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

Understand a volatility swap's linear realized-volatility payoff, quotation and sampling terms, convexity adjustment, and differences from variance swaps and listed options.

Canonical: https://wiki.fcontext.com/options/volatility-swap/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## 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.

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## 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.

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## 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.

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## 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.

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## 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.

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## Related topics

- [Variance Swap](/options/variance-swap/)
- [Realized Variance](/options/realized-variance/)
- [Vega Notional](/options/vega-notional/)
- [Variance Swap Replication](/options/variance-swap-replication/)

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## Authoritative sources

- [Robust Replication of Volatility Derivatives](https://doi.org/10.1111/j.1467-9965.2008.00341.x) — Peter Carr and Roger Lee, *Mathematical Finance*
- [Volatility Swaps Made Simple](https://doi.org/10.1111/1467-9965.00085) — Oliver Brockhaus and Douglas Long, *Risk*
- [Cboe Volatility Index Methodology](https://cdn.cboe.com/api/global/us_indices/governance/VIX_Methodology.pdf) — Cboe Global Indices
- [Characteristics and Risks of Standardized Options](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document) — Options Clearing Corporation