Why is volatility in a positive direction clubbed in the same risk category as volatility in a negative direction?

Question

Let's say a stock's price has been steadily rising for the last 6 months.

It's going to have (a relatively) high volatility.

Let's say another stock's price has been falling steadily for the last 6 months.

It's going to have high (a relatively) volatility.

High volatility = High risk.

Assuming both stocks have the exact same volatility, ceteris paribus, both will have the same risk.

To my (perhaps non-sophisticated) brain, this doesn't make sense.

Is there a measure of volatility that weighs movement in the positive direction differently than movement in the negative direction? Shouldn't there be one?

Or am I thinking this wrong?

Answer 1

Mostly, when an equity's price rises, its statistical and implied volatilities fall and vice versa.

The reason why is a mathematical phenomenon mixed with the reality that a unceasingly falling asset price will soon not exist, skewing the results with survivorship bias.

Since volatility is standard deviation of price indexes, a security that changes in price by the same amount every day will have lower volatility, so a rising price will have lower implied volatility because its mostly experiencing positive daily price change while a recently falling price will have higher volatility because factored together with the positive price changes, the negative price changes will widen the standard deviation of the securities price index.

Quantitatively, any change, in or out of one's favor, is a risk because change is uncertain, and any uncertainty is a risk.

This quantitative interpretation while valid runs almost totally counter to the value opinion, that a lower price relative to value is a lower risk than a higher price relative to value, but both have their place in time.

Over long time periods, it's best to use the value interpretation, quantitative for shorter. Using the opposite has hastily destroyed many a fund manager.