# 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?

• If anyone needs additional information to answer this question, please comment, I will provide them here. Jan 5, 2022 at 11:46
• That sqrt(90/t) sounds like a cargo-culted version of Taleb's "weighted vega" -- see his Dynamic Hedging book. He goes into more detail there, including some additional terms. Feb 5, 2022 at 18:54

$$\mathrm{Vega}\equiv \frac{\partial O }{\partial \sigma}= Sn\left(d_1\right)\sqrt{T}$$ 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.