The prices of the call and put options can be quickly calculated using many methods using the form of the characteristic function. But how to calibrate a model when we don't know the characteristic function? Actually, what models do not provide us a characteristic function?
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$\begingroup$ I think as(for any random variable) the CF always exists for real argument, we should be able to obtain a CF for any option pricing model / distribution assumption. Whether or not that may be cumbersome is a different story, though... $\endgroup$– KermittfrogDec 7, 2020 at 17:57
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You are likely thinking of affine stochastic volatility (SV) or Levy models which have characteristic functions that can be obtained via semi-analytic expression. For these types of models Fourier approaches are of the most efficient known approaches.
There are other types of non-affine models that are used for pricing that don't have known characteristic functions, but have their own pricing/calibration algorithms.
For instance local volatility models, where the volatility component is calibrated using approaches like Dupire's method. There are also local SV (LSV) models with their own calibration method. Another example would be in the SABR approach which prices options using a singular perturbation approach (not CF)
