Statistical Arbitrage in MQL5: Implementing Rolling OLS Cointegration & Mean-Reversion Half-Life

Retail trading robots fail at pairs trading because they rely on Pearson correlation—a non-stationary metric prone to spurious regression. This guide provides the complete mathematical derivation and zero-allocation MQL5 C++ implementation of Engle-Granger cointegration, rolling ordinary least squares ($\beta$), and Ornstein-Uhlenbeck half-life estimation.

1. The Correlation vs. Cointegration Trap (Granger-Newbold Fallacy)

The most common catastrophic failure in algorithmic pairs trading is confusing Pearson correlation ($r$) with cointegration. Pearson correlation measures instantaneous co-movement between two series:

Financial asset prices are non-stationary, integrated of order one: $X_t, Y_t \sim I(1)$. Regressing two non-stationary series yields statistically significant correlation ($r > 0.85$) and high $t$-statistics purely due to shared macro drift (Granger & Newbold, 1974). If you trade pairs based on high correlation, when the macro drift separates, the spread diverges infinitely, blowing up the account.

2. Rolling Ordinary Least Squares (OLS) Closed-Form Beta

Fixed hedge ratios ($\beta = 1.0$) fail in foreign exchange and commodities because relative instrument volatilities and cross-currency purchasing power shift over time. We implement rolling closed-form OLS over a rolling window $N$:

Once the dynamic hedge ratio $\beta$ is computed, the synthetic spread $S_t$ is established. Sizing is scaled so that for every $1.0$ lot of Leg A traded, exactly $\beta \cdot (\text{TickValue}_A / \text{TickValue}_B)$ lots of Leg B are simultaneously hedged to create a dollar-neutral portfolio.

3. Ornstein-Uhlenbeck (OU) Mean-Reversion Half-Life Estimation

A cointegrated spread follows a continuous-time Ornstein-Uhlenbeck stochastic differential equation:

Where:

• $\theta > 0$ represents the rate of mean reversion (speed at which spread snaps back).
• $\mu$ is the long-term equilibrium mean of the synthetic spread.
• $\sigma$ is the instantaneous diffusion volatility of the spread residuals.
• $W_t$ is standard Brownian motion.

Discretizing via an $\text{AR}(1)$ delta regression ($\Delta S_t = a + b S_{t-1} + u_t$) gives the direct discrete mapping:

The Half-Life ($\tau_{1/2}$) represents the exact expected time required for a spread divergence to decay by 50%. In institutional execution, if $\tau_{1/2} > \frac{N}{2}$ or $b \ge 0$, the series is diagnosed as non-stationary, and all entry signals are strictly gated.

4. Production-Ready Zero-Allocation MQL5 Class

High-frequency tick evaluation demands that no heap allocations occur in the OnTick() critical path. The following production-ready MQL5 class encapsulates pre-allocated ring buffers, analytical OLS calculation, and Ornstein-Uhlenbeck half-life validation:

5. Market Microstructure: Spread Drag & Execution Slippage

Statistical arbitrage strategies live or die on execution efficiency. Because each trade involves two concurrent legs (opening Leg A and Leg B simultaneously), you pay double the bid-ask friction.

Frequently Asked Questions (Quant Engineering)