Journal of Computational Finance

An exact and efficient method for computing cross-Gammas of Bermudan swaptions and cancelable swaps under the Libor market model

Dan Zhu and Mark S. Joshi

  • Using the output from the first pass regression, we introduce the HOMC method for computing the Hessians of Bermudan swaptions and cancellable swaps. 
  • The key to our approach is to perform a measure change at each exercise point if the path is near the boundary to ensure that the first-order derivatives of the pathwise estimate of the price are Lipschitz continuous. 
  • The resulting pathwise estimator of the Hessian is unbiased and accurate.
  • Our numerical results suggest that the HOMC algorithm outperforms the pathwise-likelihood ratio method for computing first-order and second-order sensitivities. It is also more broadly applicable and can be used for the reduced factor LIBOR market model.
  • HOMC can also be applied to more general callable LIBOR derivatives with straight-forward modifications. 


We introduce a new simulation algorithm for computing the Hessians of Bermudan swaptions and cancelable swaps. The resulting pathwise estimates are unbiased and accurate. Given the exercise strategy, the pathwise angularities are removed by a sequence of measure changes. The change of measure at each exercise time is chosen to be optimal in terms of minimizing the variance of the likelihood ratio terms. Numerical results for the Hessian of cancelable swaps are presented to demonstrate the speed and efficacy of the method.

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