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

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