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.
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Payoff Definition [link]
The fixed leg of a correlation swap pays the notional Failed to parse (Missing texvc executable; please see math/README to configure.): N_{\text{corr}}
times the agreed strike Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_{\text{strike}}
, while the floating leg pays the realized correlation Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_{\text{realized }} . The contract value at expiration from the pay-fixed perspective is therefore
- Failed to parse (Missing texvc executable; please see math/README to configure.): N_{\text{corr}} (\rho_{\text{realized}}-\rho_{\text{strike}})
Given a set of nonnegative weights Failed to parse (Missing texvc executable; please see math/README to configure.): w_i
on Failed to parse (Missing texvc executable; please see math/README to configure.): n
securities, the realized correlation is defined as the weighted average of all pairwise correlation coefficients Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_{i,j}
- Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_{\text{realized }} := \frac{\sum_{i\neq j}{w_i w_j \rho_{i,j}}}{\sum_{i\neq j}{w_i w_j}}
Typically Failed to parse (Missing texvc executable; please see math/README to configure.): \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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_{\text{realized }} = \frac{2}{n(n-1)}\sum_{i < j}{\rho_{i,j}}
Pricing and valuation [link]
No industry-standard models yet exist that have stochastic correlation and are arbitrage-free.
See also [link]
References [link]
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