# Technical Studies Reference

### Historical Volatility Ratio

This study calculates and displays the Historical Volatility Ratio of the Closing Prices.

Let $$C$$ be a random variable denoting the Closing Price, and let $$C_t$$ be the value of the Closing Price at Index $$t$$. Then we denote the Logarithmic Return at Index $$t$$ for the Closing Prices as $$LR_t$$, and we compute it for $$t > 0$$ as follows.

$$\displaystyle{LR_t = \ln\left(\frac{C_t}{C_{t-1}}\right)}$$

Let the Inputs Short Length and Long Length be denoted as $$n_S$$ and $$n_L$$, respectively, and let $$LR$$ be a random variable denoting the Logarithmic Return for $$C$$. The Standard Deviations of $$LR$$ with Lengths $$n_S$$ and $$n_L$$ at Index $$t$$ are denoted as $$\sigma_t\left(LR,n_S\right)$$ and $$\sigma_t\left(LR,n_L\right)$$, respectively. $$\sigma_t\left(LR,n_S\right)$$ is calculated for $$t \geq n_S$$, and $$\sigma_t\left(LR,n_L\right)$$ is computed for $$t \geq n_L$$.

The Historical Volatility Ratio at Index $$t$$ for the Closing Prices and the given Lengths is denoted as $$HVR_t\left(n_S,n_L\right)$$, and we compute it in terms of these Standard Deviations for $$t \geq n_L$$ as follows.

$$\displaystyle{HVR_t\left(n_S,n_L\right) = \frac{\sigma_t\left(LR,n_S\right)}{\sigma_t\left(LR,n_L\right)}}$$