Zorana Grbac: Interest rate derivative valuation after the credit crisis
Source: Seminar for probability, statistics, and financial mathematics
In the aftermath of the 2007-09 financial crisis, a variety of spreads have developed between quantities that had been essentially the same until then, notably LIBOR-OIS spreads, LIBOR-OIS swap spreads, and basis swap spreads. By the end of 2011, with the sovereign credit crisis, these spreads were again significant.
In this work we study the valuation of the LIBOR interest rate derivatives in this multiple-curve setup. To account for the post-crisis discrepancy between a risk-free discount curve and a LIBOR fixing curve of the interest rate derivatives, we resort to a defaultable HJM methodology.
We propose a tractable model with Vasicek volatility structure and non-negative time-inhomogeneous Lévy processes as driving processes and derive valuation formulas for common interest rate derivatives in this setup. In view of the calibration of the model we present numerical results illustrating the flexibility of the model in producing a wide range of LIBOR-OIS swap spreads and basis swap spreads.
In the next step, we account for counterparty risk and funding issues in a contract by using the counterparty clean value process of interest rate derivatives computed in the above model, as an underlier to an option called Contingent Credit Default Swap (CCDS), which prices the correction in value, known as the Credit Valuation Adjustment (CVA), to the contract due to the counterparty risk under funding constraints. We follow a reduced-form methodology through which the problem of pricing and hedging counterparty risk and funding costs can be reduced to low-dimensional Markovian pre-default CVA BSDEs, or equivalent semi-linear PDEs.