Fast Stochastic Forward Sensitivities in Monte-Carlo Simulations Using Stochastic Automatic Differentiation (with Applications to Initial Margin Valuation Adjustments (MVA))

27 Pages Posted: 15 Aug 2017 Last revised: 22 Jan 2018

See all articles by Christian P. Fries

Christian P. Fries

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

Date Written: August 12, 2017

Abstract

In this note we apply the stochastic (backward) automatic differentiation to calculate stochastic forward sensitivities. A forward sensitivity is a sensitivity at a future point in time, conditional to the future states (i.e., it is a random variable). A typical application of stochastic forward sensitivities is the exact calculation of an initial margin valuation adjustment (MVA), assuming that the initial margin is determined from a sensitivity based risk model. The ISDA SIMM model is an example of such a model.

We demonstrate that these forward sensitivities can be obtained in a single stochastic (backward) automatic differentiation sweep with an additional conditional expectation step. Although the additional conditional expectation step represents a burden, it enables us to utilize the expected stochastic (backward) automatic differentiation - a modified version of the stochastic (backward) automatic differentiation.

As a test case we consider a hedge simulation requiring the numerical calculation of 5 million sensitivities. This calculation, showing the accuracy of the sensitivities, requires approximately 10 seconds on a 2014 laptop. However, in real application the performance may even be more impressive since 90% of the computation time is consumed by the conditional expectation regression, which does not scale with the number of products.

Keywords: Automatic Differentiation, Adjoint Automatic Differentiation, Monte Carlo Simulation, American Monte Carlo, Conditional Expectation, Forward Sensitivities, Initial Margin Simulation, MVA, ISDA SINM

JEL Classification: C15, G13

Suggested Citation

Fries, Christian P., Fast Stochastic Forward Sensitivities in Monte-Carlo Simulations Using Stochastic Automatic Differentiation (with Applications to Initial Margin Valuation Adjustments (MVA)) (August 12, 2017). Available at SSRN: https://ssrn.com/abstract=3018165 or http://dx.doi.org/10.2139/ssrn.3018165

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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