  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.
