Journal of Computational Finance

Risk.net

Automatic differentiation for diffusion operator integral variance reduction

Johan Auster

  • We demonstrate applications of automatic differentiation techniques in the Heath–Platen diffusion operator integral framework, which enables large variance reductions via approximating models with semianalytical solutions.
  • By utilizing nested dual numbers, the exact value function sensitivities can be computed numerically in an efficient way, relying only on the solution to the pricing problem in the approximating model.
  • The estimator is applied to various pricing problems in the Heston model. For the at-the-money European call option, we achieve time savings by a factor of more than 2 000 for the same expected relative error compared with standard Monte Carlo methods.
  • Further applications to discretely monitored barrier options and floating-strike lookback put options showcase the ease of extending the approach to complex options without requiring manual derivation of sensitivities.

This paper demonstrates applications of automatic differentiation with nested dual numbers in the diffusion operator integral variance-reduction framework originally proposed by Heath and Platen. Combining this estimator with automatic differentiation techniques for computing value function sensitivities allows for a flexible implementation without trade-offs in numerical stability or accuracy. This fully mitigates a key practical shortcoming of the original estimator, as we remove the dependency on error-prone and problem-specific manual calculations. We perform a relative error analysis of the estimator and standard Monte Carlo estimation against the numerical integration solution of the European call option in the Heston model and find computational time savings in excess of three orders of magnitude for the same expected relative errors for an at-the-money option. The implementation is further extended to the valuation of discrete down-and-out barrier call options and floating-strike lookback put options, demonstrating the relative ease of applying the automatic differentiation approach to path-dependent options with monitoring bias corrections.

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