Variance swap

A variance swap is an over-the-counter financial derivative that allows one to speculate on or hedge risks associated with the magnitude of movement, i.e. volatility, of some underlying product, like an exchange rate, interest rate, or stock index.

One leg of the swap will pay an amount based upon the realised variance of the price changes of the underlying product. Conventionally, these price changes will be daily log returns, based upon the most commonly used closing price. The other leg of the swap will pay a fixed amount, which is the strike, quoted at the deal's inception. Thus the net payoff to the counterparties will be the difference between these two and will be settled in cash at the expiration of the deal, though some cash payments will likely be made along the way by one or the other counterparty to maintain agreed upon margin.

tructure and features

The features of a variance swap include:
* the variance strike
* the realised variance
* the vega notional: Like other swaps, 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 vega, to convert the payoff into dollar terms.

The payoff of a variance swap is given as follows:

:$N\_\{\; ext\{var(sigma\_\{\; ext\{realised^2-sigma\_\{\; ext\{strike^2)$

where:
*$N\_\{\; ext\{var$ = variance notional (a.k.a. variance units),
*$sigma\_\{\; ext\{realised^2$ = annualised realised variance, and
*$sigma\_\{\; ext\{strike^2$ = variance strike.cite web
last =
first =
authorlink =
coauthors =
title = Variance and Volatility Swaps
work =
publisher = FinancialCAD Corporation
date =
url = http://www.fincad.net/support/developerFunc/mathref/VarianceSwaps.htm
format =
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accessdate = 2009-09-29 ]

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 n+1 sample points $S\_\{\; ext\{0,S\_\{\; ext\{1,\; ...,\; S\_\{\; ext\{n.$Define, for i=1 to n, $R\_\{\; ext\{i\; =\; ln(S\_\{\; ext\{i/S\_\{\; ext\{i-1),$ the log returns.Then
*$sigma\_\{\; ext\{realised^2\; =\; A\; sum\_\{\; ext\{i=1^\{\; ext\{n\; R\_\{\; ext\{i^2/n$

where $A$ is an annualisation factor normally chosen to be approximately the number of sampling points in a year (commonly 252). It can be seen that subtracting the mean return will decrease the realised variance. If this is done, it is common to use $n+1$ as the divisor rather than $n$, corresponding to an unbiased estimate of the sample variance.

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

:$N\_\{\; ext\{var=frac\{N\_\{\; ext\{vol\}\{2sigma\_\{\; ext\{strike\}$

where $N\_\{vol\}$ is the corresponding vega notional for a volatility swap.cite web
last =
first =
authorlink =
coauthors =
title = Variance and Volatility Swaps
work =
publisher = FinancialCAD Corporation
date =
url = http://www.fincad.net/support/developerFunc/mathref/VarianceSwaps.htm
format =
doi =
accessdate = 2009-09-29 ] This makes the payoff of a variance swap comparable to that of a volatility swap, another less popular instrument used to trade volatility.

Pricing and valuation

The variance swap may be hedged and hence priced using a portfolio of European call and put options with weights inversely proportional to the square of strikecite web
last = Demeterfi, Derman, Kamal, Zou
first =
authorlink =
coauthors =
title = More Than You Ever Wanted To Know About Volatility Swaps
work =
publisher = Goldman Sachs Quantitative Strategies Research Notes
date = 1999
url = http://www.ederman.com/new/docs/gs-volatility_swaps.pdf
format =
doi =
accessdate = ] cite web
last = Bossu, Strasser, Guichard
first =
authorlink =
coauthors =
title = Just What You Need To Know About Variance Swaps
work =
publisher = JPMorgan Equity Derivatives report
date = 2005
url = http://math.uchicago.edu/~sbossu/VarSwaps.pdf
format =
doi =
accessdate = ] .

Any volatility smile model which prices vanilla options can therefore be used to price the variance swap. For example, using the Heston model, a closed-form solution can be derived for the fair variance swap rate. Care must be taken with the behaviour of the smile model in the wings as this can have a disproportionate effect on the price.

Uses

Many find variance swaps interesting or useful for their purity. An alternative way of speculating on volatility is with an option, but if one only has interest in volatility risk, this strategy will require constant delta hedging, so that direction risk of the underlying security is approximately removed. What is more, a replicating portfolio of a variance swap would require an entire strip of options, which would be very costly to execute. Finally, one might often find the need to be regularly rolling this entire strip of options so that it remains centered around the current price of the underlying
security.

The advantage of variance swaps is that they provide pure exposure to the volatility of the underlying price, as opposed to call and put options which may carry directional risk (delta). The profit and loss from a variance swap depends directly on the difference between realized and implied volatility. [cite web
last = Curnutt
first = Dean
authorlink =
coauthors =
title = The Art of the Variance Swap
work =
publisher = Derivatives Strategy
date = 2000-02
url = http://www.derivativesstrategy.com/magazine/archive/2000/0200col.asp
format =
doi =
accessdate = 2009-09-29 ]

Another aspect that some speculators may find interesting is that the quoted strike is determined by the implied volatility smile in the options market, whereas the ultimate payout will be based upon actual realized variance. These two are not usually the same and thus creates an opportunity for volatility arbitrage. For the same reason, these swaps can be used to hedge Options on Realized Variance.

Related instruments

Closely related strategies include straddle, volatility swap, correlation swap, gamma swap, conditional variance swap, corridor variance swap, forward-start variance swap, option on realized variance and correlation trading.

References