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Lay out how a correlation swap is built, what it is used for and what it pays.

FRM® Part II · Market Risk Measurement and Management · Correlation Basics: Definitions, Applications, and Terminology · FRM2.MR.8.d

The idea

A correlation swap exchanges a fixed correlation, the swap rate, for the realised average pairwise correlation of a set of assets over the life of the contract. The swap rate is set so the swap has no value at inception, and no principal changes hands. The buyer receives notional × (realised − fixed) at maturity and the seller pays it, so the buyer is long correlation. With n assets there are n(n − 1) ÷ 2 pairs. The realised average over all of those pairs is given by Average = 2 ÷ (n² − n) × the sum of the pairwise correlations. The swap is used to hedge or to speculate on correlation. A dealer who has sold basket options and a dispersion trader both lose when correlation rises, so both buy the swap. Common slips are dividing by n² and reversing buyer and seller. Exam point: the buyer is long correlation.

Formulas

Payoff to the buyer of a correlation swap

payoff = N × (ρrealised − ρfixed)

N
the notional
ρrealised
the realised average pairwise correlation, 2 ÷ (n² − n) × the sum of pairwise correlations
ρfixed
the fixed swap rate set at inception

The buyer is long correlation and the seller pays when realised exceeds the fixed rate. Average over n(n − 1) ÷ 2 pairs, not n² or n(n + 1) ÷ 2.

A practice question

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

A correlation swap is primarily used to:

  1. Hedge the volatility of a single stock.Why not: That is the role of a variance or volatility swap; a correlation swap needs several assets.
  2. Transfer the default risk of a reference entity.Why not: That is a credit default swap; a correlation swap pays on correlation, not on default.
  3. Hedge or take a view on the future average correlation among a set of assets. - correct
  4. Exchange fixed interest payments for floating payments.Why not: That is an interest rate swap; the floating leg of a correlation swap is realised correlation.

The swap isolates correlation as the traded variable, either to hedge exposures such as basket options and dispersion trades or to speculate on correlation.

A harder one

A correlation swap on five stocks has a fixed rate of 45% and a notional of USD 3 million. At maturity the ten pairwise realised correlations sum to 5.2, and realised correlation is defined as their average. What is the cash flow for the seller of the swap?

  1. The seller receives USD 726,000.Why not: Divides the sum by n² = 25 instead of the number of pairs, 10.
  2. The seller pays USD 210,000. - correct
  3. The seller receives USD 310,000.Why not: Divides by 15 = n(n+1)/2, wrongly counting each asset's correlation with itself as a pair.
  4. The seller receives USD 210,000.Why not: Gets the average right (0.52) but reverses the sign: the seller pays when realised exceeds the strike.

Realised = 5.2 / 10 = 0.52. Payoff to the buyer = 3,000,000 x (0.52 - 0.45) = USD 210,000, which the seller pays.

The hardest tier

A correlation swap on three assets has a strike of 50% and pays USD 50,000 per percentage point by which the realised average pairwise correlation exceeds the strike, received by the buyer. The three pairwise realised correlations are 0.62, 0.48 and 0.55. What is the outcome for the buyer?

  1. Pays USD 666,667.Why not: Averages with 2 / n² = 2 / 9 instead of 2 / (n² - n), giving 36.7%.
  2. Receives USD 600,000.Why not: Uses only the highest pairwise correlation instead of the average.
  3. Receives USD 250,000. - correct
  4. Receives USD 2,500.Why not: Treats the 0.05 difference as 0.05 of a point instead of 5 percentage points.

Average = (0.62 + 0.48 + 0.55) / 3 = 0.55, which is 5 percentage points above the 50% strike. Payoff = 5 x USD 50,000 = USD 250,000 to the buyer.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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