# 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.