Why do we need to calibrate vega?

I was going through some paid video on options. The tutor in the video asked the following question:

Person $A$ has the following portfolio at the start of April

• Portfolio of options with vega $20,000$ expiring end of April.
• Portfolio of options with vega $-40,000$ expiring end of May.
• Portfolio of options with vega $15,000$ expiring end of June.

Now if the monthly implied volatility increases from $\sigma$ % to $(\sigma+1)$%, is it good for person $A$, what is his exposure.

The naive approach is to add all vega's to get $-5,000$ and say with increase in volatility he makes a loss. The tutor goes on to explain that this approach is not correct and one needs to calibrate vegas as time of expiry is different. He says one can add $20,000 + (-40,000/(\sqrt{2})) + (15,000/\sqrt{3})$.

My doubt is why is the naive approach wrong. Vega means change in options price with $1$% change in implied volatility. Doesn't vega (if obtained from pricing models like Black Scholes) itself incorporate the time to expiry factor ? Would it be wrong to say portfolio of second month changes by $-40,000*\sqrt{252}$ ( taking annualized volatility).

PS : I know I am missing something. Being a beginner please excuse me if I used any wrong terms.

• It seems like he is assuming that the shorter term volatilities change more than the longer term ones and the relatively sensitivity is proportional to $1 / \sqrt{T}$. This is not an uncommon assumption and the corresponding vegas are often referred to as "time weighted vegas". Sep 19 '18 at 9:55
• can you give this formally as answer. Would be really helpful. Thanks. Sep 19 '18 at 12:12

It seems like he is assuming that the shorter term volatilities change more than the longer term ones and the relatively sensitivity is proportional to $1 / \sqrt{T}$. Thus, this hedge is not against a parallel shift of the surface. This is not an uncommon assumption and the corresponding vegas are often referred to as "time weighted vegas".
• Typically, the short-term implied volatility is more volatility than the long-term implied volatility. Using the $1 / \sqrt{T}$ approach e.g. assumes that when the 1-month implied volatility increases by 1%, then the 2-month implied volatility increases by ~0.7%. You are hedged against this scenario, which is not a parallel shift. Sep 19 '18 at 12:33