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 realized 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.

Structure and features

The features of a variance swap include:

• the variance strike
• the realized 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:

${\displaystyle N_{\operatorname {var} }(\sigma _{\text{realised}}^{2}-\sigma _{\text{strike}}^{2})}$

where:

• ${\displaystyle N_{\operatorname {var} }}$ = variance notional (a.k.a. variance units),
• ${\displaystyle \sigma _{\text{realised}}^{2}}$ = annualised realised variance, and
• ${\displaystyle \sigma _{\text{strike}}^{2}}$ = variance strike.^[1]

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 ${\displaystyle n+1}$ observed prices ${\displaystyle S_{t_{0}},S_{t_{1}},...,S_{t_{n}}}$ where ${\displaystyle 0\leq t_{i-1}<t_{i}\leq T}$ for ${\displaystyle i=1}$ to ${\displaystyle n}$. Define ${\displaystyle R_{i}=\ln(S_{t_{i}}/S_{t_{i-1}}),}$ the natural log returns. Then

• ${\displaystyle \sigma _{\text{realised}}^{2}={\frac {A}{n}}\sum _{i=1}^{n}R_{i}^{2}}$

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

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

${\displaystyle N_{\operatorname {var} }={\frac {N_{\text{vol}}}{2\sigma _{\text{strike}}}}}$

where ${\displaystyle N_{\text{vol}}}$ is the corresponding vega notional for a volatility swap.^[1] 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 strike.^[2][3]

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.

We can derive the payoff of a variance swap using Ito's Lemma. We first assume that the underlying stock is described as follows:

${\displaystyle {\frac {dS_{t}}{S_{t}}}\ =\mu \,dt+\sigma \,dZ_{t}}$

Applying Ito's formula, we get:

${\displaystyle d(\log S_{t})=\left(\mu -{\frac {\sigma ^{2}}{2}}\ \right)\,dt+\sigma \,dZ_{t}}$

${\displaystyle {\frac {dS_{t}}{S_{t}}}\ -d(\log S_{t})={\frac {\sigma ^{2}}{2}}\ dt}$

Taking integrals, the total variance is:

${\displaystyle {\text{Variance}}={\frac {1}{T}}\ \int \limits _{0}^{T}\sigma ^{2}\,dt\ ={\frac {2}{T}}\ \left(\int \limits _{0}^{T}{\frac {dS_{t}}{S_{t}}}\ \ -\ln \left({\frac {S_{T}}{S_{0}}}\ \right)\right)}$

We can see that the total variance consists of a rebalanced hedge of ${\displaystyle {\frac {1}{S_{t}}}\ }$ and short a log contract.
Using a static replication argument,^[4] i.e., any twice continuously differentiable contract can be replicated using a bond, a future and infinitely many puts and calls, we can show that a short log contract position is equal to being short a futures contract and a collection of puts and calls:

${\displaystyle -<!--\; -\; -->\ln <!--\; \; -->\left(\frac{S_T}{S^{\ast <!--\; \ast \; -->}}\right)=-<!--\; -\; -->\frac{S_T-<!--\; -\; -->S^{\ast <!--\; \ast \; -->}}{S^{\ast <!--\; \ast \; -->}}+\underset{K\le <!--\; \le \; -->S^{\ast <!--\; \ast \; -->}}{\int <!--\; \int \; -->}(K-<!--\; -\; -->S_T{)}^{+}\frac{dK}{K^{}}}$Zdroj:https://en.wikipedia.org?pojem=Variance_swap
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