Journal of Computational Finance

Risk.net

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

Christian Fries

  • In this paper, we apply the stochastic (backward) algorithmic differentiation to calculate stochastic forward sensitivities, i.e., the random variable representing sensitivities at a future points in time.
  • 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.
  • We demonstrate that these forward sensitivities can be obtained in a  single stochastic automatic differentiation sweep. Our test case generates 5 million sensitivities in seconds.

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.

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