11. Simulation/Regression Pricing Schemes in Pure Jump Setups

In this chapter we devise simulation/regression numerical schemes in pure jump models. There the idea is to perform the nonlinear regressions, used for computing conditional expectations, in the time variable for a given state of the model rather than in the space variables at a given time in the diffusive setups of Chap. 10. This idea is stated in the form of a generic lemma that is valid in any continuous-time Markov chain model. This is then tested in the context of two credit risk applications, the first of which values the sensitivities of a CDO tranche in a homogeneous groups model of portfolio credit risk by Monte Carlo without resimulation. The second computes by Monte Carlo the CVA on a CDO tranche in a common shock model of counterparty credit risk. CVA stands for credit valuation adjustment, the correction in value to a derivative accounting for the default risk of your counterparty, a topical issue since the crisis. But wait: are you perfect yourself? Isn’t it so that most Western banks nowadays quote at a few hundreds of basis points of credit spread? This means that you should also account for your own default risk in the valuation, otherwise I doubt many clients would agree to deal with you—which implies the related nonlinear funding struggle that if you are credit risky, the funding of your position will involve (at least) two rates, a lending and a borrowing one. Now, quiz to the reader (not answered in this chapter, and in fact nowhere else either): how would you price nonlinear funding costs on a very high-dimensional and discrete underlying like a CDO tranche?