Ornstein-Uhlenbeck Mean-Reversion Calibrator

Commodity prices, credit spreads, and pairs-trade residuals all revert toward some long-run level, eventually. This calibrates exactly how fast, and shows you honestly how much noise sits in that estimate.

How To Use This Model

Reading This Tool

Set a true mean-reversion speed, long-run mean, and volatility, then simulate a path and calibrate it back.

The tool runs OLS regression of daily changes on levels to recover θ, μ, and σ. Compare calibrated to true values, the gap is genuine estimation noise, not an error.

True Process Parameters

True Mean-Reversion Speed (θ) 2.00

True Long-Run Mean (μ) 100

True Volatility (σ) 5.0

Starting Value (X₀) 110

Sample Length (Trading Days) 1000

Simulation Seed 42

Simulation uses daily steps (Δt = 1/252). Try increasing sample length and watch the calibrated θ converge closer to the true value, a direct demonstration of estimation consistency.

True Vs. Calibrated Parameters

θ: True → Calibrated

μ: True → Calibrated

σ: True → Calibrated

Half-Life Of Reversion

Simulated Path vs. Long-Run Mean

Continuous-Time Stochastic Processes

The Core Formulas

dX_t = θ(μ−X_t)dt + σdW_t
Discretized regression: ΔX_t = a + b·X_t + ε, where b = −θΔt, a = θμΔt
Half-life = ln(2) / θ

θ, μ are recovered from the OLS slope and intercept of regressing period-over-period changes on levels; σ is recovered from the regression's residual standard deviation, rescaled by √Δt.

When To Actually Use This Model

Modeling mean-reverting commodity prices, interest rate spreads, or convenience yields in a derivatives or commodities course.
Calibrating a pairs-trading or statistical arbitrage strategy's entry and exit bands around a reverting spread.
Teaching the Vasicek short-rate model, which uses this exact process for the instantaneous interest rate.
Demonstrating small-sample estimation bias and consistency in continuous-time econometrics.

Key Assumptions & Limitations

Assumes constant θ, μ, and σ over the full sample; real spreads often show regime changes.
OLS-based calibration is simple but known to be biased in small samples specifically for θ; MLE or more advanced estimators reduce but don't eliminate this.
Assumes Gaussian innovations; many real financial series exhibit fatter tails than this implies.
A negative or implausibly large calibrated θ from real data usually signals the series isn't genuinely mean-reverting at all.

Foundational Reference

Uhlenbeck, G. E., & Ornstein, L. S. (1930). On the Theory of the Brownian Motion. Physical Review, 36(5), 823-841.

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