At this link I have asked what is the market standard when pricing options in different asset classes. Based on the answers, the standard for FX and equities seems to be the local-stochastic volatility models. My question is how can such a model be calibrated in practice for FX. The reason why I mention FX specifically is that I can see that only 3 strikes are quoted in the market (those embedded in the ATM straddle and in the 25 delta risk reversal and butterfly). If also the 10 delta are quoted, there are 5 quotes. But how can that be enough to successfully calibrate on a FX surface? Is there any example that shows how to calibrate, for example, Heston local volatility model?
Have a look at this paper.
This is a rather exhaustive paper summarizing 9 models to be used as Local-Stochastic Volatility (LSV) models.
It describes various aspects of Calibration and Pricing of LSV models with the associated references, so that you can dig deeper in the topic should you find a suitable model for your needs.
The beginning of the abstract goes as follows:
We analyze in detail calibration and pricing performed within the framework of local stochastic volatility LSV models, which have become the industry market standard for FX and equity markets. We present the main arguments for the need of having such models, and address the question whether jumps have to be included. We include a comprehensive literature overview, and focus our exposition on important details related to calibration procedures and option pricing using PDEs or PIDEs derived from LSV models.