Federico Graceffa, Yiran Cui, Sebastian Del Baño Rollin, Guido Germano

We investigate the consistency of classes of local-stochastic volatility (LSV) models with respect to spot inversion and multiplication, and hence their applicability in the foreign exchange market. We consider two main classes, one based on the Heston model and one based on the SABR model. Then we embed them in a more general superclass of LSV models. We give general conditions the models in the superclass must satisfy to be invariant with respect to inversion and check these conditions for a collection of popular LSV models. We also investigate affine diffusion processes, showing that the symmetry conditions for inversion are automatically fulfilled. We draw conclusions on the arbitrage opportunity in variance swaps. With respect to multiplication, we consider a Heston-based class only and show that, in order for consistency to be preserved, an adjustment in the drift of the exchange rate dynamics is required.