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Volatility Swap: Payoff, Convexity, and Variance-Swap Differences

For educational purposes only; not investment advice.

Direct answer

Section titled “Direct answer”

A volatility swap is a derivative whose settlement is approximately linear in an agreed measure of realized volatility:

Payoff to volatility buyer = N_vol × (σ_realized − K_vol)

K_vol is the volatility strike and N_vol is currency per volatility point, often called Vega notional in the contract. If realized volatility finishes above the strike, the buyer receives; if below, the buyer pays, subject to the agreement’s cap, floor, disruption, collateral, and settlement terms.

The product is usually customized and over the counter, not one standardized U.S. listed equity-option contract. Its legal confirmation controls the sampling dates, return definition, annualization factor, treatment of dividends and corporate actions, calculation agent, payment date, and early termination.

Why volatility is not variance

Section titled “Why volatility is not variance”

Realized variance commonly takes a form such as:

V_realized = A/N × Σ[ln(S_i/S_(i−1))]²

and realized volatility is σ_realized = √V_realized. The exact A, observations, holidays, missing prices, and return adjustments come from the contract. A one-point move means one percentage point, so 30% − 25% = 5 points, not 0.05 points.

A variance swap pays linearly in V_realized − K_var, while a volatility swap pays linearly in √V_realized − K_vol. Because square root is nonlinear, taking the square root of a fair variance strike generally does not produce a fair volatility strike:

E[√V] ≠ √E[V]

This gap is often described through a convexity adjustment. Its size depends on the distribution of future realized variance, including volatility-of-volatility and jumps. A strip of options can underpin idealized variance replication, but the square-root payoff introduces additional model and dynamic-hedging risk for a volatility swap.

Do not confuse contract Vega notional with an option’s local Greek. N_vol directly converts settlement points into cash; option Vega changes with market state and model inputs.

Settlement and notional conversion

Section titled “Settlement and notional conversion”

Assume a long volatility swap has:

  • volatility strike K_vol = 25%;
  • notional N_vol = $10,000 per volatility point;
  • realized volatility at maturity σ_realized = 30%.

Then:

Payoff = $10,000 × (30 − 25) = $50,000

If realized volatility were 21%, the uncapped buyer payoff would be $10,000 × (21 − 25) = −$40,000. Percentages are converted to points according to the confirmation; multiplying $10,000 × 0.05 would understate the first payoff by a factor of 100.

For comparison, 25%² = 625 variance points and 30%² = 900 variance points. A variance swap would settle on a 275 variance-point difference under a matching simplified convention, but its variance notional is a different unit. Equal-looking notionals do not create equal cash risk.

Contract and risk checklist

Section titled “Contract and risk checklist”
  • Read the confirmation: underlying, observation window, close source, holidays, annualization, return type, caps, floors, and payment mechanics.
  • Verify whether the quote and payoff use decimal volatility or volatility points before converting cash.
  • Separate volatility notional, variance notional, option Vega, and gross contractual notional.
  • Recalculate realized volatility independently from the contract’s exact price series and adjustment rules.
  • Stress jumps, clustered returns, volatility-of-volatility, missing observations, market disruption, and corporate actions.
  • Model the convexity adjustment; do not set K_vol = √K_var without assumptions and evidence.
  • Include skew, wings, discrete strikes, transaction costs, and rebalancing error in any option-based hedge.
  • Assess counterparty, collateral, close-out, valuation-dispute, liquidity, funding, and legal-documentation risk.
  • Check mark-to-market exposure before maturity; a linear terminal payoff does not imply a stable interim value.
  • Do not assume OCC clearing protections apply to a bilateral OTC swap.
  • Compare alternatives only after normalizing horizon, sampling, annualization, caps, and cash units.

Common misconceptions

Section titled “Common misconceptions”
  • “A volatility swap is a listed option strategy.” It is generally a separately documented OTC derivative.
  • “The fair volatility strike is the square root of the variance strike.” The expectation and square root do not commute.
  • “Vega notional is the same as option Vega.” One is a contractual cash conversion; the other is a local model sensitivity.
  • “Realized volatility is universal.” Sampling, annualization, holidays, adjustments, and price source change the result.
  • “Linear payoff means low risk.” Large volatility-point moves can create large uncapped cash obligations.
  • “Variance replication perfectly hedges volatility.” The square-root payoff, jumps, discrete trading, wings, and costs leave basis risk.
  • “25% to 30% is a 0.05-point move.” It is five volatility points under standard point quotation.
  • “A model mark is an executable exit.” Customized swaps can be illiquid and subject to valuation disputes.
Section titled “Related topics”

Authoritative sources

Section titled “Authoritative sources”