Valuing Credit Risk - Variance Reduction Techniques for Monte Carlo Methods

This paper deals with the valuation of credit risk derivatives on the basis of Monte
Carlo simulation methods with the main viewpoint on variance reduction techniques.
Therefore, first an overview on credit risk derivatives like credit default swaps and first
to default baskets is given. It turns out that modelling of the joint distribution of
dependent credit default times proves to be the crucial element. Once obtained, any
credit derivative can be valued. A convenient way of achieving this is by use of the
copula concept, which migrates marginal distributions of credit default times obtained
from a credit curve into a joint distribution incorporating any kind of desired dependency
structure. A section devoted to this concept provides the necessary background
and properties. Next, the general Monte Carlo concept is introduced in detail and carefully
adapted to the valuation of credit derivatives, following the path of constructing
dependent uniform random variables from dependent normal random variables. At the
same time, first insight is gained in the field of variance reduction which is intensified
in chapter four, where a series of techniques including antithetic sampling and control
variates is presented. The main focus shall lie from there on on importance sampling.
In order to increase the efficiency of Monte Carlo methods, sampling is restricted to the
region of importance where the function to be evaluated - here: the indicator function
of the credit default times - does not vanish. This technique is applied and examined
in detail in the final chapter for the one- and multi-credit case. Exponential as well as
normal importance sampling densities are derived.