  We develop a stochastic volatility framework for modeling multiple currencies
based on CBI-time-changed L\'evy processes. The proposed framework captures the
typical risk characteristics of FX markets and is coherent with the symmetries
of FX rates. Moreover, due to the self-exciting behavior of CBI processes, the
volatilities of FX rates exhibit self-exciting dynamics. By relying on the
theory of affine processes, we show that our approach is analytically tractable
and that the model structure is invariant under a suitable class of
risk-neutral measures. A semi-closed pricing formula for currency options is
obtained by Fourier methods. We propose two calibration methods, also by
relying on deep-learning techniques, and show that a simple specification of
the model can achieve a good fit to market data on a currency triangle.
