Abstracts

The multi-curve framework has become the standard interest rate derivative pricing framework; collateral discounting is becoming a standard in presence of collateral agreements. We described a generalised collateral framework that covers cash collateral, foreign currency collateral, collateral by assets, collateral with haircut and their combinations. The formulas obtained is similar to the standard discounting with collateral cash account formulas but with a generalised meaning of rate. The new rate includes repo rate of collateral assets, haircut factors and interest actually paid. Generalised definitions of pseudo-discount factors are introduced.
The collateral framework is applied to interest rate derivatives, leading to a unified multi-curve and collateral framework. The pricing of standard interest rate instruments based on Ibor-like indexes are presented in the framework. The exact details under which the framework can be applied are analysed, in particular we explicit the exact meaning of ``OIS discounting''. A generic approach to curve calibration in the framework is proposed. The market instruments containing informations about the theoretical quantities like pseudo-discount factors and collateral index forward rates are detailed.
The collateral forward rates depend of the collateral agreement. In practice it is important to be able to compute the forward rate for different collateral agreements from the forward rates associated to the standard collateral rule. Ways to achieve those conversions using market instruments are proposed.
In the last part, HJM-like dynamic of the collateral curves and the main results of the model are described. In that model the pricing of STIR futures under stochastic spread between the collateral and the forward curves is presented.
Based on the HJM dynamic, we also present convexity adjustments for forward rates resulting from change of collateral. Those theoretical results are important to estimate the magnitude of the change when the market is not providing enough information to estimate them directly.