# Technical Studies Reference

### Hurst Bands

This study calculates and displays Hurst Bands for the data specified by the Input Data Input. This indicator was developed by Jim Hurst.

Let $$X$$ be a random variable denoting the Input Data, and let the Inputs Moving Average Length, Inner Band Multiplier, Outer Band Multiplier, and Extreme Band Multiplier be denoted as $$n$$, $$v_I$$, $$v_O$$, and $$v_E$$, respectively. We denote the Displaced Price at Index $$t$$ as $$X^{(D)}_t(X,n)$$, and we compute it as follows.

$$\displaystyle{X^{(D)}_t(X,n) = X_{t - \left\lfloor\frac{n}{2}\right\rfloor - 1}}$$

For an explanation of the floor function ($$\left\lfloor{\space\space}\right\rfloor$$), refer to our description here.

There are six Hurst Bands: two Inner Bands, two Outer Bands, and two Extreme Bands. These are computed in terms of a Simple Moving Average of the Displaced Price for $$t \geq n + \left\lfloor\frac{n}{2}\right\rfloor$$ as follows.

Top and Bottom Inner Hurst Bands:

$$\displaystyle{TIB^{(H)}(X,n,v_I) = SMA_t\left(X^{(D)}(X,n),n\right) + \frac{v_I}{100} \cdot SMA_t\left(X^{(D)}(X,n),n\right)}$$
$$\displaystyle{BIB^{(H)}(X,n,v_I) = SMA_t\left(X^{(D)}(X,n),n\right) - \frac{v_I}{100} \cdot SMA_t\left(X^{(D)}(X,n),n\right)}$$

Top and Bottom Outer Hurst Bands:

$$\displaystyle{TOB^{(H)}(X,n,v_O) = SMA_t\left(X^{(D)}(X,n),n\right) + \frac{v_O}{100} \cdot SMA_t\left(X^{(D)}(X,n),n\right)}$$
$$\displaystyle{BOB^{(H)}(X,n,v_O) = SMA_t\left(X^{(D)}(X,n),n\right) - \frac{v_O}{100} \cdot SMA_t\left(X^{(D)}(X,n),n\right)}$$

Top and Bottom Extreme Hurst Bands:

$$\displaystyle{TEB^{(H)}(X,n,v_E) = SMA_t\left(X^{(D)}(X,n),n\right) + \frac{v_E}{100} \cdot SMA_t\left(X^{(D)}(X,n),n\right)}$$
$$\displaystyle{BEB^{(H)}(X,n,v_E) = SMA_t\left(X^{(D)}(X,n),n\right) - \frac{v_E}{100} \cdot SMA_t\left(X^{(D)}(X,n),n\right)}$$

In addition to these six bands, the moving average $$SMA_t\left(X^{(D)}(X,n),n\right)$$ is also displayed.

Note: Depending on the setting of the Input Moving Average Type, the Simple Moving Averages in the above formulas could be replaced with Exponential Moving Averages, Linear Regression Moving Averages, Weighted Moving Averages, Wilders Moving Averages, Simple Moving Averages - Skip Zeros, or Smoothed Moving Averages.

Note: The default settings for the Band Multipliers are $$v_E = 0.05$$, $$v_O = 0.025$$, and $$v_I = 0.0125$$, which are ideal for Intraday charts with time scales up to about 10 minutes. As the time scale increases, the values of these multipliers must be manually increased for best results. For Daily charts, the recommended settings are $$v_E = 4.2$$, $$v_O = 2.6$$, and $$v_E = 1.6$$. The spreadsheet test for this study was done on a Daily chart with these settings.