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Given a specified maturity is there a way to compare Vegas between different strikes? Surely the Vega of an ATM option will be very different from the same Vega of an OTM option

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    $\begingroup$ What do you mean by ‘different’? Pls $\endgroup$
    – dm63
    Mar 14, 2022 at 14:29
  • $\begingroup$ For a given underlying you will have different strikes each with it's own volatility (shown in the volatility smile). As far as I understand, when you compute Vega you compute it using the sensitivity to these different strike volatilities. So an ATM Vega is the sensitivity to the ATM Vol while OTM Vega is the sensitivity to the the OTM Vol. My question is, is there a way to compare these different Vegas? Is this even important? The easiest way out is to bucket these different Vegas separately, is there a better way? $\endgroup$ Mar 15, 2022 at 7:50
  • $\begingroup$ @Socrates231 . The way you are describing the starting situation is very clear. What is however unclear is what comparing the vegas means. They are numbers and therefore can be compared, bucketed, and what not. $\endgroup$
    – Kurt G.
    Mar 15, 2022 at 10:32
  • $\begingroup$ As a concrete example I mean something like this: say I look at the AAPL option chain for the March 18 expiry and I focus on calls. My underlying is at 155 and I have +1000 Vega in the 140 Strike (ITM), -500 Vega in the 155 Strike (ATM) and -500 Vega in the 170 Strike (OTM). What's my total Vega in this case? I don't think you can say 0 since those Vegas are computed to different volatilities, or am I missing something? $\endgroup$ Mar 16, 2022 at 10:58
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    $\begingroup$ @Socrates231 . Correct, those vegas are computed to different volatilities. When the vol changes that were used to calculate them were small enough one can apply a simple result from multivariable calculus which says that the sum of the three vegas (in your case zero) is the portfolio vega when all vols are changed in parallel by the same absolute amounts. In reality this should approximately be true. The sum of bucketed vegas should always be compared to a single vega you get from a parallel vol shift. This is an important sanity check in practice. $\endgroup$
    – Kurt G.
    Mar 16, 2022 at 14:34

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Vega is the measure of the rate of change of option's value with every percentage change in volatility, i.e., dV/dσ.

  • First of all, we need an option valuation model to get the function of option price (V) wrt underlying security price, strike price, time to maturity, risk-free interest rate & volatility.

  • Once we have an option pricing model, take the partial derivative of option price (V) wrt volatility (σ) to get vega (ν).

  • If we consider Black Scholes model for valuation, Vega (ν) = ∂V/∂σ = e−qTS√T φ(d1), where d1 is a intermediate parameter in black scholes model.

  • On putting all the values in the function of Vega (ν) except Strike price, we get a function describing the dependency of vega for different strike prices. We can even plot the graph like the below example to compare vega for different strike prices & different times to maturity.

Vega vs Strike Price Graph

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  • $\begingroup$ Your plot illustrates the fact that when the option is deep in the money or deep out of the money, then a small change in the implied volatility affects the value of the option less, than the small change in the implied volatility affects an option close to being at the money. $\endgroup$ Aug 14, 2022 at 22:30

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