# Technical Studies Reference

### Chande Momentum Oscillator

This study calculates and displays the Chande Momentum Oscillator of the data specified by the Input Data Input, as well as three horizontal lines determined by the user.

Let $$X$$ be a random variable denoting the Input Data, and let $$X_t$$ be the value of the Input Data at Index $$t$$. The Up Change and Down Change of the Input Data at Index $$t$$ are denoted as $$\Delta X_t^{(+)}$$ and $$-\Delta X_t^{(-)}$$, respectively, and we compute them for $$t > 0$$ as follows.

$$\displaystyle{\Delta X_t^{(+)} =\left\{ \begin{matrix} X_t - X_{t - 1} & X_t \geq X_{t - 1} \\ 0 & X_t < X_{t - 1} \end{matrix}\right .}$$

$$\displaystyle{-\Delta X_t^{(-)} =\left\{ \begin{matrix} 0 & X_t \geq X_{t - 1} \\ X_{t - 1} - X_t & X_t < X_{t - 1} \end{matrix}\right .}$$

Let the Input CMO Length be denoted as $$n_{CMO}$$. Then we denote the Up Sum and Down Sum at Index $$t$$ for the given Inputs as $$S_t^{(+)}(X,n_{CMO})$$ and $$S_t^{(-)}(X,n_{CMO})$$, respectively, and we compute them for $$t \geq n_{CMO}$$ as follows.

$$S_t^{(+)}(X,n_{CMO}) = \displaystyle{\sum_{i = t - n_{CMO} + 1}^t \Delta X_i^{(+)}}$$

$$S_t^{(-)}(X,n_{CMO}) = \displaystyle{\sum_{i = t - n_{CMO} + 1}^t \left(-\Delta X_i^{(-)}\right)}$$

For an explanation of the Sigma ($$\Sigma$$) notation for summation, refer to our description here.

We denote the Chande Momentum Oscillator at Index $$t$$ for the given Inputs as $$CMO_t(X,n_{CMO})$$, and we compute it for $$t \geq n_{CMO}$$ as follows.

$$\displaystyle{CMO_t(X,n_{CMO}) =\left\{ \begin{matrix} 100 \cdot \frac{S_t^{(+)}(X,n_{CMO}) - S_t^{(-)}(X,n_{CMO})}{S_t^{(+)}(X,n_{CMO}) + S_t^{(-)}(X,n_{CMO})} & S_t^{(+)}(X,n_{CMO}) + S_t^{(-)}(X,n_{CMO}) \neq 0 \\ CMO_{n - 1}(X,n_{CMO}) & S_t^{(+)}(X,n_{CMO}) + S_t^{(-)}(X,n_{CMO}) = 0 \end{matrix}\right .}$$

Finally, the Line 1, Line 2, and Line 3 Inputs determine the levels of three horizontal lines that are displayed with $$CMO_t(X,n_{CMO})$$.