Automatic Backward Differentiation for American Monte-Carlo Algorithms (Conditional Expectation)

TL;DR

This paper develops a method for automatic differentiation of algorithms involving conditional expectations, simplifying the computation of sensitivities in American Monte Carlo methods, especially for Bermudan option valuation.

Contribution

It introduces a novel approach to backward differentiation with conditional expectations, enabling flexible and straightforward implementation in valuation algorithms.

Findings

Allows differentiation of algorithms with conditional expectations

Enables use of different estimators for valuation and differentiation

Simplifies implementation of sensitivities in American Monte Carlo methods

Abstract

In this note we derive the backward (automatic) differentiation (adjoint [automatic] differentiation) for an algorithm containing a conditional expectation operator. As an example we consider the backward algorithm as it is used in Bermudan product valuation, but the method is applicable in full generality. The method relies on three simple properties: 1) a forward or backward (automatic) differentiation of an algorithm containing a conditional expectation operator results in a linear combination of the conditional expectation operators; 2) the differential of an expectation is the expectation of the differential $\frac{d}{dx} E(Y) = E(\frac{d}{dx}Y)$; 3) if we are only interested in the expectation of the final result (as we are in all valuation problems), we may use $E(A \cdot E(B\vert\mathcal{F})) = E(E(A\vert\mathcal{F}) \cdot B)$, i.e., instead of applying the (conditional) expectation operator to a function of the underlying random variable (continuation values), it may be applied to the adjoint differential. \end{enumerate} The methodology not only allows for a very clean and simple implementation, but also offers the ability to use different conditional expectation estimators in the valuation and the differentiation.