Risk-neutral valuation under differential funding costs, defaults and collateralization

We develop a unified valuation theory that incorporates credit risk (defaults), collateralization and funding costs, by expanding the replication approach to a generality that has not yet been studied previously and reaching valuation when replication is not assumed. This unifying theoretical framework clarifies the relationship between the two valuation approaches: the adjusted cash flows approach pioneered for example by Brigo, Pallavicini and co-authors ([12, 13, 34]) and the classic replication approach illustrated for example by Bielecki and Rutkowski and co-authors ([3, 8]). In particular, results of this work cover most previous papers where the authors studied specific replication models.