The Normal Copula and Correlation
In this chapter we concentrate on the theoretical aspects of pricing models and, in particular, the ‘Normal Copula’. In Chapter 21 we discuss application and some of the failings of the model and how to get round them.
The reader probably believes that the default correlation and correlated spreads have the same source so that correlating spreads is the most sensible model to develop. While this is true, it is impractical at present - we shall see later (from the pricing of F2D baskets using alternative models) that spreads have to be modelled as a jump process in order to replicate market prices with any sensible levels of spread volatility. Jump processes are difficult to handle - efficient methods have not yet been developed or accepted to incorporate jump processes in a way that can allow the models to be easily calibrated to market data and also lead to fast run times. As a consequence, the model that is in common usage is a correlated default time model.
20.1 DEFAULT TIME CORRELATION
We shall explain the concept of default time correlation by considering simulated default times.
20.1.1 Generating Correlated Default Times
First consider two names which have a default time correlation, ρ . Section 6.3 showed how to obtain the Cholesky decomposition from the correlation matrix and how to generate correlated random Normal numbers using this decomposition. Chapter 18 showed how to convert a normal random number into a default time given ...