Correlation swap

A correlation swap is an over-the-counter financial derivative that allows one to speculate on or hedge risks associated with the observed average correlation, of a collection of underlying products, where each product has periodically observable prices, as with a commodity, exchange rate, interest rate, or stock index.

Payoff Definition
The fixed leg of a correlation swap pays the notional $$N_{\text{corr}}$$ times the agreed strike $$\rho_{\text{strike}}$$, while the floating leg pays the realized correlation $$\rho_{\text{realized }}$$. The contract value at expiration from the pay-fixed perspective is therefore
 * $$N_{\text{corr}} (\rho_{\text{realized}}-\rho_{\text{strike}})$$

Given a set of nonnegative weights $$w_i$$ on $$n$$ securities, the realized correlation is defined as the weighted average of all pairwise correlation coefficients $$\rho_{i,j}$$:
 * $$\rho_{\text{realized }} := \frac{\sum_{i\neq j}{w_i w_j \rho_{i,j}}}{\sum_{i\neq j}{w_i w_j}}$$

Typically $$\rho_{i,j}$$ would be calculated as the Pearson correlation coefficient between the daily log-returns of assets i and j, possibly under zero-mean assumption.

Most correlation swaps trade using equal weights, in which case the realized correlation formula simplifies to:
 * $$\rho_{\text{realized }} = \frac{2}{n(n-1)}\sum_{i > j}{\rho_{i,j}}$$

The specificity of correlation swaps is somewhat counterintuitive, as the protection buyer pays the fixed, unlike in usual swaps.

Pricing and valuation
No industry-standard models yet exist that have stochastic correlation and are arbitrage-free.