Finance: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]\displaystyle{ N_{\text{corr}} }[/math] times the agreed strike [math]\displaystyle{ \rho_{\text{strike}} }[/math], while the floating leg pays the realized correlation [math]\displaystyle{ \rho_{\text{realized }} }[/math]. The contract value at expiration from the pay-fixed perspective is therefore
- [math]\displaystyle{ N_{\text{corr}} (\rho_{\text{realized}}-\rho_{\text{strike}}) }[/math]
Given a set of nonnegative weights [math]\displaystyle{ w_i }[/math] on [math]\displaystyle{ n }[/math] securities, the realized correlation is defined as the weighted average of all pairwise correlation coefficients [math]\displaystyle{ \rho_{i,j} }[/math]:
- [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \rho_{\text{realized }} = \frac{2}{n(n-1)}\sum_{i \gt j}{\rho_{i,j}} }[/math]
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
Sources
- Meissner, Gunter (2014). Correlation risk modeling and management : an applied guide including the Basel III correlation framework-- with interactive models in Excel/VBA. Wiley. p. 11. ISBN 111879690X.
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