The valuation of vulnerable European options with risky collateral

This paper presents a model for valuing vulnerable European options with risky collateral under the assumption that the holder of vulnerable options could recover a proportion of the option value using the collateral account when default occurs. We describe the underlying asset and the risky collateral using correlated geometric Brownian motions and consider default risk in a reduced form model. An integral pricing formula of call options is derived when the default intensity follows an Ornstein–Uhlenbeck process. For practical purposes, we work under the default intensity captured by Cox–Ingersoll–Ross and Ornstein–Uhlenbeck processes respectively, and numerical results show that the differences in the values of vulnerable options under these two intensity processes are tiny. The impacts of risky collateral and default risk on option prices are illustrated. Specially, the effect of wrong (right) way risk can be reflected by the correlation between the underlying asset and default intensity.