mql5/Indicators/MyIndicators/Quant/PairsTrading_Pro.md

Pairs Trading Cointegration Pro Suite (Standard & MTF)

1. Summary

The Pairs Trading Cointegration Pro Suite is an institutional-grade, high-performance statistical arbitrage trading suite comprising four advanced indicators:

• PairsTrading_Pro (Z-Score separate window oscillator)
• PairsTrading_Bands_Pro (7-line main-chart overlay bands)
• PairsTrading_MTF_Pro (Multi-Timeframe separate window oscillator)
• PairsTrading_Bands_MTF_Pro (Multi-Timeframe 7-line 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), this 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.

The Single-Symbol Paradigm & O(1) Memory Access

To maximize trading speed and simplify execution panels, the suite automatically determines Symbol A from the chart's native symbol (_Symbol). The trader only needs to input Symbol B (InpSecondSymbol) on the parameters window.

By binding Symbol A strictly to the current chart, the calculation engine accesses prices directly via the native close[] array inside the OnCalculate() function. This completely eliminates O(N) lookup overheads (like iBarShift and iClose) for Symbol A, reducing tick processing time to a true O(1) constant time complexity.

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2. Mathematical Foundations and Calculation Logic

The statistical calculations operate on synchronized close prices for Asset A (P_{A,t}, representing the native _Symbol) and Asset B (P_{B,t}, representing InpSecondSymbol) 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. Volatility-Normalized 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}}}

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3. Cointegration Bands (Main Chart Projection)

By rearranging the spread equation back to the price space of Asset A (the chart's active symbol), the suite projects the dynamic statistical boundaries as a 7-channel corridor directly onto the main price chart:

A. Equilibrium Core (Z = 0)

The Center Line represents the cointegrated fair value price of Asset A relative to Asset B:

\text{Center Line (Equilibrium):} \quad \hat{P}_{A,i} = \beta_i P_{B,i} + \alpha_i

B. Warning Bands (Z = +-1.5 / Inner Channel)

Triggers early scale-in warning zones for potential mean-reversion trades:

\text{Upper Inner Band:} \quad \text{Band}_{\text{up, inner}} = \hat{P}_{A,i} + M_{\text{inner}} \times \sigma_{\text{spread}} \text{Lower Inner Band:} \quad \text{Band}_{\text{low, inner}} = \hat{P}_{A,i} - M_{\text{inner}} \times \sigma_{\text{spread}}

C. Core Entry Bands (Z = +-2.0 / Outer Channel)

Statistically represents a 95.4% probability of price containment under a normal distribution. This is the optimal entry boundary for statistical arbitrage:

\text{Upper Outer Band:} \quad \text{Band}_{\text{up, outer}} = \hat{P}_{A,i} + M_{\text{outer}} \times \sigma_{\text{spread}} \text{Lower Outer Band:} \quad \text{Band}_{\text{low, outer}} = \hat{P}_{A,i} - M_{\text{outer}} \times \sigma_{\text{spread}}

D. Extreme Stop-Out Bands (Z = +-2.5 / Capitulation Channel)

A high-volatility cushion representing a 98.8% probability limit. Reaching this zone suggests a severe cointegration breakdown or macro capitulation. Useful for absolute stop-losses or hyper-aggressive reversal entries:

\text{Upper Extreme Band:} \quad \text{Band}_{\text{up, extreme}} = \hat{P}_{A,i} + M_{\text{extreme}} \times \sigma_{\text{spread}} \text{Lower Extreme Band:} \quad \text{Band}_{\text{low, extreme}} = \hat{P}_{A,i} - M_{\text{extreme}} \times \sigma_{\text{spread}}

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4. The Statistically Pure Cutoff (Session-Start Noise Filtering)

When employing session anchors (ANCHOR_SESSION or ANCHOR_CUSTOM_SESSION), the active window size resets to 1 at the beginning of each active period and increments bar-by-bar.

During the first 14 bars of a session, running OLS is statistically invalid due to severe degrees-of-freedom limitations. Calculating standard deviations on 3, 5, or 8 samples yields highly erratic, spiked, and squeezed channels that create false signals and compress the chart's vertical scale.

A. The EMPTY_VALUE Cutoff Solution

To maintain institutional quantitative standards, the bands strictly enforce a 15-bar minimum cutoff.

• The Rule: If N_{\text{active}} < 15 or standard deviation is \le 0, all 7 band buffers are populated with EMPTY_VALUE.
• The Visual Benefit: Instead of collapsing the channels onto the raw price line (which creates a messy, overlapping web of lines at the start of every session), the channels simply remain invisible during the initialization phase, rendering only when the mathematical model has stabilized.

B. Standard vs. MTF Cutoff Widths

You will observe that Multi-Timeframe (MTF) charts display a wider blank zone at the beginning of each session compared to standard single-timeframe charts. This is a mathematically correct scaling consequence:

• On a local M15 chart, 15 bars require exactly 3.75 hours of market activity to stabilize.
• On an H1 (MTF) chart, 15 bars require exactly 15 hours of market activity to stabilize.
• Because the higher timeframe requires longer historical duration to construct its OLS sample pool, the MTF version correctly maintains a wider blank zone, shielding the trader from pre-stabilization noise on the macro level.

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5. The Pure vs. Hybrid MTF Dilemma

When trading in a Multi-Timeframe (MTF) environment (e.g. tracking M5 cointegration on an M1 chart), a distinct structural divergence occurs between the main-chart bands and the separate-window oscillator:

A. The Discrepancy

You may observe the lower timeframe price (M1) pierce the M5 outer band on the main chart, while the separate-window MTF Z-Score remains neutral (Gray).

• The Reason: The main chart compares the live, real-time lower-timeframe price (P_{A, \text{ltf}}) against the static higher-timeframe band. If the price spikes violently during the 5-minute interval, it will visually pierce the band. However, the Pure MTF Oscillator computes the Z-Score using the closed higher-timeframe price (P_{A, \text{htf}}). Since the 5-minute candle hasn't closed yet or its average close is lower, the pure HTF Z-Score remains neutral.

B. Pure MTF vs. Hybrid MTF Configuration

• Pure MTF (Default): Calculates everything strictly on the higher timeframe. It provides the highest statistical stability and filters out intraday/micro-timeframe false breakouts.
• Hybrid MTF (Optional Custom Setup): Uses the higher timeframe's stable structural parameters (\beta_{\text{htf}}, \alpha_{\text{htf}}, and \sigma_{\text{spread, htf}}), but computes the Z-Score numerator using the live lower-timeframe price (P_{A, \text{ltf}}). \text{Z}_{\text{hybrid}} = \frac{P_{A, \text{ltf}} - \beta_{\text{htf}} P_{B, \text{ltf}} - \alpha_{\text{htf}}}{\sigma_{\text{spread, htf}}} Under this hybrid model, the separate window Z-Score is mathematically guaranteed to cross the \pm 2.0 boundaries at the exact second the price pierces the bands on the main chart.

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6. Parameters

A. Common Parameters

• Comparison Symbol (InpSecondSymbol): The benchmark asset to correlate with (Symbol B). Symbol A is automatically set to the chart's native _Symbol.
• Anchor Reset (InpAnchor): The reset anchor period (None, Session, Week, Month, Custom Session).
• Lookback (InpLookback): The rolling regression window size (Used if Anchor = None). Default: 120.
• 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 OrangeRed/DeepSkyBlue Core Entry Bands (Z=\pm 2.0).
• Outer Band Multiplier (InpOuterMultiplier): The Z-Score multiplier for the outer bands (Default: 2.0).
• Draw Extreme Bands (InpDrawExtremeBands): Toggle to draw the solid Crimson/DodgerBlue Stop/Reversal Bands (Z=\pm 2.5).
• Extreme Band Multiplier (InpExtremeMultiplier): The Z-Score multiplier for the extreme bands (Default: 2.5).

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7. 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 (Symbol A is set as the chart's main active asset):

Asset Class	Symbol A (Chart)	Symbol B (InpSecondSymbol)	Recommended TF	Lookback / Anchor	Inner / Outer / Extreme Mult	Trading Style & Concept
Energies	UKOIL (Brent)	USOIL (WTI)	M5 / M15	120 / ANCHOR_NONE	1.5 / 2.0 / 2.5	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 / 2.5	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 / 2.5	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 / 2.5	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 / 2.5	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 / 2.5	Group Arbitrage. High European index cointegration due to shared Eurozone macro factors.