# 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!

• I guess another way to ask the question is, given a volatility-spot curve, for any spot value S, if I sum the vega or variance vega of each one of the options in the replication portfolio using Black Scholes formula, I don't think I'll get the same value for all S values? – RAY Feb 15 '15 at 16:09
• You may be confused by the fact that the vega as you used it only applies in the BS model whereas your portfolio valuation is not based on a model but on market values. When considering a non parametrical skewed implied volatility the notion of vega is not trivial (no parameter to derive w.r.t.). The naive way to do it is to consider a parallel shift of the implied volatility slice at the maturity of the variance swap. This is however not arbitrage free and not very meaningful so we use instead good parametrical functions and derive the shift by bumping one or more parameter of the curve. – vanna Feb 22 '15 at 0:18

## 3 Answers

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...

• "The variance swap's Vega that is equal to the variance notional refers to the realized variance." I'm not sure I agree. The value of the variance swap at any given time is determined by the sum of variance already realized and variance to be realized. The variance to be realized is "implied" by the current value of the swap. – RAY Feb 15 '15 at 15:27
• @RAY: When you say, the variance Vega is equal to the variance notional, it is because you derive it from Price = VarNotional * (Variance - Strike). However the Variance in the formula is an estimation of what the realized variance will be at the end of the swap. Whether you estimate this variance with option implied volatilities is up to you. – d--b Feb 15 '15 at 17:25
• What I mean, is that if you decide to use the replication argument to price the variance swap, the calculation will give you a price for the variance swap (as the sum of replicating options), which in turn will give you an option-implied estimation of the realized variance at the end of the swap. That estimation does depend on the implied vols you used to price the options. – d--b Feb 15 '15 at 17:30

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.

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.