Vulnerable European and American options in a hazard-process model

Vulnerable options of European and American style with a possible occurrence of an exogenous termination are studied under market incompleteness in a hazard-process setup. It is proved that the reduced upper price of a vulnerable European option coincides with the unique price of an American option with a properly defined payoff and holder’s exercise times constrained to the random set given by the right support of the hazard process. For a vulnerable American option , it is shown that the reduced upper price equals the price of a specific American option with unrestricted exercise times, whereas the reduced lower price coincides with the price of a particular game option in which the issuer’s exercise times are constrained to the above-mentioned random set.