Automatic Backward Differentiation for American Monte-Carlo Algorithms - ADD for Conditional Expectations and Indicator Functions

In this note we derive a modified backward automatic differentiation (a.k.a. adjoint automatic differentiation, adjoint algorithmic differentiation) for algorithms containing conditional expectation operators and/or indicator functions. Bermudan option and xVA valuation are prototypical examples. We consider the Bermudan product valuation, but the method is applicable in full generality. Featuring a clean and simple implementation, the method improves accuracy and performance. For conditional expectation operators it offers the ability to use different estimators in the valuation and the differentiation. For the indicator function, the method allows to use "per-operator"-differentiation of the indicator function, enabling an accurate treatment of each individual exercise boundary - which is not possible in a classic finite difference applied to the Bermudan valuation.