# Technical Studies Reference

### 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. Then we denote the two indicators for KD - Fast at Index $$t$$ for the given Inputs as $$Fast\% K_t(X^{(High)},X^{(Low)},X^{(Close)},n_{FastK})$$ and $$Fast\% D_t(X^{(High)},X^{(Low)},X^{(Close)},n_{FastK},n_{FastD})$$, respectively, and we compute them for $$t \geq n_{FastK} + n_{FastD}$$ as follows.

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

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

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

In the above formula, $$Fast\% K(X^{(High)},X^{(Low)},X^{(Close)},n_{FastK})$$ is a random variable denoting the Fast %K for the aforementioned Inputs.

Note: Fast %D is also known as Slow %K.

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.