# Correlation Swaps

> A correlation swap is a transaction in which two parties exchange, at maturity, the difference between realized correlation and a fixed correlation (the strike) set at inception, multiplied by a notional amount. Realized correlation is the average of the correlations among the constituents of a stock index, weighted by their index weights. A variance swap trades the size of moves, while a correlation swap trades how closely the stocks move together.

- URL: https://www.derivatives-compass.org/en/equity/correlation-swaps/
- Area: Equity Derivatives › Equity Exotics
- Last reviewed: 2026-10-22
- 日本語版: https://www.derivatives-compass.org/ja/equity/correlation-swaps/index.md

## Key points

- The buyer receives the notional amount times the difference between realized correlation and the strike, and pays the same amount if the difference is negative.
- Realized correlation is the average of the pairwise correlations among stocks, weighted by index weights.
- Realized correlation cannot exceed 1, so unlike a variance swap, the seller's loss is also limited.
- Realized correlation can be approximated from the variance swaps on the index and on its constituents.
- The strike of a correlation swap can differ from the correlation implied by a dispersion trade.

A dealer that sells products linked to several stocks sees its profit and loss move when the stocks' co-movement changes, so it looks for ways to [hedge](https://www.derivatives-compass.org/en/glossary/hedging/) that. A correlation swap turns that co-movement itself into a payment, and it comes up when managing [correlation](https://www.derivatives-compass.org/en/glossary/correlation/) risk.

## What is exchanged

A [correlation swap](https://www.derivatives-compass.org/en/glossary/correlation-swap/) is a transaction in which two parties exchange, once at maturity, the difference between realized correlation and the strike. It is similar to a [variance swap](https://www.derivatives-compass.org/en/glossary/variance-swap/), and at maturity the notional amount times the difference between realized correlation and the strike is paid[^js-def]. The buyer's receipt is as follows.

$$
N \times \left(\rho_R - K\right)
$$

$$\rho_R$$ is the realized correlation over the contract period, $$K$$ is the strike set at inception, and $$N$$ is the notional amount. When the receipt is negative, the buyer pays the seller.

Realized correlation is computed from the $$n$$ stocks in the index. For each pair of stocks, take the correlation $$\rho_{ij}$$, and average them using the product of the index weights $$w_i$$ and $$w_j$$ as weights[^js-def].

$$
\rho_R = \frac{\sum_{i<j} w_i w_j \rho_{ij}}{\sum_{i<j} w_i w_j}
$$

If all weights are equal, this is the simple average of the pairwise correlations.

## An example of the payment at maturity

Consider a hypothetical example in which Bank X buys a correlation swap from Securities Firm Y. The index holds three stocks, A, B, and C, in equal weights, the strike is 0.5, and the notional amount is 100 million yen. Because the weights are equal, realized correlation is the simple average of the correlations of three pairs (A and B, A and C, B and C).

If the realized pairwise correlations are 0.7, 0.6, and 0.5, realized correlation is 0.6, and Bank X receives 100 million yen × (0.6 − 0.5) = 10 million yen. If they are 0.3, 0.4, and 0.5, realized correlation is 0.4, and Bank X receives 100 million yen × (0.4 − 0.5) = −10 million yen, that is, it pays 10 million yen. Securities Firm Y's profit and loss has the opposite sign. If realized correlation is above the strike, the buyer gains and the seller loses; if it is below, the reverse holds.

| Realized correlation | Receipt of Bank X (buyer) |
| -------------------- | ------------------------- |
| 0.0                  | −50 million yen           |
| 0.5                  | 0 yen                     |
| 1.0                  | 50 million yen            |

**Figure 1: Realized correlation and the buyer's receipt**

The amount the buyer receives at maturity for a correlation swap with a strike of 0.5 and a notional of 100 million yen. The horizontal axis is realized correlation (0 to 1), and the vertical axis is the buyer's receipt in million yen. The line rises to the right. At a realized correlation of 0 the buyer pays 50 million yen, and at 1 the buyer receives 50 million yen. The sign of the receipt switches on either side of the strike of 0.5.

Realized correlation is an average of pairwise correlations, so it does not exceed 1. In this example, the seller's largest payment is 50 million yen, when realized correlation is 1, because the difference from the strike of 0.5 can be at most 0.5.

The buyer's side has a limit too. Take an index of three stocks in equal weights. The average of the three correlations cannot fall below −0.5, which can be calculated from the properties of a correlation matrix. The buyer's largest payment is therefore 100 million yen × (−0.5 − 0.5) = −100 million yen, that is, it pays 100 million yen.

By contrast, the loss of a variance swap seller has no upper limit in the formula. The payoff of the variance swap is covered on the [Variance Swaps and Volatility Swaps](https://www.derivatives-compass.org/en/equity/variance-swaps/) page.

## How realized correlation is calculated

Besides computing every pairwise correlation, there is an approximation from the realized volatilities of the index and its constituents. Realized correlation can be approximated by the ratio of the index variance to the average of the constituents' variances weighted by index weights. Bossu showed this approximation[^js-approx].

$$
\rho_R \approx \frac{\sigma_I^2}{\sum_i w_i \sigma_i^2}
$$

$$\sigma_I$$ is the realized volatility of the index and $$\sigma_i$$ is that of stock $$i$$. The numerator corresponds to the index variance swap, and the denominator to the weighted sum of the constituents' variance swaps. Realized correlation can therefore be seen as the ratio of two traded variance swaps. In its original form, the denominator is the square of the weighted average volatility. The approximation above replaces it with the weighted average of variances, which is smaller than the original value[^js-approx].

## Differences from a variance swap

| Item                 | Variance swap                                          | Correlation swap                                                                                         |
| -------------------- | ------------------------------------------------------ | -------------------------------------------------------------------------------------------------------- |
| Basis of the payment | Realized variance (the square of the size of moves)    | Realized correlation (how closely stocks move together)                                                  |
| Form of the receipt  | Notional amount × (realized variance − strike)[^js-vs] | Notional amount × (realized correlation − strike)                                                        |
| Limit on loss        | The seller's loss has no upper limit in the formula    | Both the seller and the buyer have a limit (previous section)                                            |
| Replication          | Can be replicated with a combination of options        | Can be approximately replicated, but the effects of liquidity and of the volatility of volatility remain |

A variance swap responds to the size of each stock's moves. A correlation swap pays differently when how closely the stocks move together changes, even if the size of the moves stays the same.

## Role in dispersion trading

[Dispersion trading](https://www.derivatives-compass.org/en/glossary/dispersion-trading/) takes opposite positions in options on an index and options on its constituents. When it is built with variance swaps, you sell the index variance swap and buy the constituents' variance swaps, which produces profit and loss from correlation. However, the profit and loss of this trade also includes a part from changes in the size of the constituents' moves.

The profit and loss is the sum of two parts: the spread between implied and realized correlation multiplied by the constituents' average variance, and a volatility part. Implied correlation is the correlation backed out from the prices of options or variance swaps. The volatility part is a term of the order of volga. Volga is the second derivative of an option's price with respect to volatility, and it expresses the curvature of the price when volatility moves[^js-pl].

The payment of a correlation swap depends on correlation alone. It has therefore been proposed as a more direct way to trade correlation[^js-intro].

## How it is priced and why it is hard to hedge

The strike of a correlation swap can differ from the implied correlation obtained from dispersion trading. According to a 2010 paper, earlier empirical work observes a difference between the strike of a correlation swap and the implied correlation of a dispersion trade made with variance swaps of the same maturity. The paper says this difference can be explained by the volga of the dispersion trade. It adds that the conclusion needs to be reviewed if the weighting scheme of the dispersion trade changes[^js-spread].

Using the relationship between realized correlation and the approximation, a correlation swap can be dynamically quasi-replicated by a dispersion trade with variance swaps. Bossu showed this method. The paper points out two issues. The liquidity of variance swaps is not necessarily sufficient in every market, or for every index and its constituents. The method also does not specify the form of volatility, so it does not take into account volatility moving randomly (the volatility of volatility)[^js-limit].

## What it is used for

According to a 2010 paper, as a result of banks selling products such as [worst-of](https://www.derivatives-compass.org/en/glossary/worst-of-option/) options, the banks hold a short correlation position, that is, a position that loses when correlation rises. The correlation swap is named as a way to offset part of it[^js-use].

Consider a hypothetical example in which Bank X has sold a worst-of call. Suppose it buys the same correlation swap as in the previous example, with a notional amount of 100 million yen and a strike of 0.5. The product is a worst-of call, and it loses 12 million yen when realized correlation is 0.1 above the strike and gains 12 million yen when it is 0.1 below.

If realized correlation is 0.6, the swap's receipt is 10 million yen and the loss on the product is 12 million yen, so the total is −2 million yen. If realized correlation is 0.4, the swap's payment is 10 million yen and the gain on the product is 12 million yen, so the total is 2 million yen. Because the sensitivities to correlation of the product and the swap are not the same, a difference remains. If the product's sensitivity is constant as in this example, a notional amount of 120 million yen makes the payment 12 million yen per 0.1, and the difference disappears.

[^js-def]: Section 3.2 (page 5) of [Antoine Jacquier and Saad Slaoui, "Variance dispersion and correlation swaps"](https://arxiv.org/pdf/1004.0125). The formula for the payment at maturity, and the formula for realized correlation as the average of pairwise correlations weighted by $$w_i w_j$$.

[^js-approx]: Section 3.2 (page 5) of the same paper. Realized correlation can be approximated by a ratio of the realized volatilities of the index and its constituents, Bossu (2006) showed this approximation, and the approximation that uses the weighted average of variances is a lower bound of the original value by Jensen's inequality.

[^js-vs]: Section 2 (page 2) of the same paper. The formula for the payoff of a variance swap, and the explanation that it can be replicated with an infinite number of calls and puts.

[^js-pl]: The abstract and chapter 5 of the same paper. The profit and loss of a dispersion trade is the sum of a term equal to the spread between implied and realized correlation times the constituents' average variance, and a volatility term of the order of volga.

[^js-intro]: Section 1 (page 1) of the same paper. Trading volatility creates positions in correlation, and correlation swaps were proposed from the need to hedge them.

[^js-spread]: The abstract and chapter 5 of the same paper. The strike of a correlation swap and the implied correlation of the same maturity are observed empirically not to match, the difference can be explained by volga, and it needs to be reviewed if the weighting scheme changes.

[^js-limit]: Section 3.2 (page 5) of the same paper. Bossu showed that a correlation swap can be dynamically quasi-replicated by a variance dispersion trade. The end of the section notes that liquidity is insufficient in some markets and that the model does not specify the form of volatility or take the volatility of volatility into account.

[^js-use]: The abstract of the same paper. As a result of selling products such as worst-of options, banks hold a short correlation position, and correlation swaps and dispersion trading have been proposed as ways to buy back part of it.

## Sources

- [Variance dispersion and correlation swaps](https://arxiv.org/pdf/1004.0125) — arXiv（Antoine Jacquier、Saad Slaoui、2010 年）