Monte Carlo Greeks for Advanced Financial Applications- Jump Diffusions and (Time-Changed) Lévy Processes based Models -
Monte Carlo Greeks for Advanced Financial Applications- Jump Diffusions and (Time-Changed) Lévy Processes based Models -

Summary/Abstract: In this paper we present a method to derive generic Monte Carlo estimators for the Greeks for models based on jump diffusion processes, Lévy processes and time-changed Lévy processes. We use proxy schemes introduced in Fries and Kampen (2005), Fries (2007), Fries and Joshi (2006). Proxy schemes are a novel finite difference and therefore purely numerical method to cope with some of numerical instabilities of the likelihood ratio method. Furthermore, the knowledge likelihood ratio coefficient which involves derivatives of the transition density is not necessary. The main idea is to use an integration by parts argument and apply it to the density. If the density is not available in closed form we apply fourier inversion to compute it from the characteristic function. The proxy method has been applied to Gaussian models. Computing Greeks in the more general setting of (time changed) Lévy models or jump processes has been addressed but only in terms of heavy mathematical machinery such as Malliavin calculus or in terms of the application of the likelihood ratio method. After reviewing recent results for the estimation of Greeks for certain Lévy models we show how our method applies to those cases. Especially, we apply the method to the Merton jump diffusion model, the Variance Gamma model and to the stochastic volatility Variance Gamma model. We estimate Greeks such as Delta, Gamma or Vega for some (discontinuous) payoffs. Finally, we show that the method is stable and applicable to a wide range of payoffs and models.

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