Hull & White Convexity Adjustments for Credit-Riskless Interest Rate Swaps Under CSA

Abstract

Using a multi-curve pricing framework has become standard market practice for investment banks. A new IBOR curve bootstrapping procedure is now in use consisting in discounting using an OIS curve. The curve obtained allows to recover market forwards corresponding to a CSA world which is the interbank world. Currently this curve is being used to calculate forwards for non-CSA trades, which represents an important approximation since it does not take into account the convexity adjustment implied when changing from CSA to non-CSA probability measure. By reducing counter-party risk using CSA contracts, new risk factors have arisen that are left unhandled.

We show in this paper how to calculate non-CSA forwards convexity adjustment. This adjustment depends on collateral and funding rates volatilities as well as correlation between both. We take into account these parameters under one factor gaussian short rate models, and give a detailed development for the Hull & White specific case. These closed formulae allow in turn fast bootstrapping procedures and therefore potential risk management for non-CSA swaps.

Keywords: CSA convexity, IRS, OIS-IBOR dynamics, Hull & White, CSA unmanaged risk