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# 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. Explicit formulas for these quantities are given below.

$$\max_t(H,n_K) = \max\{H_{t - n_K + 1},...,H_t\}$$

$$\min_t(L,n_K) = \min\{L_{t - n_K + 1},...,L_t\}$$

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 $$Range_t(n_K)$$ and $$RelRange_t(n_K)$$, and we compute these for $$t \geq \max\{n_K,n_D\} as follows. \(Range_t(n_K) = \max_t(H,n_K) - \min_t(L,n_K)$$

$$RelRange_t(n_K) = C_t - \displaystyle{\frac{\max_t(H,n_K) + \min_t(L,n_K)}{2}}$$

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(EMA(RelRange(n_K),n_D),n_D)}{EMA_t(EMA(Range(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.