Variance Swap
A variance swap is an over-the-counter derivative contract in which two parties exchange the realized variance of an underlying asset's returns over a specified period against a fixed strike (the variance strike), with payoff determined by the difference between realized variance and the pre-agreed strike multiplied by a notional vega amount. It provides pure, direct exposure to volatility without the delta-hedging complexity of standard options positions.
Key takeaways
- The payoff of a variance swap is: (Realized Variance - Variance Strike) × Notional Vega, where realized variance is typically measured as the annualized sum of squared daily log returns.
- Unlike options, variance swaps have no delta and do not require ongoing delta-hedging — they provide 'clean' volatility exposure.
- Variance strikes are typically quoted in volatility-squared terms (e.g., 20² = 400), though practitioners often convert to vol-squared for pricing convenience.
- Realized variance is calculated daily, and the convexity of variance versus volatility means variance swaps are worth more than the square of the expected volatility.
- Sellers of variance swaps (typically banks and hedge funds) replicate via a static portfolio of options across all strikes, making the volatility surface central to pricing.
Explanation
Variance swaps emerged in the late 1990s as a mechanism for investors to express directional views on volatility without the complications of managing delta exposure. In a standard options position, delta changes continuously as the underlying price moves, requiring constant rebalancing to maintain a pure volatility exposure. A variance swap eliminates this problem entirely: the buyer receives realized variance and pays a fixed variance strike, with no delta to manage. This makes variance swaps the instrument of choice for hedge funds seeking to express views that volatility will be higher or lower than the market-implied level, and for asset managers seeking to hedge volatility risk in their portfolios.
The mechanics of a variance swap are straightforward. At initiation, the two parties agree on the variance strike (Kvar), the notional vega amount (N), and the observation period (typically one month, three months, or one year). Realized variance (RV) is computed at maturity as the annualized variance of daily log returns: RV = (252 / n) × Σ[ln(Sᵢ/Sᵢ₋₁)]², where n is the number of daily observations. The payoff to the buyer is N × (RV - Kvar). If realized variance exceeds the strike, the buyer profits; if realized variance falls below the strike, the seller profits. The variance strike is set at inception so that the fair value of the swap is zero — it equals the market's expectation of future realized variance.
Pricing variance swaps is intimately linked to the entire implied volatility surface. The theoretical replication of a variance swap requires a portfolio of options at every strike from zero to infinity, weighted by 1/K² (the square of the strike price). In practice, this replication is approximated by holding options at available strikes on the listed options chain. The fair variance strike is therefore a function of the entire volatility surface, not just the at-the-money implied volatility. This explains why variance swaps are systematically priced at a premium to the square of ATM implied volatility — the contribution of out-of-the-money puts (which have high implied vol due to the volatility skew) pushes the fair variance strike above the square of ATM vol.
Convexity is a defining feature of variance swaps versus volatility swaps. Because variance is the square of volatility, the relationship is convex: expected variance = (expected vol)² + variance of volatility. This means that buyers of variance swaps benefit from high volatility-of-volatility (vol-of-vol) environments, while sellers are hurt. In turbulent markets — like the 2008 global financial crisis or the March 2020 COVID shock — realized variance can dramatically overshoot the variance strike, producing enormous payoffs for variance swap buyers and equally large losses for sellers, typically dealers who have written protection.
The variance swap market serves important functions in financial markets. It allows banks to hedge their vega exposure accumulated from selling options to corporate clients, and provides hedge funds a liquid vehicle for expressing volatility views. The VIX index, widely reported as 'the fear index,' is conceptually and mathematically linked to the fair variance strike of variance swaps on the S&P 500 — it represents the square root of the 30-day variance swap rate, making the VIX a direct read on the variance swap market.
Formula
Payoff = N × (RV - Kvar); RV = (252/n) × Σ[ln(Sᵢ/Sᵢ₋₁)]²; Kvar = (σ_ATM)² + Convexity Adjustment
Example
A macro hedge fund believes S&P 500 realized volatility over the next three months will exceed the market-implied level. The desk enters a 3-month variance swap as buyer: variance strike (Kvar) = 400 (equivalent to 20% volatility squared), notional vega = $100,000. This means a 1-point move in variance generates a $100,000 payoff. Over the three months, the S&P 500 experiences a sharp correction, with realized daily log returns averaging 1.8% per day. Realized variance = 252 × (0.018)² = 252 × 0.000324 = 0.0816, or 816 in variance points. Payoff = $100,000 × (816 - 400) = $100,000 × 416 = $41,600,000. The fund profits $41.6 million. If volatility had been benign at 15% annualized (RV = 225), the fund would have lost $100,000 × (225 - 400) = -$17,500,000.
Related terms
Asian Option At The Money Automatic Exercise Convexity Delta Exchange Financial Crisis Hedge Fund Hedging Implied Volatility Implied Volatility Surface Options Chain