FRM Exam Part I · Exotic Options

Volatility and Variance Swaps: Payoffs and Pricing

Updated 11 October 2026 · Fact-checked

A volatility swap pays the difference between realized volatility and a fixed strike, times a notional per volatility point. A variance swap pays the difference between realized variance and a variance strike, times a variance notional. Variance swaps can be replicated with options, so they are easier to price and hedge.

Understand Volatility and Variance Swaps

A volatility swap is a forward contract on realized volatility. At maturity, one side pays the other based on how much the underlying actually moved over the life of the swap, compared with a fixed strike agreed at the start. You use it when you want pure exposure to volatility without the price-direction risk of an option.

A variance swap works the same way, but it settles on realized variance, which is the square of volatility. The strike is a variance strike, usually written as the square of a volatility quote. For example, a strike of 20 means a variance strike of 400 in volatility points squared, or 0.04 in decimals.

Why do variance swaps matter more? Realized variance is the average of squared daily returns. A portfolio of options across all strikes, weighted by 1 ÷ K², replicates this payoff. So the fair variance strike can be found from option prices. A volatility swap has no such exact static replication, because volatility is the square root of variance. It needs a dynamic hedge and a model.

The VIX is built on this idea. It is computed from a strip of out-of-the-money option prices on the S&P 500, and its squared value approximates the fair variance strike for a 30-day swap. So VIX² ÷ 10,000 is roughly the 30-day variance swap rate in decimal terms.

Two features matter for the exam. First, a variance swap has convexity: the payoff is linear in variance but convex in volatility. A long variance position gains more from a volatility rise than it loses from an equal fall. Second, because of Jensen's inequality, the fair (no-arbitrage) volatility strike is below the square root of the fair variance strike. This statement is about the fair strikes. It does not hold for any quoted strike you may be given.

Key formulas to remember

Variance swap payoff: Payoff = N_var × (σ²_realized − K²_var); Use consistent units. N_var is the variance notional; σ²_realized and K²_var must be in the same units (decimals or volatility points squared).
Volatility swap payoff: Payoff = N_vol × (σ_realized − K_vol); N_vol is the notional per one volatility point. Linear in volatility.
Vega notional to variance notional: N_var = N_vega ÷ (2 × K_vol); Used when volatility points are quoted. K_vol is the volatility strike in volatility points. Gives the variance notional per point of variance.
Realized variance (annualized): σ²_realized = (252 ÷ n) × Σ rᵢ²; Daily log returns rᵢ, n observations, mean assumed zero in the market convention. Realized volatility is the square root.
Variance replication weights: Option weight at strike K ∝ 1 ÷ K²; Out-of-the-money puts and calls across all strikes. This is why the fair variance strike comes from the option strip.
VIX link: Variance strike ≈ (VIX ÷ 100)²; VIX is quoted in percentage points for a 30-day horizon. It approximates the fair variance strike.

How to solve Volatility and Variance Swaps questions

Use this sequence for any question on volatility or variance swaps. Most mistakes come from units, so fix them first.

1Identify the swap type: volatility or variance. Note whether you are long (receive realized) or short (pay realized).
2Write down the strike and the notional. Check whether the strike is given as volatility or variance, and whether the notional is vega, variance, or per volatility point.
3Put all quantities in the same units. If the strike is quoted as 20, either use 20 and 20² = 400 with volatility points, or convert to 0.20 and 0.04.
4Compute realized volatility or variance from the data given, annualizing if necessary (multiply daily variance by 252).
5Apply the correct payoff formula: N_var × (σ²_realized − K²) or N_vol × (σ_realized − K).
6Check the sign and the party. A long position gains when realized exceeds the strike.
7For conceptual parts, recall the key facts: variance swaps are replicable by options, volatility swaps are not, and variance payoffs are convex in volatility.

Quickest way: Volatility points shortcut

When to use it: When the question quotes strikes and realized values in volatility points (for example 20 and 24) and gives a vega notional.

Convert vega notional to variance notional: N_var = N_vega ÷ (2 × K).
Compute σ²_realized − K² in volatility points squared, for example 24² − 20² = 176.
Multiply by N_var.
Cross-check against the approximation: payoff ≈ N_vega × (σ − K) for small differences. Because of convexity, the variance payoff is larger than the vega-equivalent volatility payoff on an upside move (realized above the strike), and it loses less on a downside move (realized below the strike).

Common mistakes in Volatility and Variance Swaps

Mixing percentage and decimal units, such as using 0.04 for the strike but 24² for realized.
Strikes are quoted as whole numbers like 20, but formulas are often taught with decimals.
Fix: Pick one convention at the start and convert every number. Check that the result has a sensible size.
Treating a variance swap payoff as linear in volatility.
Students remember that the swap is on volatility exposure and forget it settles on the square.
Fix: Remember that payoff is linear in variance, so it is convex in volatility. A rise in volatility gains more than an equal fall loses.
Saying the volatility swap can be replicated exactly with a static option portfolio.
Variance and volatility are confused.
Fix: Only variance has the 1 ÷ K² option replication. Volatility needs a dynamic hedge and a model.
Setting the volatility strike equal to the square root of the variance strike.
It looks like a natural conversion.
Fix: By Jensen's inequality the fair (no-arbitrage) volatility strike is lower than the square root of the fair variance strike, because the square root is concave. This applies to the fair strikes, not to any quoted strike.
Forgetting to annualize realized variance, or using the wrong number of trading days.
Daily returns are given and the formula needs an annual figure.
Fix: Multiply the average daily squared return by 252 unless the question states another convention. Take the square root for volatility.
Getting the sign wrong for the long and short sides.
Students think of the long as paying the fixed rate, as in interest rate swaps.
Fix: The long side receives realized and pays the strike. It profits when realized variance or volatility exceeds the strike.

Worked examples

Example 1

A trader is long a variance swap with a variance notional of USD 2,500 per volatility point squared and a volatility strike of 20. At maturity, realized volatility is 24. What is the payoff?

Show the solution

Strike in variance terms: 20² = 400.
Realized variance: 24² = 576.
Difference: 576 − 400 = 176 volatility points squared.
Payoff = 2,500 × 176 = USD 440,000.

Answer: The long position receives USD 440,000.

Example 2

A volatility swap has a vega notional of USD 100,000 per volatility point and a strike of 25. The equivalent variance swap uses the same strike. (a) Find the variance notional that matches the vega notional. (b) If realized volatility is 20, find the variance swap payoff to the long.

Show the solution

(a) N_var = N_vega ÷ (2 × K) = 100,000 ÷ (2 × 25) = 100,000 ÷ 50 = 2,000.
(b) Strike variance: 25² = 625. Realized variance: 20² = 400.
Difference: 400 − 625 = −225.
Payoff = 2,000 × (−225) = −USD 450,000.
Compare with the volatility swap: 100,000 × (20 − 25) = −USD 500,000. On this downside move the variance swap loses less (USD 450,000 against USD 500,000) because of convexity. On an upside move it would gain more than the vega-equivalent volatility swap.

Answer: (a) Variance notional = USD 2,000 per volatility point squared. (b) The long variance swap loses USD 450,000.

Exam tips

Read the units first. Many questions are designed so that a wrong unit gives a plausible but incorrect option.
Memorize N_var = N_vega ÷ (2K). It shows up in calculation questions.
For conceptual questions, link variance swaps to option replication with 1 ÷ K² weights and to the VIX.
Expect a convexity question: a long variance position benefits more from a volatility rise than it loses on an equal fall.
Use a financial calculator for squares and products to avoid arithmetic slips, but keep the units written down.

Practice questions from Exotic Options

Volatility and Variance Swaps: frequently asked questions

What is the difference between a variance swap and a volatility swap?

A variance swap settles on realized variance against a variance strike, while a volatility swap settles on realized volatility against a volatility strike. Variance swaps can be replicated with a static strip of options. Volatility swaps need dynamic hedging and a model.

How does the VIX relate to variance swaps?

The VIX is calculated from out-of-the-money S&P 500 option prices using the same replication logic as a variance swap. VIX squared approximates the fair 30-day variance strike. This is why the VIX is described as a measure of expected variance under risk-neutral pricing.

Why is the volatility swap strike below the square root of the variance strike?

The square root is a concave function. By Jensen's inequality, the expected value of volatility is less than the square root of the expected variance. So the fair volatility strike is lower.

Who uses variance swaps and why?

Traders use them to take a view on volatility, or to trade the gap between implied and realized volatility. Hedgers use them to protect against volatility spikes without taking direction risk. A variance swap is a cleaner volatility instrument than a delta-hedged option.