ABSTRACT
Contents 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 General Valuation of Counterparty Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3 Modeling Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.1 G2++ Interest Rate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3.2 CIR++ Stochastic Intensity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3.3 CIR++ Model: CDS Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3.4 Interest-Rate/Credit-Spread Correlation . . . . . . . . . . . . . . . . . . . . . . 71
4.4 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4.1 Discretization Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4.2 Forward Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4.3 Callable Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.5.1 Single Interest Rate Swaps (IRS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.5.2 Netted Portfolios of IRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.5.3 European Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.5.4 Bermudan Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.5.5 CMS Spread Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.5.6 CMS Spread Ratchets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6 Results Interpretation and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Appendix: Detailed Output Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Abstract: We consider counterparty risk for interest rate payoffs in the presence of correlation between the default event and interest rates. The previous analysis of Brigo and Masetti (2006), assuming independence, is further extended to interest-rate payoffs different from simple swap portfolios. A stochastic intensity model is adopted for the default event. We find that correlation between interest rates and default has a relevant impact on the positive adjustment to be subtracted from the default-free price to take into account counterparty risk. We analyze the pattern of such impacts as product characteristics and tenor structures change through some relevant numerical examples. We find the counterparty risk adjustment to decrease with the correlation for receiver payoffs, while the analogous adjustment for payer payoffs increases. The impact of correlation decreases when the default probability increases.
