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On the consistency of jump-diffusion dynamics for FX rates under inversion

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  • Federico Graceffa
  • Damiano Brigo
  • Andrea Pallavicini

Abstract

In this note we investigate the consistency under inversion of jump diffusion processes in the Foreign Exchange (FX) market. In other terms, if the EUR/USD FX rate follows a given type of dynamics, under which conditions will USD/EUR follow the same type of dynamics? In order to give a numerical description of this property, we first calibrate a Heston model and a SABR model to market data, plotting their smiles together with the smiles of the reciprocal processes. Secondly, we determine a suitable local volatility structure ensuring consistency. We subsequently introduce jumps and analyze both constant jump size (Poisson process) and random jump size (compound Poisson process). In the first scenario, we find that consistency is automatically satisfied, for the jump size of the inverted process is a constant as well. The second case is more delicate, since we need to make sure that the distribution of jumps in the domestic measure is the same as the distribution of jumps in the foreign measure. We determine a fairly general class of admissible densities for the jump size in the domestic measure satisfying the condition.

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  • Federico Graceffa & Damiano Brigo & Andrea Pallavicini, 2019. "On the consistency of jump-diffusion dynamics for FX rates under inversion," Papers 1905.05310, arXiv.org, revised Jul 2019.
    • Handle: RePEc:arx:papers:1905.05310
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    1. Damiano Brigo & Nicola Pede & Andrea Petrelli, 2019. "Multi-Currency Credit Default Swaps," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(04), pages 1-35, June.
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    Cited by:

    1. Claudio Fontana & Alessandro Gnoatto & Guillaume Szulda, 2021. "CBI-time-changed L\'evy processes for multi-currency modeling," Papers 2112.02440, arXiv.org, revised Jul 2022.

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