Rolling Window Geometric Brownian Motion Projections

2 Jan

📊 Rolling GBM Projections + EV & Adjustable Confidence Bands
Overview

The Rolling GBM Projections + EV & Adjustable Confidence Bands indicator provides traders with a robust, dynamic tool to model and project future price movements using Geometric Brownian Motion (GBM). By combining GBM-based simulations, expected value (EV) calculations, and customizable confidence bands, this indicator offers valuable insights for decision-making and risk management.
Key Features

Rolling GBM Projections: Simulate potential future price paths based on drift (μμ) and volatility (σσ).
Expected Value (EV) Line: Represents the average projection of simulated price paths.
Confidence Bands: Define ranges where the price is expected to remain, adjustable from 51% to 99%.
Simulation Lines: Visualize individual GBM paths for detailed analysis.
EV of EV Line: A smoothed trend of the EV, offering additional clarity on price dynamics.
Customizable Lookback Periods: Adjust the rolling lookback periods for drift and volatility calculations.

Mathematical Foundation
1. Geometric Brownian Motion (GBM)

GBM is a mathematical model used to simulate the random movement of asset prices, described by the following stochastic differential equation:
dSt=μStdt+σStdWt
dSt​=μSt​dt+σSt​dWt​

Where:

StSt​: Price at time tt
μμ: Drift term (expected return)
σσ: Volatility (standard deviation of returns)
dWtdWt​: Wiener process (standard Brownian motion)

2. Drift (μμ) and Volatility (σσ)

Drift (μμ): Represents the average logarithmic return of the asset. Calculated using a simple moving average (SMA) over a rolling lookback period.
μ=SMA(ln⁡(St/St−1),Lookback Drift)
μ=SMA(ln(St​/St−1​),Lookback Drift)

Volatility (σσ): Measures the standard deviation of logarithmic returns over a rolling lookback period.
σ=STD(ln⁡(St/St−1),Lookback Volatility)
σ=STD(ln(St​/St−1​),Lookback Volatility)

3. Price Simulation Using GBM

The GBM formula for simulating future prices is:
St+Δt=St×e(μ−12σ2)Δt+σϵΔt
St+Δt​=St​×e(μ−21​σ2)Δt+σϵΔt
​

Where:

ϵϵ: Random variable from a standard normal distribution (N(0,1)N(0,1)).

4. Confidence Bands

Confidence bands are determined using the Z-score corresponding to a user-defined confidence percentage (CC):
Upper Band=EV+Z⋅σ
Upper Band=EV+Z⋅σ
Lower Band=EV−Z⋅σ
Lower Band=EV−Z⋅σ

The Z-score is computed using an inverse normal distribution function, approximating the relationship between confidence and standard deviations.
Methodology

Rolling Drift and Volatility:
Drift and volatility are calculated using logarithmic returns over user-defined rolling lookback periods (default: μ=20μ=20, σ=16σ=16).
Drift defines the overall directional tendency, while volatility determines the randomness and variability of price movements.

Simulations:
Multiple GBM paths (default: 30) are generated for a specified number of projection candles (default: 12).
Each path is influenced by the current drift and volatility, incorporating random shocks to simulate real-world price dynamics.

Expected Value (EV):
The EV is calculated as the average of all simulated paths for each projection step, offering a statistical mean of potential price outcomes.

Confidence Bands:
The upper and lower bounds of the confidence bands are derived using the Z-score corresponding to the selected confidence percentage (e.g., 68%, 95%).

EV of EV:
A running average of the EV values, providing a smoothed perspective of price trends over the projection horizon.

Indicator Functionality

User Inputs:
Drift Lookback (Bars): Define the number of bars for rolling drift calculation (default: 20).
Volatility Lookback (Bars): Define the number of bars for rolling volatility calculation (default: 16).
Projection Candles (Bars): Set the number of bars to project future prices (default: 12).
Number of Simulations: Specify the number of GBM paths to simulate (default: 30).
Confidence Percentage: Input the desired confidence level for bands (default: 68%, adjustable from 51% to 99%).

Visualization Components:
Simulation Lines (Blue): Display individual GBM paths to visualize potential price scenarios.
Expected Value (EV) Line (Orange): Highlight the mean projection of all simulated paths.
Confidence Bands (Green & Red): Show the upper and lower confidence limits.
EV of EV Line (Orange Dashed): Provide a smoothed trendline of the EV values.
Current Price (White): Overlay the real-time price for context.

Display Toggles:
Enable or disable components (e.g., simulation lines, EV line, confidence bands) based on preference.

Practical Applications

Risk Management:
Utilize confidence bands to set stop-loss levels and manage trade risk effectively.
Use narrower confidence intervals (e.g., 50%) for aggressive strategies or wider intervals (e.g., 95%) for conservative approaches.

Trend Analysis:
Observe the EV and EV of EV lines to identify overarching trends and potential reversals.

Scenario Planning:
Analyze simulation lines to explore potential outcomes under varying market conditions.

Statistical Insights:
Leverage confidence bands to understand the statistical likelihood of price movements.

How to Use

Add the Indicator:
Copy the script into the TradingView Pine Editor, save it, and apply it to your chart.

Customize Settings:
Adjust the lookback periods for drift and volatility.
Define the number of projection candles and simulations.
Set the confidence percentage to tailor the bands to your strategy.

Interpret the Visualization:
Use the EV and confidence bands to guide trade entry, exit, and position sizing decisions.
Combine with other indicators for a holistic trading strategy.

Disclaimer

This indicator is a mathematical and statistical tool. It does not guarantee future performance.
Use it in conjunction with other forms of analysis and always trade responsibly.

Happy Trading! 🚀

3 hari yang lalu

Catatan Rilis

Tradingview mods didn't like my chart. So I'm trying again. Literally nothing different in the code.

kemarin dulu

Catatan Rilis

This update presents a significant enhancement to the script. In this version, the author has implemented adjustable skewness and kurtosis elements, complete with damping mechanisms that allow for refined control over the tail behavior and asymmetry in the distribution of price movements.
Key Enhancements

Adjustable Skewness and Kurtosis
The updated model now includes parameters to adjust skewness and kurtosis with corresponding lookback windows. These parameters are extraordinarily sensitive to market dynamics, particularly gap ups and gap downs. The author has provided the following means of modulation:
Skewness/Kurtosis Lookback Window: The default is set to 30 bars. If the signal exhibits instability — for example, projections that become excessively outlandish — it is recommended to reduce the lookback window by increments of 5, with the minimum allowable window being ten bars.
Damping Mechanism: The damping factor applied to the skewness and kurtosis elements can be adjusted. A lower numerical value (e.g., 0.001 instead of 0.003) increases the dampening effect, ensuring that overly volatile signals are moderated.

Adaptive Stability Measures
Recognizing that both skewness and kurtosis are highly responsive to abrupt market changes, additional suggestions include:
If adjusting the lookback window and damping parameters is insufficient, consider reducing the volatility lookback periods along with the overall volatility settings.
Users should note that each adjustment significantly affects projection outputs; therefore, careful tuning is necessary to maintain the utility of the indicator under highly volatile market conditions.

Mathematical Underpinnings
Skewness

Skewness is a measure of asymmetry in the distribution of returns. Mathematically, for a sample {x1,x2,…,xn}{x1​,x2​,…,xn​} with mean μμ and standard deviation σσ, skewness is defined as:
Skewness=1n∑i=1n(xi−μσ)3
Skewness=n1​i=1∑n​(σxi​−μ​)3

A positive skew indicates a longer right tail, while a negative skew signifies a longer left tail. Adjusting the skewness in the model allows it to better capture and reflect asymmetries in market behavior.
Kurtosis

Kurtosis measures the “tailedness” of the distribution compared to a normal distribution. Its formula for a sample is:
Kurtosis=1n∑i=1n(xi−μσ)4
Kurtosis=n1​i=1∑n​(σxi​−μ​)4

A normal distribution has a kurtosis of 3 (excess kurtosis of 0 when subtracting 3). Adjusting kurtosis is crucial for managing the risk associated with extreme price moves or “fat tails.”
Distribution Insights

Normal Distribution: A symmetric distribution with a kurtosis of 3, where most values cluster around the mean.
Leptokurtic Distribution: Characterized by a sharper peak and fatter tails (excess kurtosis > 0). This phenomenon indicates a higher probability of extreme outcomes, often referred to in the financial industry as “fat tails.” Proper modeling of this behavior is important to manage the risk of substantial deviations from the average trend.
Platykurtic Distribution: Exhibits a flatter peak and thinner tails (excess kurtosis < 0), suggesting fewer extreme outcomes relative to a normal distribution.

The author's update ensures that the model can dynamically adjust to reflect these distribution characteristics. Notably, the implementation of these adjustments has revealed a distinct "kurtosis smile" in the chart—an observation where the curvature of the kurtosis plot becomes more pronounced when accounting for adjusted skewness and kurtosis. This phenomenon aligns with market behaviors where distribution tails and peaks vary nonlinearly, offering deeper insights into market volatility.
Practical Adjustments & Usage

For optimal performance:

Instability Issues: When projections become unstable (e.g., disappearing due to outlandish price projections), try reducing the skewness/kurtosis lookback window by 5 bars (ensuring it does not drop below 10 bars).
Increasing Dampening: If the lookback adjustment does not stabilize the signal, enhance the damping effect by reducing the damping constant from 0.003 to 0.001.
Further Adjustments: If neither modification suffices, consider reducing the volatility lookback periods to accommodate highly volatile market conditions where the indicator might otherwise lose its predictive utility.

Conclusion

This update offers a more nuanced and dynamic approach to price projection modeling by incorporating damped, adjustable skewness and kurtosis. The refined control over these parameters provides traders with the ability to adjust their models to various market conditions—particularly when facing sudden price gaps or increased volatility. As always, thorough testing and individual tuning are essential for aligning the model with one's specific trading strategy and market observations. NOTE: The default settings have been set to only show the EV, with 55 bars in to the future. If you would like to continue to see that far, be sure to add more instances of the indicator to your chart. After adding a new instance of the indicator, turn off the EV line and turn on a different line, such as upper or lower confidence bands. TradingView will not display any more than 55 bars ahead, so only one line at a time can be turned on when looking that far ahead, but adding more instances of the indicator makes getting around this easier. Enjoy! Anchored version will receive updates soon.