Joint pricing of default-free and defaultable claims in a reduced-form model featuring a martingale part

Abstract

Structural and reduced-form approaches form the main classes of default models. The second family can be classified in two sub-groups, using the shape of the Azéma supermartingale, G. Those for which G is continuously decreasing coincide with Cox models, and have been extensively studied and applied to many practical problems. They are handy because they display the so-called immersion property (also known as the H hypothesis) according to which, using standard notations, every F-martingale is a G-martingale. In contrast, those for which G features a martingale part – hence, can be locally increasing – have been studied from a theoretical perspective but have been used in few applications only, in which case the adopted setup was quite sophisticated. This can be explained because of the theoretical challenges associated with the design of a market model that could accommodate both default-free and defaultable assets as well as of a lack of comprehension about how to simulate default times correlated with other risk factors in such a setup. This paper fills this gap by proposing a concrete example of arbitrage-free market model of this class and relying on the Φ-martingale process. The latter exhibits a similar complexity as Cox but has the advantage to be automatically calibrated to a given default probability curve and CDS option quotes, easy to simulate, and able to display stronger dependence effects compared to state-of-the-art intensity models. Explicit calculations and numerical applications featuring counterparty risk and, in particular, credit valuation adjustment (CVA) with wrong-way risk, illustrate the results.