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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.

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    $\begingroup$ 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". $\endgroup$ – LocalVolatility Sep 19 '18 at 9:55
  • $\begingroup$ can you give this formally as answer. Would be really helpful. Thanks. $\endgroup$ – sashas Sep 19 '18 at 12:12
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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".

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  • $\begingroup$ I am sorry. But what do you mean by "assuming that the shorter term volatilities change more than the longer term ones". As I mentioned isn't the time to expiry already incorporated into vega ? Really sorry I'm a beginner. Also what do you mean by "this hedge is not against a parallel shift of the surface" ? $\endgroup$ – sashas Sep 19 '18 at 12:17
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    $\begingroup$ 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. $\endgroup$ – LocalVolatility Sep 19 '18 at 12:33
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    $\begingroup$ Each maturity has a different ATM IV, you can think of it as the "term structure of IV" (just like the term structure of interest rates). A "parallel shift" means the IVs would all increase by the same amount. More commonly, in response to an event, the near term IVs increase more than the long term. $\endgroup$ – Alex C Sep 19 '18 at 21:37
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Your tutor is calculating the increase in total variance. The black-scholes model has the variance term of sigma^2 * Time-to-expiry.

Hence, when the monthly volatility increases by 1%, the effective increase for the 3mth option is sqrt(3) * 1%, the 2mth option is sqrt(2) * 1% etc. He explicitly assumes the vega is relative to the total variance - i.e. the vega is due to an increase in the sqrt(variance = sigma^2 * T).

I personally don't think he is doing this correctly, as the industry standard of defining vega, is literally the change in price due to a change in % volatility.

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