# Correlation swap

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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.