Journal of Computational Finance

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Importance sampling applied to Greeks for jump–diffusion models with stochastic volatility

Sergio De Diego, Eva Ferreira and Eulàlia Nualart

In this paper:

  • We develop a variance reduction technique, based on importance sampling and the stochastic Robbins-Monro algorithm, for option prices of jump-diffusion models with stochastic volatility.
  • The technique is applied to compute numerically the derivative price sensitivities (Greeks).
  • We improve the numerical computation of the Greeks formulas obtained via the Malliavin calculus, with no need for any localization procedure, since the localization optimization proposals are not suitable to compute the Greeks.
  • The results show that the variance is significantly reduced.

We develop a variance reduction technique, based on importance sampling in con- junction with the stochastic Robbins–Monro algorithm, for option prices of jump– diffusion models with stochastic volatility. This is done by combining the work developed by Arouna for pricing diffusion  models, and extended by Kawai for Lévy processes without a Brownian  component. We apply this technique to improve the numerical computation of derivative price sensitivities for general Lévy processes, allowing  both Brownian  and jump parts. Numerical  examples are performed for both the Black–Scholes and Heston models with jumps and for the Barndorff–Nielsen–Shephard model to illustrate the efficiency  of this numerical technique. The numerical results support that the proposed methodology improves the efficiency of the usual Monte Carlo procedures.

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