Interpreting Time-Scaled Vega for Options
Summary
The document examines a rule for scaling options vega by the square root of 90 divided by time to expiration. The response interprets it as a rough comparison convention for options on a fixed underlying, rather than a claim that longer-dated options have lower raw price sensitivity to volatility. In the Black–Scholes setting described, vega contains a square-root-of-time factor. Multiplying by the stated scale cancels that factor when time is expressed consistently, leaving a quantity proportional to the normal density of d1.
The 90 is interpreted as an approximate count of trading days in a quarter; in year fractions, the corresponding factor uses 0.25. This makes the adjustment a chosen horizon normalization, and a different reference horizon would produce a different scale. The response calls the approach quick and rough, and its formula assumes the stated Black–Scholes setup while disregarding dividends. It does not establish that the scaling is an appropriate risk measure for every portfolio or market context.
Key ideas
- Black–Scholes vega includes a square-root-of-time factor.
- The stated time scaling cancels that factor when units are expressed consistently.
- The 90-day constant represents an approximate quarterly trading horizon.
- The resulting comparison is a normalization convention, not proof that raw longer-dated vega is smaller.
- The derivation assumes a Black–Scholes setting and disregards dividends.
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Full text
# Why Do I Need to Scale Options Vega w.r.t T (Time till Expiration)
# Why Do I Need to Scale Options Vega w.r.t T (Time till Expiration)
In the book that I am using, it said that I need scale vega according time with this formula: $\sqrt{90/T}$ to get the weight of the vega w.r.t t. The reasoning it offered is as follows:
"Because the vega of an one year option is larger than the one with an one month option, we assume that the longer term one has a larger risk. This assumption is incorrect. Longer term options' changes in volatility is often much smaller than near-term options. Therefore we shal compare them after weighting each option." The book then goes on to introduce the previous formula that I wrote in the first paragraph.
I am really confused at what it's trying to say. Vomma states the derivative of vega w.r.t to volatility. And vomma is generally larger as time till expiration goes larger, which means that option vegas are more sensitive to volatility when I am further from expiration. If it means the tendency of longer-term options Implied Volatility to change, that is not of my concern. I don't predict future volatility, I just use vega compare the tendency of their options' prices to change w.r.t to volatility.
Vomma w.r.t Time Red for bought options and blue for Sold ones with other colors for different volatility
Secondly, can anyone prove that formula please, I am genuinely confused at where the 90 comes from?
## Answer by Kermittfrog (score 2, accepted)
https://quant.stackexchange.com/a/69421
To me, this looks like a (very?) quick-and-dirty way to compare options' sensitivities for a fixed underlying asset: Disregarding dividends, the Black-Scholes Vega is calculated as
$$ \mathrm{Vega}\equiv \frac{\partial O }{\partial \sigma}= Sn\left(d_1\right)\sqrt{T} $$ where $d_1=\frac{\ln S-\ln X+(r+0.5\sigma^2)T}{\sigma\sqrt{T}}$, and $T$ is the time to maturity in year fractions. In the stated scaling factor, $\sqrt{90/T}$, the 90 relates to trading days (approx. 1/4 of a year). Restating the factor in year fractions $\sqrt{0.25/T}$, we get scaled vega as
$$ \mathrm{scaled Vega}\equiv \sqrt{0.25/T}\times\mathrm{Vega}=Sn\left(d_1\right)\sqrt{T}\sqrt{0.25/T}=Sn\left(d_1\right)\sqrt{0.25},$$
or $0.5Sn(d_1)$. I think they could use any other scaling factor like 360 (=1 year) or the like - in effect, we only compare $Sn(d_1)$ across options, now.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.