Pricing vulnerable European options with stochastic default barriers

CF Lo and KC Ku Institute of Theoretical Physics and Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China Email: cho-hoi_hui{at}hkma.gov.hk Received on 31 July 2006. Accepted on 15 March 2007. This paper develops a valuation model of European options incorporatinga stochastic default barrier, which extends a constant defaultbarrier proposed in the Hull–White model. The defaultbarrier is considered as an option writer's liability. Closed-formsolutions of vulnerable European option values based on themodel are derived to study the impact of the stochastic defaultbarriers on option values. The numerical results show that negativecorrelation between the firm values and the stochastic defaultbarriers of option writers gives material reductions in optionvalues where the options are written by firms with leverageratios corresponding to BBB or BB ratings.     

On the existence of conically self-similar free-vortex solutions to the Navier-Stokes equations

In this paper a proof of existence and non-existence of theconically self-similar free-vortex solutions to the Navier-Stokesequations, originally found by Yih et al. (1982, Phys. Fluids.25, 2147-2158), is presented. This proof clearly establishesthat these solutions do not have any kind of singularity atthe symmetry axis. This analysis gives considerably improvedexistence and non-existence bounds and it is shown that thesebounds are close to optimal in the low-swirling limit. Thisapproach links the questions of existence and non-existencefor the swirling case and for the non-swirling case. The proof,which is an extension of techniques developed by Serrin (1972,Phil. Trans. R. Soc. Lond. 271, 325-360), is based on Schauder'sFixed Point Theorem and is, therefore, non-constructive. Therefore,the paper ends with a brief discussion of the question of howto compute the conically self-similar free-vortex solutionsto the Navier-Stokes equations.