abbreviations for "standard deviation" when used as an informal unit

"Standard deviation" is often abbreviated as "Std Dev" or "SD" in statistics, standard deviation providing an indication of how far data deviate from the mean.

Examples:

• So the SD(X) is the √Var(X). (citation)

• So the Std Dev of X is the sqrt of the Var of X. (citation)

You can also use a small sigma (i.e., σ), but gauge your audience. If you are writing for an audience of mathematicians or statisticians, they'll get it, but if you're writing for an audidence of students, especially students taking finite math or statistics for the first time, you would do better to use "SD" or "Std Dev" because beginner students likely won't be knowledgeable enough or have enough of a handle on statistical jargon to be able to readily infer what "σ" means, "Std Dev" probably being the most reader-friendly abbreviation, albeit also the least brief abbreviation.

For an audience of pragmatic finance folks, writing in the style of the Investopedia would be a safe bet.

For example, in this article, there is a section on “the Greeks”,

https://www.investopedia.com/trading/getting-to-know-the-greeks/

As described by the Investopedia, the main greeks are vega, theta, delta and gamma. They are used like ordinary English words.

Volatility is represented symbolically by the Greek letter sigma. However, in finance writing, when written out, it is almost always referred to as volatility. It’s either the symbol or the word.

https://www.investopedia.com/terms/v/volatility.asp

This poses a problem for the OP’s specific need, since the “obvious” symbol for standard deviation has already been taken.

However, if the underlying need is to characterize the bid-ask spread, there are other order book metrics, such as width, breadth and depth. These are also defined on the Investopedia.

However, if there’s no alternative to using a metric based on the standard deviation of some other price variable, it might be safest to define it as the ratio between the observed price gap and reference value. It could then be referred to in the text as the “gap ratio” and denoted by whatever symbol is convenient.

Another possibility lies in the definition of the reference value as a standard deviation. It might be possible to refer to it as X-volatility, where X is a suitable descriptive term.

For a good example of how to write about these topics, the best reference is Option Volatility and Pricing by Sheldon Natenberg.

https://www.amazon.com/dp/0071818774/ref=cm_sw_r_cp_awdb_imm_7PKNPG2F2AFHFSVTM0D4