Variance Swap

Unlike vanilla options, a variance swap provides direct, linear exposure to realized variance without the path-dependency or delta-hedging complexities. The payoff is based solely on the squared log returns of the underlying asset over the contract's life.

• Delta-neutral at inception: No initial exposure to the underlying price direction.
• Convex payoff profile: The profit is proportional to the difference between realized variance and the variance strike squared.
• No strike price selection: Eliminates the need to choose a specific option strike, simplifying the volatility view.

The payoff is determined by the spread between the variance strike (K²) agreed upon at inception and the annualized realized variance (σ²_realized) calculated at expiry.

• Realized variance calculation: Typically uses daily closing prices, summing squared log returns and annualizing with a 252-day factor.
• Payoff formula: Notional × (σ²_realized - K²).
• Capped payoffs: Many OTC variance swaps include a cap on maximum realized variance to limit the seller's tail risk exposure.

Variance swaps are quoted using vega notional, which represents the dollar profit or loss for a 1% change in realized volatility. This convention simplifies comparison with option vega.

• Vega notional to variance notional: Variance Notional = Vega Notional / (2 × K), where K is the variance strike.
• Constant gamma: A variance swap provides a constant dollar gamma profile, unlike options where gamma peaks near the strike.
• No theta decay: The holder does not suffer from time decay, making it a cleaner instrument for long volatility positions.

The variance swap market is a primary mechanism for harvesting the volatility risk premium (VRP)—the persistent spread between implied and subsequent realized volatility.

• Short variance strategies: Selling variance swaps systematically captures the VRP, as implied volatility tends to exceed realized volatility over time.
• Crisis alpha: Long variance positions provide significant convex payoffs during market crashes, serving as portfolio tail-risk hedges.
• Roll yield: In VIX futures term structure, the typical contango allows short positions to earn a positive roll yield as futures converge to spot.

During its life, a variance swap's value is marked-to-market based on the realized variance accrued to date and the implied variance of the remaining term.

• Vega sensitivity: The position's vega is proportional to the remaining time to maturity.
• Gamma sensitivity: The dollar gamma is constant and proportional to the variance notional.
• Theta profile: Unlike options, a variance swap has no explicit theta; its value changes only with realized variance accumulation and shifts in forward implied variance.

The actual historical fluctuation of an asset's price over the contract period, calculated as the annualized sum of squared log returns. This is the floating leg of the swap. The payoff is determined by the difference between this realized figure and the pre-agreed strike.

• Typically calculated using daily closing prices
• Sensitive to intraday jumps and gap risk
• Convexity adjustment required for fair pricing

The fixed leg of the swap, representing the level of variance bought or sold at contract inception. It is set such that the initial value of the swap is zero. The strike is not simply the square of the implied volatility; it includes an adjustment for the volatility skew.

• Quoted in volatility points for convenience
• Determined by replicating a log contract
• Directly comparable to the VIX index methodology

The dollar amount used to approximate the profit or loss for a 1% move in realized volatility. It linearizes the convex payoff of the variance swap for intuitive trade sizing. The actual variance notional is derived by dividing the vega notional by twice the strike volatility.

• Formula: Variance Notional = Vega Notional / (2 * K_vol)
• Simplifies risk management for traders
• P&L is strictly convex in variance terms

A forward contract on future realized volatility (the square root of variance). Unlike a variance swap, its payoff is linear in volatility. However, it is not purely statically replicable, making hedging and pricing model-dependent due to the concavity of the square root function.

• Subject to volatility of volatility risk
• Requires an approximation like the Brockhaus-Long approximation
• Simpler payoff structure for directional vol views

The correction applied to the fair strike of a variance swap to account for the fact that the payoff is a function of variance (squared volatility) rather than volatility. This adjustment is proportional to the variance of the realized volatility process.

• Ensures no-arbitrage pricing relative to volatility swaps
• Increases the fair variance strike above the squared ATM implied vol
• Magnitude depends on the volatility of volatility parameter