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 (default: Brent vs. WTI Crude Oil) 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 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.

• 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].

• 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.

• 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.