Finance:Correlation swap

From HandWiki

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.