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Ornstein-uhlenbeck

Statistical Mean-Reversion Engine [SMRE]## Statistical Mean-Reversion Engine (SMRE) SMRE is an open-source mean-reversion indicator that combines a rigorous statistical core with up to eight optional confirmation layers, designed primarily for index-futures trading on intraday timeframes (1-minute through 1-hour). ### What it does For every bar, SMRE fits an Ornstein-Uhlenbeck (OU) process to the recent price series via linear regression on lag-1 prices, yielding four outputs: - **μ (the mean)** — the equilibrium price the series is reverting to - **θ (mean-reversion speed)** — how strongly the series pulls back to μ - **HL (half-life)** — how many bars it takes to revert halfway - **σ_eq (stationary residual variance)** — used to z-score the current price The current price's z-score against μ (the "OU Z") is the primary signal. When |OU Z| exceeds a configurable threshold, a mean-reversion entry is considered — but only after the script also confirms that the recent price series is genuinely stationary using three orthogonal statistical tests: - **Hurst exponent** must be below 0.55 (i.e., the series is not persistently trending) - **Augmented Dickey-Fuller** t-statistic must be below -2.86 (rejects unit root) - **Variance Ratio** test at q=4 must be below 1.0 (variance grows sub-linearly with horizon) If all four conditions pass, the L1 (statistical core) signal fires. ### Why the multi-layer structure (mashup justification) A single OU-based mean-reversion signal works well in stationary regimes but degrades in trending or volatile conditions. SMRE addresses this by validating each potential entry through up to eight orthogonal confirmation channels, each measuring something the others do not: - **L2 — Volatility Regime (6-state):** Classifies market state via VIX, ADX, and realized volatility. Suppresses signals during high-trend conditions (regime 6, "Spike") where mean-reversion historically fails. - **L3 — Spot-Futures Basis (Kalman filter):** Tracks the deviation between actual and theoretical futures pricing. Statistically significant basis dislocations often resolve via mean-reversion. - **L4 — Options Surface:** Computes ATM implied volatility from straddle pricing and a skew z-score from OTM put/call ratio. Optional; requires user to provide option symbols. - **L5 — Microstructure:** Blends rolling VWAP and session-anchored VWAP z-scores with VPIN (a volume-clock toxicity proxy) and order-flow imbalance. Captures flow-based exhaustion. - **L6 — Gamma Walls (GEX) OR Put-Call Ratio:** Two mutually exclusive options. GEX requires OI symbols at five strikes; PCR requires a single broker-published PCR feed. Both detect option-driven price magnets. - **L7 — Dispersion:** Rolling correlation of index returns with its top 5 constituent stocks' returns. High dispersion (low correlation) penalizes signals; high cohesion boosts them. - **L7b — Residual Dispersion:** Idiosyncratic residual z-scores (β-adjusted) per constituent. If 3 of 5 stocks show same-sign extreme residuals, the index is detached from constituents — strong mean-reversion candidate. - **L9 — Cross-Asset Stress:** Sigma-normalized stress across USD/INR, DXY, and crude oil. Penalizes signals during cross-asset hedging cascades. Each layer outputs a {direction, strength} pair. The Layer 8 fusion engine combines these via a weighted composite score (default weights: L1=0.28, L5=0.22, L3=0.18, L4=0.12, L6/L7b=0.10), then applies a regime multiplier (L2 × L7 × VRP × cross-asset × expiry), clamped to to prevent extreme compounding. If the absolute composite score crosses one of three thresholds (0.25 / 0.40 / 0.45 by default), a signal is fired at Scalp / Swing / Session horizon respectively. A TCA cost filter then validates that the expected move (distance to μ) exceeds estimated round-trip transaction cost; otherwise the signal is suppressed. ### Originality The author is not aware of any other public Pine script that implements the full OU-fit chain (mean, mean-reversion speed, half-life, stationary variance) together with all three stationarity tests (Hurst, ADF, Variance Ratio) directly in Pine v6 — every step is computed natively, no external library calls. Additionally, the session-anchored VWAP with running volume-weighted sigma bands, the rolling-beta residual dispersion across multiple constituents, and the Kalman-filtered futures-basis residual are original Pine implementations. The signal telemetry module (a 200-signal FIFO ring buffer with horizon × composite-magnitude bucket attribution) is also an original diagnostic tool. ### How to use 1. **Apply to an index futures chart.** Defaults are pre-configured for NSE NIFTY1! futures, but inputs allow any index — change the VIX symbol, spot/futures symbols, constituent symbols, and currency pairs. 2. **Read the compact dashboard.** It's a single 9-row table (default position: middle-right) showing only what you need to evaluate a setup: | Row | What it shows | What it means | |---|---|---| | Title | Profile + OU window in use | Confirms which calibration is active | | OU Z-Score | Z-score with half-life (HL) | How extended price is + how long mean-reversion typically takes | | Stat Validity | H / ADF / VR pass-fail | Whether the recent series is actually stationary (all 3 must pass) | | Regime | Volatility state + VIX value | Whether market conditions favor mean-reversion | | Composite | Fused score × regime multiplier | The unified signal strength | | Confluence | Layers agreeing (out of 6) | How many orthogonal signals support the direction | | TCA Edge | Expected move in bps + PASS/FAIL | Whether the trade clears transaction costs | | E / SL / TP | Entry, Stop, Target + Risk:Reward | The trade levels if a signal fires | | **DECISION** | Direction · Horizon · Side | The actionable output (green=long, red=short, gray=neutral) | 3. **Trade levels and markers.** When a signal fires, entry/stop/target lines auto-plot on the chart. Stop is ATR-based (default 1.2× ATR); target is min(OU mean μ, entry + 2× ATR). Triangle markers plot below (long) or above (short) the bar — small for Scalp, medium for Swing, large for Session. 4. **Optional diagnostic.** A separate Signal Telemetry table (disabled by default; enable via the "Show Telemetry Dashboard" input) tracks the last 200 signals' outcomes (win = price touched μ, loss = stop hit, expired = timeout) and reports hit rate by horizon × composite-magnitude bucket. This is a backward-looking diagnostic, not a backtest. ### Recommended chart and timeframe This indicator was developed and parameter-tested primarily on NIFTY1! futures. The OU window auto-mapping (1m→32, 2m→20, 5m→12, 15m→32, 30m→20, 1h→24) was selected empirically through parameter sweeps. Users on other instruments should expect to tune the OU window manually or accept the auto-mapped default as a starting point. The indicator works on any timeframe between 1 minute and daily, though intraday timeframes (1m through 1h) are where the multi-layer confluence adds the most value. ### Important notes - This is an **indicator**, not a strategy — no backtest equity curve is produced. The telemetry table is a descriptive measure of recent signal outcomes only. - Many layers are **optional**. If you don't have symbols for options OI, just leave those inputs blank; the script will redistribute composite weight naturally across the active layers. - Signals can fluctuate intra-bar before bar close, especially in real-time mode. For consistent behavior, evaluate signals on closed bars only. - The default constituents (top-5 NIFTY weights) need to be changed in the L7 inputs to use this on a different index. ### Disclaimer This indicator is published for educational and research purposes only. It is not financial advice, not an investment recommendation, and not a solicitation to trade. Past behavior of signals does not guarantee future results. Trading futures, options, and equities carries substantial risk of loss. You are solely responsible for your trading decisions. The author makes no representations about the accuracy, completeness, or suitability of this indicator for any particular purpose. Use at your own risk, and always consult a qualified financial professional before trading.
Pine Script® indicator
by Market_Logic_India
1
Ornstein-Uhlenbeck Mean Reversion Probability Bands [UAlgo]Ornstein-Uhlenbeck Mean Reversion Probability Bands is a statistical mean reversion indicator that models price as a mean reverting process and projects dynamic probability style zones around an estimated equilibrium mean. The script uses a rolling lookback of closing prices, fits an Ornstein-Uhlenbeck inspired parameter set from recent behavior, and then converts that estimate into inner and outer deviation bands around the current mean. The indicator runs directly on price ( overlay=true ) and is built to help traders identify when price is stretched away from its estimated equilibrium. Instead of using a fixed moving average and static standard deviation, the script attempts to infer a mean reverting structure from the data itself. It estimates the long term mean, the speed of reversion, and an equilibrium style dispersion measure, then plots two upside and two downside mean reversion zones. When price pushes into the upper or lower band regions, the script calculates a standardized distance from the estimated mean and displays a probability style label with both the percentage score and the current z score. This gives the user a quick visual read of how statistically extended price is relative to the model. A key strength of this script is that it combines: A rolling Ornstein-Uhlenbeck style parameter estimation Adaptive mean reversion zones Probability style stretch labels at band events A clean overlay presentation with visible upper and lower probability regions Important note: The percentage label in this script is a normal distribution coverage style score derived from the current z score. It is best understood as a probabilistic stretch measure, not a literal exact OU first passage probability. 🔹 Features 🔸 1) Ornstein-Uhlenbeck Inspired Mean Reversion Model The script estimates a mean reverting process from recent closing prices instead of relying only on a moving average. It uses a rolling regression style approach on consecutive price observations, then converts those estimates into Ornstein-Uhlenbeck style parameters. This makes the indicator more model driven than a standard band tool. 🔸 2) Rolling Adaptive Mean Line The central mean line is not a fixed average only. It is the estimated equilibrium level ( mu ) of the fitted process. As the rolling price sample changes, the model updates and the mean shifts with changing market structure. The mean line also changes color depending on whether current price is above or below that estimated equilibrium. 🔸 3) Dual Mean Reversion Zones (Inner and Outer) The script builds two sets of reversion bands around the mean: Inner bands using the inner multiplier Outer bands using the outer multiplier This creates a layered framework where the inner zone marks an early stretch area and the outer zone marks a more extreme statistical extension. 🔸 4) Probability Style Stretch Labels When price crosses into the upper or lower band regions, the script calculates a z score based on current distance from the estimated mean and converts it into a percentage style probability score. The label shows: A directional marker The probability style percentage The current z score This gives the user both a visual event trigger and a numeric measure of extension. 🔸 5) Visual Zone Based Design The indicator uses filled upper and lower zones rather than emphasizing the band lines themselves. This creates a cleaner chart display where the mean line stays visible and the stretch regions are highlighted as colored areas above and below it. This makes the indicator easy to read during fast chart scanning. 🔸 6) Configurable Lookback, Time Step, and Band Width Users can customize: The rolling lookback period used for model estimation The time step parameter ( dt ) used in OU conversion The inner band multiplier The outer band multiplier This makes the script adaptable to different timeframes, instruments, and preferred sensitivity levels. 🔸 7) Built In Estimation Safeguards The parameter estimation logic includes fallback protections. If the inferred model parameters are unstable or unrealistic, the script falls back to simpler sample statistics. This helps prevent unusable outputs during difficult market regimes or low quality fits. 🔸 8) Directional Touch Event Logic The script tracks both upper side and lower side band interaction: Upper side events can signal statistically stretched bullish price movement Lower side events can signal statistically stretched bearish price movement Labels are only created on crossing events, which helps reduce repeated prints while price remains outside the band. 🔹 Calculations 1) Rolling Price Queue Management The script stores recent closing prices in an array with a fixed maximum length: price_array.update_queue(close, length_input) The queue update method behaves differently depending on bar state: On a new bar, it pushes the latest value On an updating live bar, it overwrites the last stored value This keeps the rolling sample aligned with the current chart state without duplicating the active bar. 2) Fallback Mean and Dispersion Estimates Before attempting the OU style fit, the script calculates simple fallback values: float fallback_mu = src_array.avg() float fallback_sigma = src_array.stdev() These act as safety defaults if the regression based OU estimate is not reliable. Important note: In this script, fallback_sigma is a simple sample standard deviation of price levels, not return volatility. 3) AR(1) Style Regression on Consecutive Prices The model estimation is built from consecutive price pairs: x = price y = price The script computes: Mean of x Mean of y Covariance between x and y Variance of x Then it estimates: float b = sum_cov / sum_var_x This creates an AR(1) style coefficient that is later translated into OU style parameters. 4) Conversion from AR(1) Form to OU Style Parameters If the estimated b is within a valid range: if b > 0.05 and b < 0.95 the script computes: float a = mean_y - b * mean_x float mu_exact = a / (1.0 - b) float theta_exact = -math.log(b) / dt Interpretation: mu_exact is the estimated long run mean. theta_exact is the implied mean reversion speed. The conversion assumes the AR(1) relation is a discrete time representation of a mean reverting process. 5) Residual Variance and Equilibrium Dispersion The script next measures residual error from the AR(1) fit: float err = y_i - (a + b * x_i) float var_err = sum_err_sq / (n - 1) Then it converts that residual variance into an equilibrium variance estimate: float var_eq = var_err / (1.0 - b * b) Finally: float calc_sigma = math.sqrt(var_eq) Important implementation note: The variable named sigma in this script is used as an equilibrium style standard deviation around the mean, not as the continuous time OU diffusion coefficient from the SDE form. 6) Stability Filter for the Estimated Sigma Even if the AR(1) fit is mathematically valid, the script only accepts the calculated sigma when it is reasonably close to the fallback sample standard deviation: if calc_sigma < fallback_sigma * 1.5 and calc_sigma > fallback_sigma * 0.5 If this test fails, the script keeps the fallback values instead. This helps avoid unstable band widths caused by bad short term fits. 7) Final Parameter Output The estimation method returns: OU_Params.new(theta, mu, sigma_eq) Where: theta is the estimated reversion speed mu is the estimated equilibrium mean sigma_eq is the accepted equilibrium dispersion measure These parameters are then used to build the bands. 8) Band Construction The script computes four band levels around the estimated mean: float up_out = mean_val + (dev_val * mult_outer) float up_in = mean_val + (dev_val * mult_inner) float dn_in = mean_val - (dev_val * mult_inner) float dn_out = mean_val - (dev_val * mult_outer) Interpretation: Inner bands represent a milder deviation from the mean. Outer bands represent a more extreme deviation from the mean. 9) Mean and Zone Visualization The mean line is explicitly plotted: p_mean = plot(ou_bands.mean, color=color_mean, linewidth=2, title="Mean") The inner and outer band plots are also created, but their colors are fully transparent: color color_inner_up = color.new(#ffb74d, 100) color color_outer_up = color.new(#ef5350, 100) ... This means the visible structure mainly comes from the zone fills: fill(p_ui, p_uo, ...) fill(p_li, p_lo, ...) So the user sees clean upper and lower probability zones rather than several bright boundary lines. 10) Touch and Crossing Logic The script first checks whether price is currently inside a stretch area: bool touch_upper = close >= ou_bands.upper_inner bool touch_lower = close <= ou_bands.lower_inner Then it checks for fresh crossings: bool cross_up_in = ta.crossover(close, ou_bands.upper_inner) bool cross_up_out = ta.crossover(close, ou_bands.upper_outer) bool cross_dn_in = ta.crossunder(close, ou_bands.lower_inner) bool cross_dn_out = ta.crossunder(close, ou_bands.lower_outer) Labels are only created when price is touching the region and a fresh crossing occurs. This avoids creating labels on every bar that remains outside the band. 11) Z Score Calculation When an event occurs, the script calculates the standardized distance from the mean: float current_z_score = dev_val != 0 ? math.abs(close - mean_val) / dev_val : 0.0 Interpretation: A z score of 1 means price is one equilibrium standard deviation away from the estimated mean. Higher values indicate a more statistically stretched condition. 12) Probability Style Score Calculation The script converts the z score into a percentage style score using an approximation of the error function: float x = math.abs(z_score) / math.sqrt(2.0) ... float prob = erf_approx * 100.0 Because erf(|z| / sqrt(2)) corresponds to the probability mass within plus or minus that z distance under a normal distribution, the output behaves like a confidence or coverage score. Important note: This is not a direct OU mean reversion probability in the strict stochastic process sense. It is a normal distribution style stretch score based on the current z distance. 13) Upper Event Label Logic When price crosses into the upper band region: if (touch_upper and cross_up_in) or (touch_upper and cross_up_out) the script prints a bearish styled label above the bar: "▼ %" + str.tostring(probability, "#.##") + " (Z:" + str.tostring(current_z_score, "#.##") + ")" This reflects the idea that price is statistically extended above the mean and may be vulnerable to reversion. 14) Lower Event Label Logic When price crosses into the lower band region: if (touch_lower and cross_dn_in) or (touch_lower and cross_dn_out) the script prints a bullish styled label below the bar: "▲ %" + str.tostring(probability, "#.##") + " (Z:" + str.tostring(current_z_score, "#.##") + ")" This reflects the idea that price is statistically extended below the mean and may be vulnerable to reversion. 15) Role of the Time Step Input The dt_input parameter affects the conversion from the AR(1) coefficient into the OU reversion speed: float theta_exact = -math.log(b) / dt A larger dt lowers the inferred theta for the same b . A smaller dt raises the inferred theta for the same b .
Pine Script® indicator
by UAlgo
1