Funding Value Adjustment for General Financial Instruments : Theory and Practice

In the first part of this paper (Antonov-Bianchetti, 2013) we developed the theoretical framework for pricing financial instruments under multiple sources of funding, leading to a non-linear pricing PDE and to Funding Value Adjustment (FVA).In this second part we develop the numerical framework for computing FVA for general financial instruments including callable features. Our main technical result is an efficient approximation of the FVA in a fully universal way. Usage of non-linear effective discounting rates permits an exact handling of all solvable special cases (the collateral as linear function of the value) by a single formula. Furthermore, the formula delivers a very accurate approximation for general instruments (barriers, Bermudans, etc.). The proof is based on our second technical result: exact calculation of prices of automatically exercisable instruments (e.g. European-style or Barrier) having different stochastic discount rates before and after exercise.We also address the implementation workflow of the FVA calculation, allowing parallel deal-by-deal computation, and we provide a concrete example of FVA calculation for a Bermudan Swaption with partial collateralization, proving that the quality of the numerical approximation is excellent