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 ${\rho }_{i,j}$:

${\rho }_{\text{realized }}:=\frac{\sum _{i\ne j}{w}_i{w}_j{\rho }_{i,j}}{\sum _{i\ne j}{w}_i{w}_j}$

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.

See also

• Variance swap
• Rainbow option