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.