  We propose Monte Carlo calibration algorithms for three models: local
volatility with stochastic interest rates, stochastic local volatility with
deterministic interest rates, and finally stochastic local volatility with
stochastic interest rates. For each model, we include detailed derivations of
the corresponding SDE systems, and list the required input data and steps for
calibration. We give conditions under which a local volatility can exist given
European option prices, stochastic interest rate model parameters, and
correlations. The models are posed in a foreign exchange setting. The drift
term for the exchange rate is given as a difference of two stochastic short
rates, domestic and foreign, each modeled by a G1++ process. For stochastic
volatility, we model the variance for the exchange rate by a CIR process. We
include tests to show the convergence and the accuracy of the proposed
algorithms.
