable flat line, the separate window's scale is fixed between -3.5 and 3.5. Outliers are simply clipped at the boundaries, maintaining a perfect, consistent visual aspect ratio across all timeframes.

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4. Parameters

A. Common Parameters

• Symbol A (InpSymbolA): The primary asset to trade (Default: "UKOIL" - Brent Crude Oil).
• Symbol B (InpSymbolB): The secondary benchmark asset (Default: "USOIL" - WTI Crude Oil).
• Anchor Reset (InpAnchor): The reset anchor period (None, Session, Week, Month, Custom Session).
• Lookback (InpLookback): The rolling regression window size (Used if Anchor = None).
• Custom Start (InpCustomStart): Session start time in format "HH:MM" (Used if Anchor = Custom).
• Custom End (InpCustomEnd): Session end time in format "HH:MM" (Used if Anchor = Custom).

B. Bands Specific Parameters

• Draw Center Line (InpDrawCenterLine): Toggle to draw the gold Equilibrium Center Line (Z=0.0).
• Draw Inner Bands (InpDrawInnerBands): Toggle to draw the dotted Coral/LightSkyBlue Warning Bands (Z=\pm 1.5).
• Inner Band Multiplier (InpInnerMultiplier): The Z-Score multiplier for the inner bands (Default: 1.5).
• Draw Outer Bands (InpDrawOuterBands): Toggle to draw the dashed Crimson/DeepSkyBlue Extreme Bands (Z=\pm 2.0).
• Outer Band Multiplier (InpOuterMultiplier): The Z-Score multiplier for the outer bands (Default: 2.0).

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5. Advanced Statistical Arbitrage Strategies

A. Classic Spread Execution (Mean Reversion)

• Buy the Spread (Z-Score \le -2.0 or Price touches Lower Outer Band): Symbol A is extremely underpriced relative to Symbol B.
  • Action: BUY Symbol A (Long) and SELL Symbol B (Short) with equal cash exposure.
• Sell the Spread (Z-Score \ge 2.0 or Price touches Upper Outer Band): Symbol A is extremely overpriced relative to Symbol B.
  • Action: SELL Symbol A (Short) and BUY Symbol B (Long) with equal cash exposure.
• The Exit (Z-Score \to 0.0 or Price touches Center Line): When the spread returns to its statistical equilibrium, close both legs simultaneously to lock in the mean-reversion profit.

B. The Quant-Grade Synergy: Cointegration + LLD Pro (Single-Leg Trading)

A major drawback of classic pairs trading is that opening two legs is capital-intensive and subject to double-broker execution slippage. By combining PairsTrading_Pro with the LLD_Pro (Lead-Lag Dominance) indicator, you can trade a single, high-probability leg:

1. Identify the Spread Extremes: PairsTrading_Pro alerts you that the spread is extremely cheap (e.g., Z = -2.5, meaning Symbol A is cheap, Symbol B is expensive).
2. Identify the Leader: Open the LLD_Pro indicator for the two symbols.
   • If Symbol B (WTI) is the Leader (leads Symbol A / Brent): WTI has already moved, and Brent (Symbol A) is mathematically guaranteed to follow to close the gap. Since Symbol A is currently too cheap, you simply BUY Symbol A (Brent) as a single directional trade!
   • This allows you to trade with half the margin requirement and zero execution hassle, leveraging the leader's predictive momentum to capture the gap-reversal.

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6. Optimized Global Multi-Asset Presets

To ensure statistical validity, only trade assets that share a fundamental, structural, or macroeconomic link. Below are the most robust, cointegrated global pairs optimized for live execution, mapped in PairsTrading_Preset_Manager.mqh:

Asset Class	Symbol A	Symbol B	Recommended TF	Lookback / Anchor	Inner / Outer Mult	Trading Style & Concept
Energies	UKOIL (Brent)	USOIL (WTI)	M5 / M15	120 / ANCHOR_NONE	1.5 / 2.0	Crude Oil Spread. Sweet/Light vs. Heavy/Sour grade arbitrage. Heavily mean-reverting.
Precious Metals	XAUUSD (Gold)	XAGUSD (Silver)	M15 / H1	120 / ANCHOR_WEEK	1.5 / 2.0	Gold-to-Silver Ratio. Decades-old commodity value parity. Highly stable weekly anchors.
Forex Majors	EURUSD	GBPUSD	M5 / M15	120 / ANCHOR_CUSTOM_SESSION (e.g., 09:00 - 18:00)	1.5 / 2.0	European Relative Value. High cointegration due to close UK-Eurozone macro ties. Custom session filters out overnight illiquidity.
Forex Commodity	AUDUSD	NZDUSD	M15 / H1	120 / ANCHOR_SESSION	1.5 / 2.0	Aussie vs. Kiwi. Commodity export-driven Oceanic currencies. Daily reset captures session shifts beautifully.
Equity Indices	US100 (Nasdaq)	US500 (S&P500)	M15 / H1	144 / ANCHOR_WEEK	1.5 / 2.0	Growth vs. Broad Market. Tech sector rotations vs. global indexing. Excellent weekly trend reversion.
Equity Indices	DE40 (DAX)	EU50 (Stoxx50)	M15 / H1	120 / ANCHOR_WEEK	1.5 / 2.0	European Equity Arbitrage. Germany's industrial core vs. broader Eurozone blue-chip baskets.
MAG7 Tech	NVDA (Nvidia)	AMD (AMD)	H1 / H4	60 / ANCHOR_NONE	1.8 / 2.5	Semiconductor Sector Spread. Extreme retail/AI hype valuation discrepancies. Higher multipliers filter stock gap volatility.