CDS index options in Markov chain models

Abstract

We study CDS index options in a credit risk model where the defaults times have intensities which are driven by a finite-state Markov chain representing the underlying economy. In this setting we derive compact computationally tractable formulas for the CDS index spread and the price of a CDS index option. In particular, the evaluation of the CDS index option is handled by translating the Cox-framework into a bivariate Markov chain. Due to the potentially very large, but extremely sparse matrices obtained in this reformulating, special treatment is needed to efficiently compute the matrix exponential arising from the Kolmogorov Equation. We provide details of these computational methods as well as numerical results. The finite-state Markov chain model is calibrated to data with perfect fits, and several numerical studies are performed. In particular we show that under same exogenous circumstances, the CDS index options prices in the Markov chain framework can be close to or sometimes larger than prices in models which assume that the CDS index spreads follows a log-normal process. We also study the different default risk components in the option prices generated by the Markov model, an investigation which is difficult to do in models where the CDS index spreads follows a log-normal process.