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Imagine I price a Variance Swap for an investor and the observation date starts tomorrow and ends in 30 days. If I use dynamic replication with options to price my variance swap do I use options with a maturity of 1 month?

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The options will form a static replication - and yes - they should expire on the same day as the variance swap. You should be sure to do all of your analytics in business time. Also, typically a year is considered 252 business days by default - even in FX where there are no holidays (though there are no WMR fixings on New Years, Christmas, Good Friday). You will still need to trade the underlying futures for the expiration date dynamically (or alternatively flip puts to calls or vice versa in your option strip as the forward moves to keep all options out of the money - a put call parity argument will explain that). The details are in Derman's famous paper about variance swaps. Skimmed wikipedia and this is probably ok too (though I didn't look carefully):

http://www.emanuelderman.com/writing/entry/more-than-you-ever-wanted-to-know-about-volatility-swaps-the-journal-of-der

https://en.wikipedia.org/wiki/Variance_swap

The wikipedia link has some good references at the bottom also.

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