# Technical Studies Reference

### Stochastic Momentum Indicator

This study calculates and displays both the Stochastic Momentum Indicator and its Exponential Moving Average.

Let $$H$$, $$L$$, and $$C$$ be random variables denoting the High, Low, and Closing Prices, respectively, and let their respective values at Index $$t$$ be $$H_t$$, $$L_t$$, and $$C_t$$. Let the Inputs %D Length, %K Length, and EMA Length be denoted as $$n_D$$, $$n_K$$, and $$n_{EMA}$$, respectively. We denote the Highest High and the Lowest Low over a sliding window of Length $$n_K$$ terminating at Index $$t$$ as $$\max_t(H,n_K)$$ and $$\min_t(L,n_K)$$, respectively.

We use these quantities to compute the Range and the Relative Range, each of which depends on $$n_K$$. The respective values of these quantities at Index $$t$$ are denoted as $$\textrm{Range}_t(H,L,n_K)$$ and $$\textrm{Range}^{(Rel)}_t(H,L,C,n_K)$$, and we compute these for $$t \geq \max\{n_K,n_D\}$$ as follows.

$$\textrm{Range}_t(H,L,n_K) = \max_t(H,n_K) - \min_t(L,n_K)$$

$$\textrm{Range}^{(Rel)}_t(H,L,C,n_K) = C_t - \displaystyle{\frac{\max_t(H,n_K) + \min_t(L,n_K)}{2}}$$

We denote the value of the Stochastic Momentum Indicator for the given Inputs at Index $$t$$ as $$SMI_t(n_K,n_D)$$, and we compute it for $$t \geq \max\{n_K,n_D\}$$ in terms of Exponential Moving Averages as follows.

$$SMI_t(n_K,n_D) = 200\cdot\displaystyle{\frac{EMA_t\left(EMA\left(\textrm{Range}^{(Rel)}(H,L,C,n_K),n_D\right),n_D\right)}{EMA_t(EMA(\textrm{Range}(H,L,n_K),n_D),n_D)}}$$

We denote the Average of the Stochastic Momentum Indicator for the given Inputs at Index $$t$$ as $$\overline{SMI}_t(n_K,n_D,n_{EMA})$$, and we compute it for $$t \geq \max\{n_K,n_D\}$$ as follows.

$$\overline{SMI}_t(n_K,n_D,n_{EMA}) = EMA_t(SMI(n_K,n_D),n_{EMA})$$

In the formulas for both the Stochastic Momentum Indicator and its Average, any Statistical Functions that are written without a subscript are understood to be Random Variables for the Statistical Functions in which they are contained.

In addition to the Subgraphs of $$SMI_t(n_K,n_D)$$ and $$\overline{SMI}_t(n_K,n_D,n_{EMA})$$, this study also displays horizontal lines whose levels are determined by the Inputs Overbought Value and Oversold Value.