# 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:

$\rho_{\text{realized }} = \frac{2}{n(n-1)}\sum_{i < j}{\rho_{i,j}}$

## Pricing and valuation

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