Variance and Volatility Swap Mechanics

Variance swaps and volatility swaps provide pure exposure to realized volatility without the directional risk of the underlying asset. These instruments pay the difference between realized volatility (or variance) and a pre-agreed strike level. Variance swaps are more common due to easier hedging, while volatility swaps require convexity adjustments.

Definition and Key Concepts

Variance Swap

Variance swap: An OTC derivative that pays the difference between realized variance and a fixed variance strike.

Payoff formula: Payoff = Variance Notional × (Realized Variance - Variance Strike)

Realized variance calculation: Variance = (252 / N) × Σ(ln(Si/Si-1))²

Where:

• N = number of observations
• Si = closing price on day i
• 252 = annualization factor

Volatility Swap

Volatility swap: An OTC derivative that pays the difference between realized volatility and a fixed volatility strike.

Payoff formula: Payoff = Vega Notional × (Realized Vol - Vol Strike)

Realized volatility: Volatility = √Variance

Key Differences

Notional Conventions

Variance swap:

• Variance notional = Vega notional / (2 × Volatility Strike)
• Example: $100K vega at 20% vol → $100K / (2 × 0.20) = $250,000 variance notional

Volatility swap:

• Vega notional directly expressed
• $100K vega = $100K per vol point

How It Works in Practice

Variance Swap Cash Flows

At inception:

• No cash exchanged
• Agree on variance strike and notional

At maturity:

• Calculate realized variance over observation period
• Settlement = Notional × (Realized - Strike)

Example:

• Strike variance: 400 (20% vol squared)
• Variance notional: $250,000
• Observation period: 3 months

If realized volatility = 25%:

• Realized variance = 625 (25²)
• Payoff = $250,000 × (625 - 400) / 10,000 = $5,625

Convention note: Variance is often quoted per 10,000 for convenience.

Volatility Swap Convexity

Issue: Volatility = √Variance, which is a concave function.

Jensen's inequality: E[√Variance] < √E[Variance]

Result: Fair volatility strike < ATM implied volatility

Convexity adjustment: Vol Strike ≈ ATM IV - (Vol of Vol)² / (8 × ATM IV)

Example:

• ATM implied vol: 20%
• Vol of vol: 80%
• Adjustment: (0.80)² / (8 × 0.20) = 0.40 vol points
• Fair vol strike: 19.6%

Variance Swap Replication

Theory: A variance swap can be replicated by a portfolio of options:

Replication formula: Variance = (2/T) × [∫₀^F P(K)/K² dK + ∫_F^∞ C(K)/K² dK]

In practice:

• Long puts at all strikes below forward
• Long calls at all strikes above forward
• Weight by 1/K²
• Delta hedge the package

This explains:

• Why variance swaps are more liquid (can be hedged)
• Why dealers prefer variance over volatility swaps

Worked Example

Variance Swap Trade

Trade details:

• Underlying: S&P 500
• Tenor: 3 months
• Variance strike: 324 (18% volatility squared)
• Vega notional: $500,000
• Variance notional: $500,000 / (2 × 0.18) = $1,388,889

Daily returns over observation period (simplified):

Realized variance calculation: Annualized variance = 252 × average(daily return²) = 252 × 0.000167 = 420 (20.5% volatility)

Settlement: Payoff = $1,388,889 × (420 - 324) / 10,000 = $1,388,889 × 0.0096 = $13,333

Equivalent vega calculation: Vol change = 20.5% - 18% = 2.5% Vega P/L ≈ $500,000 × 2.5 / 100 = $12,500

The difference from the vega approximation reflects convexity.

Volatility Swap Comparison

Same trade as volatility swap:

• Vol strike: 17.5% (adjusted for convexity)
• Vega notional: $500,000
• Realized vol: 20.5%

Settlement: Payoff = $500,000 × (20.5 - 17.5) / 100 = $15,000

Comparison:

Scenario Analysis

Variance swap P/L across volatility outcomes:

Note: Payoff is not linear in volatility—higher realized vol produces increasingly larger gains due to variance (vol²) exposure.

Risks, Limitations, and Tradeoffs

Gap Risk

Issue: Large single-day moves significantly impact variance.

Example:

• 64 trading days with 1% daily moves → 16% annualized vol
• 1 day with 8% move → adds 10+ vol points to realized

A single gap can dominate the entire variance calculation.

Tail Risk for Short Variance

Short variance has unlimited loss potential.

Dividend and Observation Risk

Common Pitfalls

Hedging Variance Swaps

Option Replication

Initial hedge:

• Calculate option weights (1/K²)
• Buy puts and calls across strike range
• Delta hedge the package

Ongoing hedge:

• Rebalance as spot moves
• Manage gamma exposure
• Roll near-dated options

Greeks of Variance Swaps

Checklist and Next Steps

Pre-trade checklist:

• Determine variance vs. volatility swap
• Calculate appropriate notional
• Verify strike quotation convention
• Assess counterparty credit
• Review observation calendar
• Document hedge accounting treatment

Documentation checklist:

• Confirm ISDA master in place
• Verify observation methodology
• Agree on market disruption provisions
• Clarify dividend adjustment treatment
• Set up collateral arrangements

Ongoing monitoring:

• Track realized variance daily
• Monitor counterparty exposure
• Assess hedge effectiveness
• Prepare for settlement calculation
• Report to risk management

Related articles:

• For VIX products, see Volatility Futures and Options (VIX) Overview
• For barrier options, see Barrier Options: Knock-In and Knock-Out Structures