What is a Variance Swap?
A variance swap is an OTC contract that pays the difference between the realized variance of an underlying over a period and a fixed "strike" agreed at inception. The cleanest way to be pure-long or pure-short volatility.
Definition
A variance swap is a forward contract on realised variance (the squared volatility). At inception, the two counterparties agree a "variance strike" K_var (sometimes quoted in vol terms as K_vol = √K_var). At maturity, the payoff is N · (realised_variance − K_var), where N is the variance notional. Because the payoff is in variance, the P&L is convex in vol: a doubling of vol from 15% to 30% generates a 4× payoff (variance went from 225 to 900). This convexity is the entire point — variance swaps are how vol traders get clean exposure to vol without the path-dependence and Greek-management overhead of options. Variance is also the underlying of the VIX index, which is itself a static-replication strip of SPX options.
Why it matters & how it's calculated
Static replication: a variance swap can be perfectly replicated by an infinite strip of OTM options weighted by 1/K², plus a delta-hedge in the underlying. In practice, you use a finite strip; the truncation error is small for the body of the distribution but grows in the tails (which is why "wing prices" matter so much in variance swap pricing). Daily realised variance is computed as Σ (ln(Sᵢ/Sᵢ₋₁))² × (252/n), with subtleties around overnight gaps and dividends. The fair strike K_var equals the expected realised variance under the risk-neutral measure — equivalently, the integral over the surface of the static-replication weights. The famous "fair-strike formula" gives K_var ≈ (2/T) · [∫ P(K)/K² dK + ∫ C(K)/K² dK], with the integrals over OTM puts and calls respectively. The wedge between K_var and ATM IV² is the "variance risk premium," and it is structurally positive (sellers of variance get paid for taking left-tail risk). Practical issue: convexity. A variance swap caps risk at 2.5× the strike (typical contract specification) to prevent unbounded losses in a crash. The cap creates a small short-call-option position embedded in the swap, which itself has Greeks.
Formula
Payoff = N · (RV² − K_var) Fair strike ≈ (2/T) [∫P(K)/K² dK + ∫C(K)/K² dK]
Worked example
You buy a 1-month SPX variance swap with K_vol = 18 (so K_var = 324), notional $10,000 per variance point. If realised vol over the month comes in at 22 (variance = 484), payoff = $10,000 × (484 − 324) = $1.6M. If realised comes in at 14 (variance = 196), you pay $10,000 × (196 − 324) = $1.28M. Note the asymmetry: realised vol of 22 (up 4 from 18) pays $1.6M; realised vol of 14 (down 4) costs $1.28M. The convexity favours the long.
Related concepts
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