Gaussian copula

Copula is simply a 2D joint distribution, or in plain language "the law on how does one swap rate behave taking into account the behaviour of some other swap rate".

If you've never heard about copula before, Thomas introduces this concept in a very simple way.

The Gaussian copula itself joints together two arbitrary complex marginal swap rate distributions that are possibly correlated. So it gives you a 2D distrubition.

For mathematical definition please check Wikipedia (Copula)

For any arbitrary distribution we always have:

Proof

Notes:

• Instead normal you might use any other distribution you want, but gaussian distribution has a good properties including correlation structure.
• we do not assume swap rates to follow normal distributions, we assume the normal correlation structure between two arbitrary market distributions.