Fast stochastic forward sensitivities in Monte Carlo simulations using stochastic automatic differentiation (with applications to initial margin valuation adjustments)

Christian Fries

Abstract

In this paper, we apply stochastic (backward) automatic differentiation to calculate stochastic forward sensitivities. A forward sensitivity is a sensitivity at a future point in time, conditional on future states (ie, it is a random variable). A typical application of stochastic forward sensitivities is the exact calculation of an initial margin valuation adjustment, assuming the initial margin is determined from a sensitivity- based risk model. The ISDA Standard Initial Margin 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 applications the performance may be even more impressive, since 90% of the computation time is consumed by the conditional expectation regression, which does not scale with the number of products.