Copulas Applied to Derivatives Pricing
In this chapter, we highlight the importance that copulas have gained in derivatives pricing in the last decade. This is due mainly to the growth seen in the credit derivatives markets. Indeed, basket default swaps, CDO tranches, and all correlation structures are often priced and risk managed using the copula technology. Copulas have been used widely in the insurance business, and the transition to the credit derivatives world was natural, as the latter is often thought of as an insurance business on the default of companies.
This chapter is organized as follows: First, we tackle copulas from a theoretical point of view by presenting various properties and families of copulas. We present as well the copula of a stochastic process highlighting the time dependency, or autocorrelation, induced by a process. Second, we look at some applications to derivatives pricing. We start by presenting the factor copula technique, which enables us to reduce dimensionality of the problem and find semiclosed-form solutions for various derivatives contracts. Last, we apply the previous approach to the pricing not only of credit derivatives but some popular multiunderlying equity derivatives, precisely collateralized debt obligations, basket default swaps, and Altiplanos.
7.2 THEORETICAL BACKGROUND OF COPULAS
Definition and Sklar Theorem
DEFINITION 7.2.1 A copula is an n-dimensional function C: [0, 1]n → [0, 1], ...