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Contrast volatility swaps with variance swaps and describe each.

FRM® Part I · Financial Markets and Products · Exotic Options · FRM1.FM.15.f

The idea

A volatility swap exchanges realised volatility for a fixed strike volatility at maturity. Volatility swap payoff = vega notional × (realised volatility − strike volatility). The vega notional is quoted per volatility point. A variance swap exchanges realised variance for a strike variance. Variance swap payoff = variance notional × (realised variance − strike variance). The variance is in squared volatility points. For small moves around the strike the two swaps are equivalent, linked by this conversion. Variance notional = vega notional ÷ (2 × strike volatility). The variance swap payoff is convex in volatility, so the long gains more when volatility rises and loses less when it falls. The fair variance strike therefore exceeds the squared fair volatility strike. Variance swaps can be replicated statically with options, whereas volatility swaps need a model. The exam favours conversions between the notionals and payoffs in volatility points.

Formulas

Payoff of a volatility swap

payoff = Nvega × (σR − σK)

Nvega
the vega notional, in money per volatility point
σR
the realised volatility, in volatility points
σK
the strike volatility, in volatility points

Paid to the receiver of realised volatility. It is linear in volatility and needs a model to replicate.

Payoff of a variance swap

payoff = Nvar × (σR² − σK²)

Nvar
the variance notional, in money per variance point
σR
the realised volatility, in volatility points
σK
the strike volatility, in volatility points

Variance is in squared volatility points. The payoff is convex in volatility, so the long gains more on a rise than it loses on an equal fall.

Variance notional equivalent to a vega notional

Nvar = Nvega ÷ (2 × σK)

Nvega
the vega notional, in money per volatility point
σK
the strike volatility, in volatility points

Makes the two swaps pay about the same for small moves around the strike. The strike is in volatility points, not a decimal.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement correctly describes a volatility swap?

  1. The payoff is a notional per variance point times the difference between realised variance and strike variance.Why not: Describes a variance swap, not a volatility swap.
  2. At maturity the payoff is a vega notional per volatility point times the difference between realised volatility and the strike volatility. - correct
  3. The payoff depends on the change in the VIX index between inception and maturity.Why not: Describes a VIX-based product; a volatility swap pays on realised volatility over its life.
  4. A fixed amount is paid only if realised volatility exceeds the strike volatility.Why not: Describes a binary payoff; a volatility swap pays in proportion to the difference.

A volatility swap exchanges realised volatility for a fixed strike volatility. Payoff = vega notional x (realised volatility - strike volatility), with the notional quoted per volatility point.

A harder one

A volatility swap has a vega notional of USD 150,000 per volatility point and a strike of 18.5. Realised volatility over the period is 21.3. The payoff to the receiver of volatility is:

  1. USD -420,000Why not: Sign error: the receiver of realised volatility gains when it exceeds the strike.
  2. USD 3,195,000Why not: Multiplies the notional by realised volatility without subtracting the strike.
  3. USD 420,000 - correct
  4. USD 16,716,000Why not: Uses the difference of squared volatilities, as for a variance swap, with a vega notional.

Payoff = 150,000 x (21.3 - 18.5) = 150,000 x 2.8 = USD 420,000.

The hardest tier

A variance swap has a variance notional of USD 3,000 per variance point and a strike volatility of 15. Realised variance is annualised as (252 / n) x the sum of squared daily returns (in percent), with n = 5 daily returns of 1.2, -0.8, 0.5, -1.5 and 1.0 percent, assuming a zero mean. The payoff to the receiver of variance is closest to:

  1. USD 5,000Why not: Applies the variance notional to the volatility difference (16.77 - 15) rather than the variance difference.
  2. USD -672,000Why not: Forgets the 252 annualisation, giving a realised variance of about 1.1 against a strike variance of 225.
  3. USD 169,000 - correct
  4. USD 380,000Why not: Uses 252 / (n - 1) instead of the 252 / n stated, overstating realised variance.

Sum of squares = 1.44 + 0.64 + 0.25 + 2.25 + 1.00 = 5.58. Realised variance = 252 / 5 x 5.58 = 281.2 (volatility 16.8). Payoff = 3,000 x (281.2 - 225) = about USD 169,000.

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