Collateral and PDE

Abstract

After Lehman default (credit crisis 2007), practitioners considered the default risk as a major risk. The regulators pushed the industry to use collateral in order to reduce the risk. In this new world, we want to see how this new considerations affect the theory related to the Partial Differential Equation (PDE) pricing methodology. First, we consider a theoretical framework introduced by Piterbarg, then we relax the perfect collateralisation assumption and establish the correct derivation of the hedgingportfolio.We do not intent to offense anyone and recognise fully the work done in the building of the Post Lehman Theory.We establish different PDE forms dependent of the treasury management strategy and also retrieve Kamtchueng results for the Funding Valuation Adjustment. Indeed there are differences between Funding charge coming from the cash flows due and the funding cost strategy resulting of cash and trading desks.