Editing Variance swap (section)

==Structure and features==
The features of a variance swap include:
* the '''variance strike'''
* the '''realized variance'''
* the '''vega notional''': Like other [[swap (finance)|swap]]s, the payoff is determined based on a [[notional amount]] that is never exchanged. However, in the case of a variance swap, the notional amount is specified in terms of [[Greeks (finance)#Vega|vega]], to convert the payoff into dollar terms.

The payoff of a variance swap is given as follows:

:<math>N_{\operatorname{var}}(\sigma_{\text{realised}}^2-\sigma_{\text{strike}}^2)</math>

where:
*<math>N_{\operatorname{var}}</math> = variance notional (a.k.a. variance units),
*<math>\sigma_{\text{realised}}^2</math> = annualised realised variance, and
*<math>\sigma_{\text{strike}}^2</math> = variance strike.<ref name="FINCAD">{{cite web
  | title = Variance and Volatility Swaps
  | publisher = FinancialCAD Corporation
  | url = http://www.fincad.com/support/developerFunc/mathref/VarianceSwaps.htm
  | archive-url = https://web.archive.org/web/20080630063102if_/https://fincad.com/support/developerFunc/mathref/VarianceSwaps.htm
  | archive-date = 2008-06-30
  | access-date = 2009-09-29 }}</ref>

The annualised realised variance is calculated based on a prespecified set of sampling points over the period. It does not always coincide with the classic statistical definition of variance as the contract terms may not subtract the mean. For example, suppose that there are <math>n+1</math> observed prices  
<math>S_{t_0},S_{t_1}, ..., S_{t_n} </math>
where <math>0\leq t_{i-1}<t_{i}\leq T</math>
for <math>i=1</math> to <math>n</math>. Define <math>R_{i} = \ln(S_{t_{i}}/S_{t_{i-1}}),</math> the natural log returns.
Then 
*<math>\sigma_{\text{realised}}^2 =  \frac{A}{n} \sum_{i=1}^n  R_i^2 </math>

where <math>A</math> is an annualisation factor normally chosen to be approximately the number of sampling points in a year (commonly 252) and <math>T</math> is set be the swaps contract life defined by the number <math>n/A</math>. It can be seen that subtracting the mean return will decrease the realised variance. If this is done, it is common to use <math>n-1</math> as the divisor rather than <math>n</math>, corresponding to an unbiased [[estimator|estimate]] of the sample variance.

It is market practice to determine the number of contract units as follows:

:<math>N_{\operatorname{var}}=\frac{N_{\text{vol}}}{2\sigma_{\text{strike}}}</math>

where <math>N_{\text{vol}}</math> is the corresponding vega notional for a [[volatility swap]].<ref name="FINCAD"/> This makes the payoff of a variance swap comparable to that of a [[volatility swap]], another less popular instrument used to trade volatility.