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I am given a list of Options positions consisting of various combinations of Underlying and Strikes. I am also given the Vega values for each go these positions.

Now, given this information, I want to calculate total Vega exposure of this portfolio. Should I just add up the individual Vegas and report that as total Vega (without considering the sign ofcourse)?

Is that approach correct at least approximately? If not, what can be the correct approach given the information I have?

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    $\begingroup$ You can if your portfolio of options only has one underlying but you said your portfolio of options have various underlyings. You can do that if the underlying is highly correlated whether positively or negatively which can be handled by multiplying the signs. Otherwise, you need to decompose the underlying volatility movement into independent components and add up the vegas for each. $\endgroup$
    – Hans
    Commented Mar 10, 2021 at 16:31
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    $\begingroup$ To add on top of @Hans comment: if the strike range varies widely (per underlying), you might need to bucket in the strike dimension as well. In the end, this depends on the correlation between your implied vol nodes per underlying and across underlyings (as per Hans comment). $\endgroup$
    – Kermittfrog
    Commented Mar 10, 2021 at 17:41
  • $\begingroup$ @Kermittfrog Could you please clarity correlation between your implied vol nodes $\endgroup$
    – Daniel
    Commented Mar 10, 2021 at 18:26
  • $\begingroup$ The strike dimension is not necessary unless you are assessing response under extreme shock which you may need to add second derivatives of vega like vanna $\endgroup$
    – Yanyi Yuan
    Commented Mar 11, 2021 at 20:47
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    $\begingroup$ Say you have $N$ OTC options on a given underlying, varying in strike space as well as in time to maturity. Each option has a vega with respect to 'its' implied volatility. Usually, you would not hedge this portfolio with other options that are back-to-back, but with a set of options that are commonly quoted, or available at an exchange or such. Further, the implied volatility surface may not move in lockstep across all strikes and tenors. Hence, one might want to attribute their vega sensitivities not to one lump sum, but to a vega per IV-node. HTH? $\endgroup$
    – Kermittfrog
    Commented Mar 17, 2021 at 7:55

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It has been pointed out in the comments (@Hans and @Kermittfrog) that unless you make the strong and fallacious assumption that the IV surface moves in parallel, you'd need to bucket the Black-Scholes vegas by strike and tenor.

My sole addition over the comments is to point out a fairly recent article by Francois and Stentoft in which they propose an in my opinion interesting new definition of vega.

Since in a flat volatility Black-Scholes world vega and gamma are related as follows: $$ v^{BS}(K,\sigma) = T\sigma S^2 \Gamma^{BS}(K,\sigma), $$ Francois and Stentoft propose to define vega in the presence of a skew analogously by $$ v^{SI}(K,I(K)) = T I(K) S^2 \Gamma^{SI}(K,I(K)), $$ where the subscript SI stands for ``smile implied'', $I(K)$ is the implied volatility corresponding to the strike $K$, and $\Gamma^{SI}(K,I(K))$ is the smile implied gamma which can be deduced/computed from market price $C^{mkt}(K)$ of options: $$ \Gamma^{SI}(K,I(K)) = \frac{ \partial^2 C^{mkt}}{ \partial S^2} (K). $$ In stochastic volatility models $\frac{ \partial^2 C^{mkt}}{ \partial S^2} (K)$ can be implied from the skew in a parameter-free manner. For other models, some additional assumptions are required.

Notice, though, that this definition does not circumvent the issue of having to specify how the IVs are correlated to each other across strike and time to maturity in order to arrive at an aggregate dollar vega of the portfolio.

It's clear that it's high time a smart person writes a paper titled "Oi guv, wots me vega?!", as this is obviously an important question without a clear and robust answer thus far.

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First try to calculate the Correlation among vega of different underlying . This exercise can be done by historical data ( year ). Calculate the Vega of portfolio by multiplying correlation and arriving at ONE short or Long Vega Number . If not Significant it is also Known as " Square Vega " . Even though you have a Square Vega the portfolio will move the next day . If OTM options are purchased in large qty , best approach is to do scenario analysis and calculating lock delta . Regards

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