Volatility Bands Using t-Distribution & Prediction IntervalsThis indicator is a statistical tool designed to project a dynamic range where the next asset price is expected to fall within a specific level of confidence.
Unlike standard volatility bands, a prediction interval is a statistical range that estimates where a single future observation will fall, given a specific probability. Because predicting a single future outcome introduces more inherent uncertainty than predicting an average, these bands are wider and mathematically tighter for forecasting next-bar anomalies.
How it Works
This indicator uses a Student's t-distribution (or an optional z-distribution for performance) to calculate historical volatility thresholds.
Finding the Estimated t-distribution
When you select a 95% Confidence Level, the script sets an error alpha of 5% (1 - 0.95). Because asset price movement can deviate to either the upside or downside, it splits this error equally into both tails (a two-tailed test). The script normalizes this, targeting an exact area of 0.025 (2.5%) in the extreme tails of the curve.
To find where that target area lies, the script reconstructs the Probability Density Function (PDF) of the Student's t-distribution. The height of the curve depends heavily on the Degrees of Freedom (df = length - 1).
Because Pine Script doesn't have a native Gamma function, the script uses double factorials to calculate the complex math coefficients. If your sample size (length) is exceptionally high (>300), the t-distribution naturally mirrors a regular normal distribution, so it switches to a standard Gaussian curve equation to save some time.
// Calculate double factorial
double_fac(int n) =>
float res = 1.0
int curr = n
while curr > 1
res := res * curr
curr := curr - 2
res
// t-distribution formula
cur_dist(float x, int cur_df) =>
float res = 0.0
if cur_df > 300
res := 0.3989422804014326779 * math.exp(-x * x * 0.5)
else
float coeff = double_fac(cur_df - 1) / (math.sqrt(cur_df) * double_fac(cur_df - 2) * (cur_df % 2 == 0 ? 2.0 : math.pi))
res := coeff * math.pow(1.0 + x * x / cur_df, -0.5 * (cur_df + 1))
res
To find the approximate t-value that corresponds to our target tail area, the script performs numerical integration using the Trapezoid Rule (trapezoid_reverse). This is the most time consuming part.
// Reverse trapezoidal to calculate t-value from area
trapezoid_reverse(float p, int cur_df) =>
float max_p = 0.5
float max_stat = 300.0
float add_prec = 1.0
float result_i = 0.0
if p == max_p
result_i := 100000000.0 // Infinity
else
float trap_sum = 0.0
float gap_width = p / (add_prec * 1250.0)
float i = 0.0
float last_func = cur_dist(0.0, cur_df)
while trap_sum < (2500.0 * add_prec) and i < max_stat
i := i + gap_width
trap_sum := trap_sum + last_func
last_func := cur_dist(i, cur_df)
trap_sum := trap_sum + last_func
result_i := i
result_i
Calculating the Prediction Interval
Once the t-critical value is found, it is then put into the standard prediction interval formula. The script uses EMA instead of SMA to improve responsiveness to current trends.
float prediction_interval_top = mean + tCrit * stdev * math.sqrt(1 + 1/length)
float prediction_interval_bot = mean - tCrit * stdev * math.sqrt(1 + 1/length)
Because the calculations for the t-distribution estimates are computationally expensive, you can select the lookback window for using the t-distribution. You can also choose to use the faster z-score for historical bars.
Note 1: The larger the sample size, the smaller you may have to set your lookback window in order to fit within the execution time limits.
You can use this for:
Overbought/Oversold, Mean reversion signals
Dynamic Stop-loss/Take-profit (SL/TP)
1. Overbought/Oversold/Mean Reversion
The indicator creates a red background to indicate that the candlestick has broken above the top 95% prediction interval band. It can be interpreted as a potential reversal.
2. Dynamic Stop-loss/Take-profit (SL/TP)
You can utilize the plotted mean and the confidence interval as entry/exit points. For example, you could put a stop-loss at the bottom band, and a take profit at the top band.
Alerts:
Price Above Top PI (Close crossed out)
Price Below Bottom PI (Close crossed out)
High Above Top PI (Wick touched/pierced top)
Low Below Bottom PI (Wick touched/pierced bottom)
Settings:
Length: The size of the sample for the standard deviation & mean calculations. 30 is recommended.
Source: Used for standard deviation & mean calculations. For example: Changing the source to 'high' would make the indicator predict the range of the next 'high'.
Confidence Level: Dictates how wide the bands should be. A 95% confidence level is default. Must be entered as a decimal between 0 and 1.
Lookback Length: Used to limit the amount of bars to back calculate the prediction interval using the t-distribution. Default is 300.
Use z-score: Use the z-score to calculated past values. Note that this will introduce error. For example, for a sample size of 30 and a 95% confidence level, using a z-score instead of a t-score will introduce a ~4.2% error in your interval width, causing your actual prediction interval to drop from 95% down to ~94.0%.
Background Colors:
Green: The low/close of the candle breached the lower band. Potential bullish reversal.
Red: The high/close of the candle breached the upper band. Potential bearish reversal.
Limitations:
The indicator assumes that the data follows a bell curve, which the markets do not.
The indicator may lag behind actual price action.
The indicator may produce false signals.
The indicator does not predict future prices.
Disclaimer: All trading decisions and responsibilities rest solely on the user of the indicator.
Индикатор Pine Script®
