A variance swap lets you bet on “realized” variance. The exchange automatically calculates realized variance for each day, so if you bet the total realized variance over the next 3 days will average to exceed 0.64 [1], then you can buy this contract. If it turns out to be 0.7812, you earn the difference of 0.1412 notional which would mean $141,200 on a million dollar notional.
[1] which means 80% vol (annualized), or roughly 5% daily realized vol (unannualized)
Standard var swap PnL is defined as
(sigma_r^{2} – K) N ….. …(1)
where
N denotes notional amount like $1,000,000
K denotes strike, which is always in terms of annualized variance
sigma_r is annualized realized Vol over the n days, actually over n1 price relatives
sigma_r^{2} is annualized realized Variance, and calculated as
252/(n1) [ ln^{2}(S2/S1) + ln^{2}(S3/S2) + … + ln^{2}(S_{n}/S_{n1}) ]
where
S2 denotes the Day 2 closing price.
ln^{2}(S2/S1) is known as daily realized Variance unannualized
ln(S2/S1) is known as daily realized Vol unannualized, or DRVol
In other words, take the n1 values of ln(PriceRelative) and find the stdev assuming 0 mean, then annualize.
A more intuitive interpretation — take the average of the n1 daily realized variances, then multiply by 252.
Now, trading often work with DRVol rather than the S2 stuff above, so there’s an equivalent PnL formula to reveal the contribution of “today’s” DRVol to a given var swap position, and also track the cumulative contribution of each day’s DRVol. Formula (1) becomes PnL ==
252N/(n1)*[ ln^{2}(S2/S1)K/252 + ln^{2}(S3/S2)K/252 + .. + ln^{2}(S_{n}/S_{n1})K/252 ], or
N/(n1)*[ 252ln^{2}(S2/S1)K + 252ln^{2}(S3/S2)K + .. + 252ln^{2}(S_{n}/S_{n1})K ]
where
N/(n1) represents the notional amount allocated to each day.
252ln^{2}(S2/S1) represents the annualized daily realized Variance on Day 2
√252 ln(S2/S1) represents the annualized DRVol, but is omitted from the formula due to clutter
In other words, for each day get the “spread” of (annualized) DRVar over strike (K), multiply it by the daily notional, and you get a daily PnL “contribution”. Add up the daily to get the total PnL. Here’s an example with daily notional = $4166666 and K = 0.09 i.e. 30% vol
closing

PR

ln PR

sqrt(252) ln PR

spread over K

daily PnL contribution

$1,200






$1,250

1.041667

0.040822

64.8029%

0.32994168

$1,374,757

$1,240

0.992

0.00803

12.7507%

0.073742023

$307,258

$1,275

1.028226

0.027835

44.1864%

0.105243561

$438,515

$1,200

0.941176

0.06062

96.2386%

0.836186882

$3,484,111

You can then add up the daily contributions, which would add up to the same total PnL by Formula in (1).