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.
The features of a variance swap include:
The payoff of a variance swap is given as follows:
N\operatorname{var
where:
N\operatorname{var
2 | |
\sigma | |
realised |
2 | |
\sigma | |
strike |
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
S | |
t0 |
,S | |
t1 |
,...,
S | |
tn |
0\leqti-1<ti\leqT
i=1
n
Ri=
ln(S | |
ti |
/S | |
ti-1 |
),
2 | |
\sigma | |
realised |
=
A | |
n |
n | |
\sum | |
i=1 |
2 | |
R | |
i |
where
A
T
n/A
n-1
n
It is market practice to determine the number of contract units as follows:
N\operatorname{var
where
Nvol
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:
dSt | |
St |
=\mudt+\sigmadZt
Applying Ito's formula, we get:
d(logSt)=\left(\mu-
\sigma2 | |
2 |
\right)dt+\sigmadZt
dSt | |
St |
-d(logSt)=
\sigma2 | |
2 |
dt
Taking integrals, the total variance is:
Variance=
1 | |
T |
T | |
\int\limits | |
0 |
\sigma2dt =
2 | |
T |
\left(
T | |
\int\limits | |
0 |
dSt | |
St |
-ln\left(
ST | |
S0 |
\right)\right)
We can see that the total variance consists of a rebalanced hedge of
1 | |
St |
-ln\left(
ST | |
S* |
\right)=-
| |||||||
S* |
+
\int\limits | |
K\leS* |
+ | |
(K-S | |
T) |
dK | |
K2 |
+
\int\limits | |
K\geS* |
+ | |
(S | |
T-K) |
dK | |
K2 |
Taking expectations and setting the value of the variance swap equal to zero, we can rearrange the formula to solve for the fair variance swap strike:
Kvar=
2 | |
T |
\left(rT-\left(
S0 | |
S* |
erT-1\right)-ln\left(
S* | |
S0 |
\right)+erT
S* | |
\int\limits | |
0 |
1 | |
K2 |
P(K)dK+erT
infty | |
\int\limits | |
S* |
1 | |
K2 |
C(K)dK\right)
where:
S0
S*>0
K
Often the cutoff
S*
S*=F0=
rT | |
S | |
0e |
Kvar=
2erT | |
T |
\left(
F0 | |
\int\limits | |
0 |
1 | |
K2 |
P(K)dK+
infty | |
\int\limits | |
F0 |
1 | |
K2 |
C(K)dK\right)
One might find discrete-sampling of the realized variance, says,
2 | |
\sigma | |
realized |
2 | |
\sigma | |
realized |
Suppose that in the risk-neutral world with a martingale measure
Q
S=(St)0\leq
dSt | |
St |
=r(t)dt+\sigma(t)dWt, S0>0
where:
T
r(t)\inR
\sigma(t)>0
W=(Wt)0\leq
(\Omega,l{F},F,Q)
F=(l{F}t)0\leq
W
Given as defined above by
2 | |
(\sigma | |
realized |
-
2 | |
\sigma | |
strike |
) x Nvar
t0
V | |
t0 |
V | |
t0 |
| ||||||||||
=e |
EQ
2 | |
[\sigma | |
realized |
-
2 | |
\sigma | |
strike |
\mid
l{F} | |
t0 |
] x Nvar.
To avoid arbitrage opportunity, there should be no cost to enter a swap contract, meaning that
V | |
t0 |
2 | |
\sigma | |
strike |
=EQ
2 | |
[\sigma | |
realized |
\mid
l{F} | |
t0 |
],
which remains to be calculated either by finding its closed-form formula or utilizing numerical methods, like Monte Carlo methods.
Many traders 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 on 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.[6]
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. Historically, implied variance has been above realized variance,[7] a phenomenon known as the variance risk premium, creating an opportunity for volatility arbitrage, in this case known as the rolling short variance trade. For the same reason, these swaps can be used to hedge options on realized variance.
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.