Bergman, piterbarg, and beyond: Pricing derivatives under collateralization and differential rates

Abstract

We extend Piterbarg's result (Piterbarg, Risk, 2:97-102, 2010) on European-style derivative pricing under collateralization by relaxing the assumption of a single unsecured funding rate. Introducing different lending and borrowing rates has the effect of producing nonlinear price functionals for general claims. Buyer and seller prices diverge, and values of derivative portfolios are not the sum of the individual deal values. Conditions under which no-arbitrage price bounds can be derived explicitly are given and numerical examples showcased.