Stochastic Algorithmic Differentiation of (Expectations of) Discontinuous Functions (Indicator Functions)

21 Pages Posted: 19 Nov 2018 Last revised: 12 Nov 2019

See all articles by Christian P. Fries

Christian P. Fries

Ludwig Maximilian University of Munich (LMU) - Faculty of Mathematics; DZ Bank AG

Date Written: November 14, 2018

Abstract

In this paper, we present a method for the accurate estimation of the derivative (aka. sensitivity) of expectations of functions involving an indicator function by combining a stochastic algorithmic differentiation and a regression.

The method is an improvement of the approach presented in [Risk Magazine April 2018].

The finite difference approximation of a partial derivative of a Monte-Carlo integral of a discontinuous function is known to exhibit a high Monte-Carlo error. The issue is evident since the Monte-Carlo approximation of a discontinuous function is just a finite sum of discontinuous functions and as such, not even differentiable.

The algorithmic differentiation of a discontinuous function is problematic. A natural approach is to replace the discontinuity by continuous functions. This is equivalent to replacing a path-wise automatic differentiation by a (local) finite difference approximation.

We present an improvement (in terms of variance reduction) by decoupling the integration of the Dirac delta and the remaining conditional expectation and estimating the two parts by separate regressions. For the algorithmic differentiation, we derive an operator that can be injected seamlessly - with minimal code changes - into the algorithm resulting in the exact result.

Keywords: Algorithmic Differentiation, Automatic Differentiation, Adjoint Automatic Differentiation, Monte Carlo Simulation, Indicator Function, Object Oriented Implementation, Variance Reduction

JEL Classification: C15, G13, C63

Suggested Citation

Fries, Christian P., Stochastic Algorithmic Differentiation of (Expectations of) Discontinuous Functions (Indicator Functions) (November 14, 2018). Available at SSRN: https://ssrn.com/abstract=3282667 or http://dx.doi.org/10.2139/ssrn.3282667

Christian P. Fries (Contact Author)

Ludwig Maximilian University of Munich (LMU) - Faculty of Mathematics ( email )

Theresienstrasse 39
Munich
Germany

DZ Bank AG ( email )

60265 Frankfurt am Main
Germany

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