Why Coarse Option Grids Overstate a Variance Swap Fair Strike

Summary

The document raises a question about replicating a variance swap fair strike from options when implied volatility is flat. The author expects the strike to equal implied volatility squared, but reports that a replication using constant volatility returns a higher value. The suggested explanation is that the option-strike grid is too coarse.

The practical lesson is that discretizing the replication integral can create approximation error, even when the volatility surface has no skew or curvature. A finer grid should improve the approximation. The post gives no derivation, numerical comparison, or convergence test, so it does not establish the precise error behavior or how fine the grid must be. The conclusion is limited to the author’s reported implementation and refers readers to a separate comment for the detailed explanation.

Key ideas

• A flat implied volatility surface motivates comparing the variance swap fair strike with implied volatility squared.
• A discrete option replication can differ from its continuous counterpart because of grid approximation.
• The reported overpricing was attributed to an option-strike grid that was not fine enough.
• The document provides no quantitative convergence analysis or general error bound.

Tags

# Fair Strike for Variance Swap with no Skew in IV Surface


# Fair Strike for Variance Swap with no Skew in IV Surface












I am reading through Derman's 1999 research notes, "More than you ever wanted to know about Volatility Swaps."

In equation B4 of Appendix B, the author takes the Taylor Series of the variance swap replication portfolio with the constant term being equal to implied volatility squared. For me, this implies that the fair strike of a variance swap is equal to implied volatility squared when there is no skew in the IV surface.

I developed a tool to replicate fair variance strike and I always get a value higher than the IV squared when I use constant volatility to price each option in the replication. I get similar results when I use my trader's pricer at my job. I doubt there is a numerical error as I am simply doing a partial sum. Hence, I would expect the error to underprice the fair strike.

Should the fair variance price be equal to IV squared in the absence of skew and curvature in the IVS? If not, why?

Thanks!

## Answer by Anthony Edward Maylath (score 0, accepted)

https://quant.stackexchange.com/a/49495

@Quantuple had a good answer in the comments. Basically, I was not using a fine enough grid in my replication.

Click here to see the original comment that answers the question.

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.