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 ${\displaystyle N_{\text{corr))}$ times the agreed strike ${\displaystyle \rho _{\text{strike))}$, while the floating leg pays the realized correlation ${\displaystyle \rho _{\text{realized ))}$. The contract value at expiration from the pay-fixed perspective is therefore

${\displaystyle N_{\text{corr))(\rho _{\text{realized))-\rho _{\text{strike)))}$

Given a set of nonnegative weights ${\displaystyle w_{i))$ on ${\displaystyle n}$ securities, the realized correlation is defined as the weighted average of all pairwise correlation coefficients ${\displaystyle \rho _{i,j))$:

${\displaystyle \rho _{\text{realized )):={\frac {\sum _{i\neq j}{w_{i}w_{j}\rho _{i,j))}{\sum _{i\neq j}{w_{i}w_{j))))}$

Typically ${\displaystyle \rho _{i,j))$ 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:

${\displaystyle \rho _{\text{realized ))={\frac {2}{n(n-1)))\sum _{i>j}{\rho _{i,j))}$

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.