## Frequent Links

# 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 <math>N_{\text{corr}}</math> times the agreed strike <math>\rho_{\text{strike}}</math>, while the floating leg pays the realized correlation <math>\rho_{\text{realized }}</math>. The contract value at expiration from the pay-fixed perspective is therefore

- <math>N_{\text{corr}} (\rho_{\text{realized}}-\rho_{\text{strike}})</math>

Given a set of nonnegative weights <math>w_i</math> on <math>n</math> securities, the realized correlation is defined as the weighted average of all pairwise correlation coefficients <math>\rho_{i,j}</math>:

- <math>\rho_{\text{realized }} := \frac{\sum_{i\neq j}{w_i w_j \rho_{i,j}}}{\sum_{i\neq j}{w_i w_j}}</math>

Typically <math>\rho_{i,j}</math> 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:

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

## Pricing and valuation

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