Variance Swap Replication and the Meaning of Vega
Summary
The discussion examines why the variance sensitivity of a variance swap may seem inconsistent with the sensitivities of the options used to replicate it. Under assumptions of continuous paths and a continuum of strikes, the question treats the option portfolio as an exact replication and asks whether integrating option vegas should yield a constant sensitivity across volatility curves.
The answers identify a key distinction: variance notional sensitivity is defined with respect to realized variance, whereas Black–Scholes vega is sensitivity to implied volatility. Differentiating a replication with respect to variance requires differentiating both sides with respect to the same quantity; the resulting option sensitivity is not ordinary vega. Another answer frames individual option sensitivities as strike-specific contributions, analogous to key-rate durations versus a portfolio's total duration. The exchange is conceptual and does not work through a full derivation; the replication premise also relies on idealized assumptions about jumps and strike availability.
Key ideas
- Variance-swap sensitivity to realized variance differs from Black–Scholes vega to implied volatility.
- A replication sensitivity must be differentiated with respect to the same variable on both sides.
- The resulting variance sensitivity of options is not their usual implied-volatility vega.
- Option sensitivities by strike can be understood as components of aggregate portfolio sensitivity.
- Exact replication assumes continuous paths and access to a full range of strikes.
Tags
Full text
# Variance swap replication and variance vega # Variance swap replication and variance vega Noob here. I've been trying to gain a better understanding of variance swaps and what better way than to replicate it with a portfolio of better understood instruments. I have read the GS 1999 paper and JPM 2005 paper and think I get how the replication works. With the assumption of no jumps and full continuous strikes, the replication using options is exact, so the variance swap and the ideal replication portfolio should be indistinguishable. Now we know that the variance swap's variance Vega (d price/d variance) at any given time is simply the variance notional regardless of spot and vol level. So it follows that if I sum (do an integral across strikes) of the options' variance vegas, I should get a constant as well. However, in BS, Vega, gamma, variance Vega are all functions of vol. And vol is NOT a constant function of strike. Does this not mean that a different vol curve would generate a different Vega variance curve? Something is obviously wrong with my argument, but where is it wrong? Thanks! ## Answer by d--b (score 2) https://quant.stackexchange.com/a/16579 The variance swap's Vega that is equal to the variance notional refers to the realized variance. The Black-Scholes vega refers to the market implied volatility. Now if you want, you can estimate the realized variance at expiry from the volatility of the options (for instance taking the atm variance arbitrarily), and that's often what people do. But that's really up to the modeler to do that. If you decide to do that, then in that model, your estimated realized variance is a function of the atm implied vol... ## Answer by RAY (score 2) https://quant.stackexchange.com/a/16603 I think I have figured this out. The key to the understanding is to think of the options' vegas as "key-strike vegas" compared to the var swap/replication portfolio's vega, which is analogous to "key rate durations of a bond portfolio" to the total effective duration of the portfolio. ## Answer by Gordon (score 2) https://quant.stackexchange.com/a/18405 A variance swap can be replicated with vanilla European options. If you take derivative with respect to variance, you need to do the same thing on both sides. That is, you need also take derivative with respect to variance on those vanilla options. However, the resulting derivative is not the vega in the usual sense, which is the derivative with respect to the employed volatility. The confusion you have is that you are comparing different objects.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.