# KD - Fast

This study calculates and displays both the Fast %K and the Fast %D stochastic indicators for the data specified by the Input Data for High, Input Data for Low, and Input Data for Last Inputs.

Let $$X^{(High)}$$, $$X^{(Low)}$$, and $$X^{(Close)}$$ be random variables denoting the Input Data for High, Input Data for Low, and Input Data for Last, respectively, and let $$X_t^{(High)}$$, $$X_t^{(Low)}$$, and $$X_t^{(Last)}$$ be their respective values at Index $$t$$. Let the Inputs %K Length and %D Length be denoted as $$n_{FastK}$$ and $$n_{FastD}$$, respectively. The two indicators for KD - Fast are Fast%K and Fast%D (aka Slow%K). We compute them for $$t \geq n_{FastK} + n_{FastD}$$ as follows.

$$Fast\% K_t(X^{(High)},X^{(Low)},X^{(Close)},n_{FastK}) = \displaystyle{\left\{ \begin{matrix} 100\cdot\frac{X_t^{(Close)} - \min_t\left(X_t^{(Low)},n_{FastK}\right)}{\max\left(X_t^{(High)},n_{FastK}\right) - \min_t\left(X_t^{(Low)},n_{FastK}\right)} & \max\left(X_t^{(High)},n_{FastK}\right) - \min_t\left(X_t^{(Low)},n_{FastK}\right) \neq 0 \\ 100 & \max\left(X_t^{(High)},n_{FastK}\right) - \min_t\left(X_t^{(Low)},n_{FastK}\right) = 0 \end{matrix}\right .}$$

For an explanation of the $$\min$$ and $$\max$$ functions in the above formula, see our descriptions of the Moving Minimum and Moving Maximum.

Fast %D is calculated in terms of a Simple Moving Average, as shown below.

$$Fast\% D_t(X^{(High)},X^{(Low)},X^{(Close)},n_{FastK},n_{FastD}) = SMA_t(Fast\% K(X^{(High)},X^{(Low)},X^{(Close)},n_{FastK}),n_{FastD})$$

Note: For the purposes of computing the Simple Moving Average in the above formula, internal calculations for Fast %K are carried out for $$n_{FastK} + 1 \leq t < n_{FastK} + n_{FastD}$$. These values are not displayed as output.

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

In addition to the graphs of Fast %K and Fast %D, this study also displays two horizontal lines whose levels are determined by the Inputs Line1 Value and Line2 Value.